POSITIVE SOLUTIONS AND MULTIPLE SOLUTIONS AT NON-RESONANCE, RESONANCE AND NEAR RESONANCE FOR HEMIVARIATIONAL INEQUALITIES WITH p-laplacian
|
|
- Flora Phelps
- 5 years ago
- Views:
Transcription
1 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 360, Number 5, May 2008, Pages S (07) Article electronically published on December 11, 2007 POSITIVE SOLUTIONS AND MULTIPLE SOLUTIONS AT NON-RESONANCE, RESONANCE AND NEAR RESONANCE FOR HEMIVARIATIONAL INEQUALITIES WITH p-laplacian D. MOTREANU, V. V. MOTREANU, AND N. S. PAPAGEORGIOU Abstract. In this paper we study eigenvalue problems for hemivariational inequalities driven by the p-laplacian differential operator. We prove the existence of positive smooth solutions for both non-resonant and resonant problems at the principal eigenvalue of the negative p-laplacian with homogeneous Dirichlet boundary condition. We also examine problems which are near resonance both from the left and from the right of the principal eigenvalue. For nearly resonant from the right problems we also prove a multiplicity result. 1. Introduction In this paper we study the following nonlinear elliptic eigenvalue problem: { div( Dx(z) (1.1) p 2 Dx(z)) λ x(z) p 2 x(z) j(z, x(z)) a.e. on, x =0, 1 <p<+. Here R N is a bounded domain with a C 1,α -boundary (0 <α<1), j(z, x) is a measurable function which is locally Lipschitz in the x R variable and j(z, x) stands for the generalized subdifferential of x j(z, x) in the sense of Clarke [5]. The solutions of (1.1) are sought in W 1,p 0 (). Problem (1.1) belongs to the class of hemivariational inequalities that are new types of variational expressions arising if one considers more realistic mechanical laws of multivalued and nonmonotone nature (see [17], [15]). The corresponding energy (Euler) functionals are nonsmooth and nonconvex. Eigenvalue problems as (1.1) enter in the stability analysis of structures whose equilibrium position is characterized by a hemivariational inequality. The mathematical theory of eigenvalue problems for hemivariational inequalities was studied in [3], [6], [9], [10], [13], [14] for semilinear problems (i.e., p = 2), whereas nonlinear eigenvalue problems driven by the p-laplacian were investigated in [7]. None of the aforementioned works addressed the problem of existence of positive solutions. Theorems on the existence of positive and multiple solutions for hemivariational inequalities involving the p-laplacian have recently been presented in [16]. In this paper we focus on the existence of positive solutions and nontrivial multiple solutions to the eigenvalue problem (1.1) under assumptions different than those of [16], including growth conditions of rate bigger than p for the nonsmooth potential j(z, x). Specifically, denoting by λ 1 the principal (or first) eigenvalue of the negative Received by the editors February 14, Mathematics Subject Classification. Primary 35J20, 35R70; Secondary 35J60, 35J85. Key words and phrases. Hemivariational inequality, eigenvalue problem, resonance c 2007 American Mathematical Society
2 2528 D. MOTREANU, V. V. MOTREANU, AND N. S. PAPAGEORGIOU p-laplacian ( p,w 1,p 0 ()) we are able to show the existence of positive solutions in the non-resonant case λ<λ 1 and in the resonant case λ = λ 1. First, under some natural assumptions set up around a weaker version of the Ambrosetti-Rabinowitz condition, it is shown that for λ<λ 1 problem (1.1) has a positive solution. If p =2, by slightly strengthening the assumptions a converse result is provided showing that generally in this setting we cannot relax the non-resonant condition λ<λ 1 to have positive solutions for problem (1.1). However, we show that keeping the generalized nonsmooth version of Ambrosetti-Rabinowitz type hypothesis but imposing a different assumption on the generalized gradient with respect to x R of the nonsmooth potential j(z, x) we still produce a result that demonstrates that in the resonant case λ = λ 1 a positive solution exists. Afterwards, dropping the assumption describing the generalized version of the Ambrosetti-Rabinowitz condition, we present another set of hypotheses on j(z, x) which still guarantee the existence of a positive solution for problem (1.1) in the case of near resonance from the left at λ 1. Knowing that in general we should not expect having positive solutions to (1.1) in the case of near resonance from the right, i.e., λ>λ 1 close to λ 1, in this situation we supply existence and multiplicity results for nontrivial solutions. Our results extend in a nonsmooth quasilinear framework several classical properties of single- or multi-valued semilinear Dirichlet boundary value problems at non-resonance, resonance and near resonance. They allow one to cover a larger area of applicability in various nonsmooth and nonconvex problems arising in mechanics and engineering (see [15], [17]). The main tools in our approach are minimax theorems, nonsmooth analysis, spectrum of the negative p-laplacian, nonlinear regularity theory and nonlinear strong maximum principle. Examples illustrate all our results. The rest of the paper is organized as follows. Section 2 deals with some mathematical preliminaries. Section 3 contains our results on positive solutions for non-resonance and resonance. Section 4 focuses on the positive solutions in the case of near resonance from the left. Section 5 is devoted to multiple solutions with near resonance from the right. 2. Mathematical background Our approach is variational based on the nonsmooth critical point theory which uses the subdifferential theory of locally Lipschitz functions. For easy reference, first we recall some basic definitions which we will need in the sequel. Let X be a Banach space and X its topological dual. By, we denote the duality bracket for the pair (X, X ). For a locally Lipschitz function ϕ : X R, the generalized directional derivative ϕ 0 (x; h) ofϕ at x X in the direction h X is defined by ϕ 0 (x; h) = lim sup x x λ 0 ϕ(x + λh) ϕ(x ) λ whereas the generalized gradient ϕ(x) ofϕ at x X is introduced as ϕ(x) ={x X : x,h ϕ 0 (x; h), h X} (see Clarke [5]). We say that x X is a critical point of ϕ if 0 ϕ(x). A locally Lipschitz function ϕ : X R satisfies the nonsmooth Palais Smale condition at level c R (nonsmooth PS c -condition for short) if every sequence {x n } X such that ϕ(x n ) c and m(x n )=inf{ x : x ϕ(x n )} 0asn,
3 NON-RESONANCE, RESONANCE AND NEAR RESONANCE 2529 has a strongly convergent subsequence. The locally Lipschitz function ϕ : X R satisfies the nonsmooth Palais Smale condition (nonsmooth PS-condition for short) if it satisfies the nonsmooth PS c -condition for every c R. Itisalsosaidthata locally Lipschitz function ϕ : X R satisfies the nonsmooth Cerami condition at level c R (nonsmooth C c -condition for short) if every sequence {x n } X such that ϕ(x n ) c and (1 + x n )m(x n ) 0asn has a strongly convergent subsequence. In this paper we use two minimax principles in the nonsmooth critical point theory. The first one is the nonsmooth version of the classical mountain pass theorem (see [4], [8], [15]). Theorem 2.1. Let X be a reflexive Banach space and let ϕ : X R be a locally Lipschitz function. Suppose that for some ρ>0 and x 1,x 2 X with x 1 x 2 >ρ one has max{ϕ(x 1 ),ϕ(x 2 )} <β:= inf{ϕ(x) : x x 1 = ρ}. If the function ϕ satisfies the nonsmooth C c -condition (in particular, the nonsmooth PS c -condition), with c =inf max ϕ(γ(t)) γ Γ t [0,1] where Γ={γ C([0, 1],X): γ(0) = x 1, γ(1) = x 2 }, then ϕ has a critical point x 0 X with ϕ(x 0 )=c β. The second minimax principle that we need is the nonsmooth variant of the Brezis-Nirenberg theorem with local linking (see [12], [19]). Theorem 2.2. If X is a reflexive Banach space admitting a direct sum decomposition X = Y V with dim Y < +, ϕ : X R is a locally Lipschitz function which is bounded below, satisfies the nonsmooth PS-condition, ϕ(0) = 0, inf ϕ<0 X and there exists ρ>0 such that { ϕ(y) 0 if y Y, y ρ, ϕ(v) 0 if v V, v ρ, then ϕ has at least two nontrivial critical points. Finally we recall some facts about the spectrum of the negative p-laplacian with Dirichlet boundary condition. Considering the nonlinear eigenvalue problem { div( Dx(z) (2.1) p 2 Dx(z)) = λ x(z) p 2 x(z) a.e. on, x =0, λ R, the least real number λ, denoted λ 1, for which problem (2.1) has a nontrivial solution in W 1,p 0 (), is called the principal eigenvalue of ( p,w 1,p 0 ()). We know that λ 1 is positive, isolated and simple. There is the following variational characterization of λ 1 > 0 using Rayleigh quotient: { Dx p } p (2.2) λ 1 =inf x p : x W 1,p 0 (), x 0. p This infimum is actually realized at the normalized eigenfunction u 1. We have that u 1 C 1,β () with0<β<1 and we may assume u 1 (z) > 0 for all z (see, e.g., [8, p. 117]). Throughout the rest of the paper we keep the notation for u 1.
4 2530 D. MOTREANU, V. V. MOTREANU, AND N. S. PAPAGEORGIOU 3. Positive solutions with non-resonance and resonance First, we establish the existence of positive smooth solutions for problem (1.1) in the non-resonant case. The hypotheses on the nonsmooth potential j(z, x) are the following: H(j) 1 j : R R is a function such that j(z, 0) = 0 a.e. on and (i) for all x R, z j(z, x) is measurable; (ii) for almost all z, x j(z, x) is locally Lipschitz; (iii) for almost all z, allx R and all u j(z, x), we have u a(z)+c x r 1 with a L () +,c>0, p<r<p { Np N p where p if N>p, = + if N p ; (iv) there exist constants µ>pand M>0such that inf j(z, M) > 0 and µj(z, x) z j0 (z, x; x) a.a.z, allx M; j(z, x) (v) lim sup x 0 x p 0 uniformly for almost all z ; + (vi) for almost all z, allx>0andallu j(z, x), we have u ξx p 1 with a constant ξ 0. Assumption H(j) 1 (iv) is a nonsmooth variant of the Ambrosetti-Rabinowitz condition in [1]. Notice that here we do not require the sign condition j(z, x) > 0for almost all z and all x M. Theorem 3.1. If hypotheses H(j) 1 hold and λ<λ 1, then problem (1.1) has a solution x C 1 0() with x(z) > 0 for all z. Proof. Let τ : R R be the Lipschitz continuous truncation function { x if x 0, (3.1) τ(x) = 0 if x<0. Let j 1 (z, x) =j(z, τ(x)). For almost all z, x j 1 (z, x) is locally Lipschitz and j 1 (z, ) has the generalized gradient { {0} if x<0, (3.2) j 1 (z, x) = and j j(z, x) if x>0 1 (z, 0) conv{{0} j(z, 0)}. We consider the functional ϕ 1,λ : W 1,p 0 () R defined by (3.3) ϕ 1,λ (x) = 1 p Dx p p λ p x p p j 1 (z, x(z)) dz, x W 1,p 0 (). By H(j) 1 (i) (iii), ϕ 1,λ is Lipschitz continuous on bounded sets, so locally Lipschitz. Claim 1. ϕ 1,λ satisfies the nonsmooth PS-condition. Let {x n } W 1,p 0 () be such that ϕ 1,λ (x n ) M 1 for some constant M 1 > 0, all n 1andm(x n ) 0asn. Since the set ϕ 1,λ (x n ) W 1,p () =W 1,p 0 () (1/p +1/p =1) is w-compact and the norm functional on a Banach space is weakly lower semicontinuous, we can find x n ϕ 1,λ (x n ) such that m(x n )= x n, n 1.
5 NON-RESONANCE, RESONANCE AND NEAR RESONANCE 2531 Let A : W 1,p 0 () W 1,p () be the negative p-laplacian that is the nonlinear operator defined by A(x),y = Dx(z) p 2 (Dx(z),Dy(z)) R N dz. It is known that A is monotone and demicontinuous, hence maximal monotone. For every n 1wehave x n = A(x n ) λ x n p 2 x n u n with u n L r (), where 1/r +1/r = 1, satisfying u n (z) j 1 (z, x n (z)) a.e. on. Thechoiceofthesequence{x n } W 1,p 0 () ensures x n,x n ε n x n with ε n 0, which implies that (3.4) Dx n p p + λ x n p p j1(z, 0 x n (z); x n (z)) dz ε n x n. We also have 1 (3.5) p Dx n p p λ p x n p p j 1 (z, x n (z)) dz M 1. Combining (3.4) and (3.5), we obtain ( ) ( ) µ µ p 1 Dx n p p λ p 1 x n p p (3.6) [ µj1 (z, x n (z)) + j1(z, 0 x n (z); x n (z)) ] dz µm 1 + ε n x n. Note that (3.2) yields j 1 (z, x n (z)) = 0 and j1(z, 0 x n (z); x n (z)) = 0 a.e. on {x n 0}. So, by hypotheses H(j) 1 (iii) and (iv), we have [ µj1 (z, x n (z)) + j1(z, 0 x n (z); x n (z)) ] dz [ = µj(z, xn (z)) + j 0 (z, x n (z); x n (z)) ] (3.7) dz {0<x n <M} [ µj(z, xn (z)) + j 0 (z, x n (z); x n (z)) ] dz β 1, n 1, {x n M} for some constant β 1 > 0. Returning to (3.6) and using (3.7), we derive ( ) ( ) µ µ p 1 Dx n p p λ p 1 x n p p µm 1 + ε n x n + β 1. In order to show the boundedness of {x n } in W 1,p 0 (), we may suppose without loss of generality that λ>0. Then by (2.2) it follows that ( ) µ (3.8) )(1 p 1 λλ1 Dx n p p µm 1 + ε n x n + β 1, n 1. Because λ < λ 1 and µ > p > 1, from (3.8) we infer that {x n } W 1,p 0 () is bounded. So by passing to a subsequence if necessary, we may assume w x n x in W 1,p 0 (), x n x in L r (), x n (z) x(z) a.e.on
6 2532 D. MOTREANU, V. V. MOTREANU, AND N. S. PAPAGEORGIOU and x n (z) k(z) a.e. on, for all n 1, with k L r (). The inequality x n,x n x ε n x n x reads A(x n),x n x λ x n p 2 x n (x n x) dz u n (x n x) dz ε n x n x. Since λ x n p 2 x n (x n x) dz 0 and u n (x n x) dz 0 as n, it results that lim n),x n x = 0. The operator A being maximal monotone, n it is generalized pseudomonotone and so A(x n ),x n A(x),x, orequivalently, w Dx n p Dx p. Recalling that Dx n Dx in L p (, R N )andl p (, R N )being uniformly convex we have Dx n Dx in L p (, R N ), which means x n x in W 1,p 0 (). This proves Claim 1. Claim 2. There exists ρ>0 such that β := inf{ϕ 1,λ (x) : x = ρ} > 0. By hypothesis H(j) 1 (v), given ε>0, we can find a number δ = δ(ε) > 0such that (3.9) j 1 (z, x) =j(z, x) ε p xp for a.a. z, x [0,δ]. On the other hand, due to hypothesis H(j) 1 (iii) and the mean value theorem for locally Lipschitz functions we can find a number c ε > 0 such that (3.10) j 1 (z, x) =j(z, x) c ε x r for a.a. z, x δ. From (3.9) and (3.10) we see that (3.11) j 1 (z, x) ε p x p + c ε x r for a.a. z, x R. It is clear that for obtaining the estimate in Claim 2 it suffices to admit λ>0. Therefore, by (3.3), (3.11) and (2.2), we have (3.12) ϕ 1,λ (x) 1 ( 1 λ + ε ) Dx p p c 1 Dx r p, x W 1,p 0 (), p λ 1 for some constant c 1 = c 1 (ε) > 0. Because λ<λ 1,choosingε>0such that λ + ε<λ 1 and taking into account that p<rand D p is an equivalent norm on W 1,p 0 (), from (3.12) we see that Claim 2 is satisfied for ρ>0 sufficiently small. Claim 3. There exists ˆx W 1,p 0 () with ˆx >ρsuch that ϕ 1,λ (ˆx) <ϕ 1,λ (0) = 0. For almost all z and all x R, the function s 1 s j(z, sx) is locally µ Lipschitz on (0, + ) and we have ( 1 s j(z, sx) s µ ) µ 1 j(z, sx)+ s µ+1 s µ xj(z, sx)x. Here by s and x we denote the generalized gradient with respect to s>0and x R, respectively. Using the mean value theorem for locally Lipschitz functions, for s>1 we can find θ (1,s) such that 1 j(z, sx) j(z, x) s µ ( µ 1 j(z, θx)+ θ µ+1 θ µ xj(z, θx)x = s 1 θ µ+1 ( µj(z, θx)+ xj(z, θx)θx). ) (s 1)
7 NON-RESONANCE, RESONANCE AND NEAR RESONANCE 2533 By hypothesis H(j) 1 (iv), for almost all z and all x M, this implies 1 s µ j 1(z, sx) j 1 (z, x) s 1 ( µj(z, θx) j 0 θ µ+1 (z, θx; θx) ) 0. Then for almost all z and all x M we have j 1 (z, x) =j 1 (z, x ( x ) µ ( x ) µ M M) j(z, M) inf j(z, M). M M z From this inequality and the first part of hypothesis H(j) 1 (iv), it is seen that given η>0, we can find M η > 0 such that (3.13) j 1 (z, x) η p xp for a.a. z, x M η. Combining (3.13) and H(j) 1 (iii) shows that for a constant c η > 0 one has (3.14) j 1 (z, x) η p xp c η for a.a. z, x 0. By (3.3), (2.2), (3.14) it turns out that ) ϕ 1,λ (tu 1 ) (1 tp λλ1 ηλ1 Du 1 p p + β 3, t >0, p with a constant β 3 > 0. Now choose η>λ 1 λ>0. Claim 3 follows because ϕ 1,λ (tu 1 ) as t +. Claims 1, 2 and 3 permit the use of Theorem 2.1. We obtain x W 1,p 0 () such that ϕ 1,λ (x) β>0=ϕ 1,λ (0), thereby x 0,and0 ϕ 1,λ (x). The last inclusion ensures (3.15) A(x) λ x p 2 x = u with u L r (), u(z) j 1 (z, x(z)) a.e. on. From nonlinear regularity theory (see, e.g., [8, p. 115]) we deduce x C0 1(). Let x =max{ x, 0} W 1,p 0 () and suppose x 0. Substituting { Dx Dx(z) a.e. on {x <0}, (z) = 0 a.e. on {x 0} in A(x), x λ x p 2 x, x = u, x (see (3.15)), we obtain on the basis of (3.2) that Dx p p λ x p p = ux dz =0. Recalling λ<λ 1 and x 0, this implies that Dx p p <λ 1 x p p, which contradicts (2.2). Hence x =0andsox 0, x 0. Then by H(j) 1 (vi), we have A(x) λx p 1 = u ξx p 1 a.e. on {x >0}, thus div ( Dx(z) p 2 Dx(z)) ( λ + ξ)x(z) p 1 a.e. on. Through the nonlinear strong maximum principle due to Vázquez [18] (see also Gasiński Papageorgiou [8, p. 116]) applied with the function therein β(u) = ( λ + ξ)u p 1 for u 0, we conclude that x(z) > 0 for all z. In view of (3.15) and (3.2) we note that x solves (1.1) which completes the proof. Example 3.2. Consider problem (1.1) with the following potential (for simplicity we drop the z-dependence): { 1 1 j(x) =max η x η, r x r c } p x p,
8 2534 D. MOTREANU, V. V. MOTREANU, AND N. S. PAPAGEORGIOU for c 0, p<η<p, p r<p and c 1ifp = r. It is readily seen that hypothesis H(j) 1 is fulfilled, so Theorem 3.1 applies to the corresponding problem (1.1). If p = 2, by strengthening assumption H(j) 1 we obtain a converse result. Namely, we state H(j) 2 j : R R is a function such that j(z, 0) = 0 a.e. on, itsatisfies H(j) 1 (i) (v) with p =2,and (vi) for almost all z, allx>0andallu j(z, x), one has u>0. Theorem 3.3. If hypotheses H(j) 2 hold, then problem (1.1) has a solution x C 1 0() with x(z) > 0 for all z if and only if λ<λ 1. Proof. The sufficiency part follows from Theorem 3.1 because assumption H(j) 2 is stronger than H(j) 1. For the necessity part let x C0() 1 be a positive solution of (1.1), thus x(z) λx(z) =u(z) a.e. on with u L r (), u(z) j(z, x(z)) a.e. on. It yields x(z)u 1 (z) dz λ x(z)u 1 (z) dz = u(z)u 1 (z) dz > 0 accordingtohypothesish(j) 2 (vi). This implies x(z) u 1 (z) dz = λ 1 x(z)u 1 (z) dz > λ which is equivalent to λ 1 >λ. x(z)u 1 (z) dz, In the following we focus on the resonant case λ = λ 1. To this end we need an auxiliary result. Lemma 3.4. If ϑ 1 L () +, ϑ 1 (z) λ 1 a.e. on with strict inequality on a set of positive measure, then there exists a constant ξ 0 > 0 such that ψ(x) := Dx p p ϑ 1 (z) x(z) p dz ξ 0 Dx p p, x W 1,p 0 (). Proof. From (2.2) and the hypothesis on ϑ 1,weseethatψ 0. Suppose that the conclusion of the lemma is not true. Exploiting the p-homogeneity of ψ, we find {x n } W 1,p 0 () such that ψ(x n ) 0asn and Dx n p =1foralln 1. By passing to a subsequence if necessary, we may assume that w x n x in W 1,p 0 (), x n x in L p (), x n (z) x(z) a.e.on and x n (z) k(z) a.e.on, fork L p (). Since the norm on a Banach space is weakly lower semicontinuous, in the limit as n we obtain (3.16) Dx p p ϑ 1 (z) x(z) p dz λ 1 x p p. By (2.2), it follows that x = tu 1, t R. If x = 0, it turns out that Dx n p 0 because ψ(x n ) 0, which contradicts Dx n p =1. Thusx = tu 1, t 0. From (3.16) we have Dx p p <λ 1 x p p contradicting (2.2).
9 NON-RESONANCE, RESONANCE AND NEAR RESONANCE 2535 Restricting H(j) 1 (with 1 <p<+ ) in a different way than was done in H(j) 2 for p = 2, in particular allowing the elements of j(z, x) to be not everywhere positive whenever x>0, we incorporate in our considerations resonant problems, that is, for λ = λ 1. We formulate the new hypotheses: H(j) 3 j : R R is a function such that j(z, 0) = 0 a.e. on, itsatisfies H(j) 1 (i) (iv), (vi), and (v) there exists ϑ L () such that ϑ(z) 0a.e.on with strict inequality on a set of positive measure and pj(z, x) lim sup x 0 x p ϑ(z) uniformly for almost all z ; + (vii) for almost all z, allx, y > 0andu j(z, x), v j(z, y) one has ( u x p 1 v ) y p 1 (x y) 0. Hypothesis H(j) 3 (vii) extends a condition used in [11] in the context of asymptotically linear problems. In order to deal with the resonant case λ = λ 1 we consider the locally Lipschitz function ϕ 1 : W 1,p 0 () R defined by (3.17) ϕ 1 (x) = 1 p Dx p p λ 1 p x p p j 1 (z, x(z)) dz, x W 1,p 0 (). Here enters the modified potential function j 1 (z, x) =j(z, τ(x)) with the truncation τ : R R in (3.1). The following auxiliary result on ϕ 1 in (3.17) extends [11, Lemma 2.4]. Lemma 3.5. Under hypotheses H(j) 3, any sequence {x n } W 1,p 0 (Ω) such that x n 0 a.e. in Ω and for which there exists x n ϕ 1 (x n ) provided x n,x n 0 as n contains a relabelled subsequence satisfying ϕ 1 (tx n ) 1+tp pn + ϕ 1(x n ), t >0, n 1. Proof. By passing to a subsequence if necessary, we may assume (3.18) 1 n x n,x n = Dx n p p λ 1 x n p p u n x n dz 1, n 1, n with u n L r () satisfying u n (z) j 1 (z, x n (z)) for a.a. z. Denoting by 1 the Lebesgue measure on R, we know that j 1 (z, ) is locally Lipschitz for all z \ D 0,where D 0 1 =0. Withfixedz \ D 0 and n 1, we introduce ζ(t) = tp p u n(z)x n (z) j 1 (z, tx n (z)), t >0. The function ζ is differentiable for almost all t>0, and from assumption H(j) 3 (vii), for a.a. t>0wehave ( d ζ (t) =t p 1 ds x n (z) u n (z) j ) { 1(z, tx n (z)) 0 if t 1, t p 1 0 if t (0, 1] because d ds j 1(z, tx n (z)) j 1 (z, tx n (z)), where d ds j 1(z, s) stands for the derivative of j 1 (z, s) with respect to s. This implies that (3.19) ζ(t) ζ(1), t >0.
10 2536 D. MOTREANU, V. V. MOTREANU, AND N. S. PAPAGEORGIOU Then, by (3.17) (3.19), for t>0andn 1wederive ( ) (3.20) ϕ 1 (tx n ) tp 1 pn + p u n(z)x n (z) j 1 (z, x n (z)) Also by (3.17) and (3.18) for every n 1wehave (3.21) ϕ 1 (x n ) 1 ( ) 1 pn + p u n(z)x n (z) j 1 (z, x n (z)) Using (3.21) and (3.20) we achieve the desired conclusion. We can now handle the resonant case. Theorem 3.6. If hypotheses H(j) 3 hold and λ = λ 1, then problem (1.1) has a solution x C 1 0() with x(z) > 0 for all z. Proof. First we show that ϕ 1 in (3.17) satisfies the nonsmooth C c -condition for any number c>0. Consider a sequence {x n } W 1,p 0 () such that ϕ 1 (x n ) c and (1 + x n )m(x n ) 0 as n. We f ind x n ϕ 1 (x n )withm(x n )= x n and x n = A(x n ) λ 1 x n p 2 x n u n for u n L r (), u n (z) j 1 (z, x n (z)) a.e. on. Weinfer (1 + x n ) x n,v ε n v, v W 1,p 0 (), where ε n 0. Writing x n = x + n x n with x + n =max{x n, 0} and x n =max{ x n, 0}, we set v = x n W 1,p 0 () to obtain via (3.2) that (3.22) Dx n p p λ 1 x n p p ε n This ensures that x n 1+ x n <ε n. ϕ 1 ( x n )= 1 p Dx n p p λ 1 p x n p p 0 as n. Since ϕ 1 (x n )=ϕ 1 (x + n )+ϕ 1 ( x n ), we obtain (3.23) ϕ 1 (x + n ) c as n. Suppose that x n along a relabelled subsequence. Having x n p = x n p + x + n p and admitting without loss of generality that x + n = Dx + n p, first we assume that x + n.let t n = (2pc) 1 p x + and y n = t n x + n. n We see that x n = vn A(x n )+λ 1 (x n ) p 1, where, according to (3.2), vn = A(x + n ) λ 1 (x + n ) p 1 u n ϕ 1 (x + n ). We deduce vn,x + n = x n,x + n 0asn, which enables us to apply Lemma 3.5 and (3.23) for obtaining along a relabelled subsequence that (3.24) ϕ 1 (y n )=ϕ 1 (t n x + n ) 1+tp n pn dz. dz. + ϕ 1(x + n ) c as n.
11 NON-RESONANCE, RESONANCE AND NEAR RESONANCE 2537 Eventually passing to a subsequence, we may suppose w y n y in W 1,p 0 (), y n y in L r (), y n (z) y(z) a.e.on and y n (z) k(z) a.e.on, for all n 1, with k L r (). If y =0,then (3.25) ϕ 1 (y n )=2c λ 1 p y n p p j 1 (z, y n (z)) dz 2c. Comparing (3.25) and (3.24), we achieve a contradiction since c>0. This proves that y 0 and clearly y 0. Thus the set C = {y >0} R N has the Lebesgue measure C N > 0. So x + n (z) + a.e. on C. On the other hand, (3.23) shows (3.26) pϕ 1 (x + n ) vn,x + n pc as n. As in Claim 3 of the proof of Theorem 3.1, using H(j) 1 (iii)-(iv) we derive j 1 (z, x) a 0 x µ ˆβ, for a.a. z, x 0, with constants a 0 > 0and ˆβ >0. Due to H(j) 1 (iv), this gives ux pj(z, x) (µ p)j(z, x) (µ p)a 0 x µ (µ p) ˆβ, for almost all z, allx M and all u j 1 (z, x). We conclude that (3.27) ux pj 1 (z, x) + as x + uniformly for a.a. z and all u j 1 (z, x). Thus we can find ˆη >0 such that ux pj 1 (z, x) 1 for a.a. z, x ˆη, u j 1 (z, x). In addition, from H(j) 1 (iii) there exists ˆη 1 > 0 satisfying ux pj 1 (z, x) ˆη 1 for a.a. z, x [0, ˆη], u j 1 (z, x). Consequently, a constant η 2 R can be found such that (3.28) ux pj 1 (z, x) η 2 for a.a. z, x 0, u j 1 (z, x). Then from (3.28) and (3.2) we have pϕ 1 (x + n ) vn,x + n = (u n (z)x + n (z) pj 1 (z, x + n (z))) dz (u n (z)x + n (z) pj 1 (z, x + n (z))) dz + η 2 \ C N. C By (3.27) and x + n (z) + a.e. on C it follows that (u n (z)x + n (z) pj 1 (z, x + n (z))) dz +. C Since this contradicts (3.26), we infer {x + n } W 1,p 0 () is bounded. Therefore we must have that x n. Set ŷ n = x n x n. We may assume w ŷ n ŷ in W 1,p 0 (), ŷ n ŷ in L p (), ŷ n (z) ŷ(z) a.e.on and ŷ n (z) ˆk(z) a.e. on, for all n 1, with ˆk L p (). Because {x + n } W 1,p 0 () is bounded, we have ŷ 0. By (3.22) we obtain (3.29) Dŷn p p λ 1 ŷn p p <ε n provided n is sufficiently large to have x n 1. Letting n gives Dŷ p p λ 1 ŷ p p. Recalling that ŷ 0, this implies ŷ = tu 1 with t 0. If ŷ =0,thenfrom
12 2538 D. MOTREANU, V. V. MOTREANU, AND N. S. PAPAGEORGIOU (3.29) we have ŷ n 0inW 1,p 0 (), which contradicts that ŷ n =1. Itremains that ŷ = tu 1 with t>0, thus x n (z) a.e. on. The choice of the sequence {x n } W 1,p 0 () shows (3.30) pϕ 1 (x n ) x n,x n = (u n (z)x n (z) pj 1 (z, x n (z))) dz pc as n. By (3.2) and the definition of the truncated potential j 1 (z, x), we obtain (u n (z)x n (z) pj 1 (z, x n (z))) dz =0, n 1. {x n <0} Since {x + n } W 1,p 0 () is bounded and {x n 0} N 0asn because x n (z) a.e. on, wehave (u n (z)x n (z) pj 1 (z, x n (z))) dz 0 as n. {x n 0} So finally we get (3.31) (u n (z)x n (z) pj 1 (z, x n (z))) dz 0 as n. Comparing (3.30) and (3.31) a contradiction is reached because c>0. This proves the boundedness of {x n } W 1,p 0 (). From now on we can proceed as in Claim 1 of the proof of Theorem 3.1 to check that ϕ 1 satisfies the nonsmooth C c -condition for c>0. From the hypotheses H(j) 1 (iii), H(j) 3 (v) and the mean value theorem for locally Lipschitz functions, given ε>0, there is c ε > 0 such that j 1 (z, x) 1 p (ϑ(z)+ε) x p + c ε x r for a.a. z, x R. Consequently, for all x W 1,p 0 (), by (3.17) it is seen that ϕ 1 (x) 1 p Dx p p 1 (λ 1 + ϑ(z)) x(z) p dz ε p p x p p c 2 Dx r p, with a constant c 2 > 0. We set ϑ 1 (z) =ϑ(z) +λ 1 λ 1 a.e. on with strict inequality on a set of positive measure in view of H(j) 3 (v). By Lemma 3.4 we find ξ 0 > 0 with the property ϕ 1 (x) 1 p (ξ 0 ε λ 1 ) Dx p p c 2 Dx r p, x W 1,p 0 (). Choosing ε<ξ 0 λ 1, because r>pthere is ρ>0 satisfying inf{ϕ 1 (x) : x = ρ} = β>0. Furthermore, proceeding as for Claim 3 in the proof of Theorem 3.1 we achieve the assertion therein. Then using the inequality inf γ Γ max ϕ 1(γ(t)) β>0, t [0,1] with Γ as in Theorem 2.1, it follows that the nonsmooth C c -condition suffices for c>0 to be applied. The rest of the proof proceeds as that of Theorem 3.1. We obtain x =0sinceifx = tu 1 with t>0, then x = tu 1 and ϕ 1 (x) =0,a contradiction.
13 NON-RESONANCE, RESONANCE AND NEAR RESONANCE 2539 Example 3.7. Consider problem (1.1) at resonance λ = λ 1 with the locally Lipschitz potential given below: ϑ(z) p x p if x 1, j(z, x) = 1 µ xµ + ϑ(z) p xp 1 if x 1, µ for p<µ<p and ϑ L () whereϑ(z) 0a.e. on with strict inequality on a set of positive measure. It is straightforward to verify that assumptions H(j) 3 are fulfilled, so Theorem 3.6 can be applied yielding a positive solution to problem (1.1) in the resonant case λ = λ 1 with j(z, x) asabove. 4. Non-negative solutions with near resonance from the left In this section the parameter λ approaches λ 1 > 0 from the left. Our hypotheses on the potential j(z, x) are the following: H(j) 4 j : R R is a function such that j(z, 0) = 0 a.e. on, itsatisfies H(j) 1 (i), (ii), (v), (vi), and (iii) for almost all z, allx R and all u j(z, x), we have u a(z)+c x p 1 with a L () +,c>0; (iv) there exists a constant ε 0 > 0 such that u lim inf x + x p 1 ε 0 uniformly for almost all z and all u j(z, x). Theorem 4.1. If hypotheses H(j) 4 hold, then for all λ (λ 1 ε 0,λ 1 ) problem (1.1) has a solution x C0() 1 with x(z) > 0 for all z. Proof. We again consider the locally Lipschitz functional ϕ 1,λ : W 1,p 0 () R defined by (3.3). Claim 1. For every λ (λ 1 ε 0,λ 1 ), ϕ 1,λ satisfies the nonsmooth PS-condition. Suppose that {x n } W 1,p 0 () is a sequence such that ϕ 1,λ (x n ) M 2 for some constant M 2 > 0, all n 1 and m(x n ) 0. We can find x n ϕ 1,λ (x n ) such that m(x n )= x n for all n 1. We see x n = A(x n ) λ x n p 2 x n u n with u n L p (), u n (z) j 1 (z, x n (z)) a.e. on. Let us show that {x n } W 1,p 0 () is bounded. Arguing indirectly, we assume that x n along a relabelled subsequence. Set y n = x n x n, n 1. Then at least for a subsequence, we have w y n y in W 1,p 0 (), y n y in L p (), y n (z) y(z) a.e.on and (4.1) y n (z) k(z) a.a.z, for all n 1andwithk L p (). Since x n W 1,p 0 (), we have x n, x n ε n x n with ε n 0, which implies by (3.2) that Dx n p p λ x n p p ε n x n. Ifλ 0, it is clear that {x n } W 1,p 0 () is bounded. If 0 <λ<λ 1,wenote ) (1 λλ1 Dx n p p ε n x n,
14 2540 D. MOTREANU, V. V. MOTREANU, AND N. S. PAPAGEORGIOU so again {x n } W 1,p 0 () is bounded. Writing y n =(1/ x n )(x + n x n ), from the boundedness of {x n } it results that (1/ x n )x + w n y in W 1,p 0 (), and up to a subsequence (1/ x n )x + n (z) y(z) a.e.z, therefore y 0. We now show that y 0. To this end we write (4.2) A(y n ),y n y λ y n p 2 u n y n (y n y) dz x n p 1 (y n y) dz ε n y n y wherewehaveused x n 1 for sufficiently large n. From (4.1) we infer (4.3) y n p 2 y n (y n y) dz 0 as n. By virtue of hypothesis H(j) 4 (iii), {(1/ x n p 1 )u n } is bounded in L p () so u n (4.4) x n p 1 (y n y) dz 0 as n. Passing to the limit in (4.2) and using the convergences in (4.3) and (4.4), we obtain lim A(y n),y n y = 0. Then as in the proof of Theorem 3.1 (see Claim 1), we n obtain y n y in W 1,p 0 (). This entails y = 1, hence y 0,y 0. We notice that by hypothesis H(j) 4 (iii) we have u lim sup c uniformly for almost all z and all u j(z, x). x + xp 1 Given ε (0,ε 0 ), with ε 0 in H(j) 4 (iv), we introduce the set E ε,n = {z : x n (z) > 0, ε 0 ε< u n(z) c + ε}, n 1. x n (z) p 1 Note that for almost all z {y > 0}, we have x n (z) +. So hypothesis H(j) 4 (iv) implies that the characteristic function χ Eε,n of E ε,n has the property χ Eε,n (z) 1a.e.on{y >0}. Assumption H(j) 4 (iii) ensures u n (z) (4.5) x n p 1 a(z) x n p 1 + c y n(z) p 1 a.e. on. Then {(1/ x n p 1 )u n } is bounded in L p (), so up to a subsequence one has 1 x n p 1 u w n h in L p () asn, for some h L p (). Since using (4.5), ( u n 1 χeε,n) x n p 1 0 in L1 ({y >0}), it follows that u n w χ Eε,n h in L 1 x n p 1 ({y >0}) asn. The definition of the sets E ε,n shows χ Eε,n (z)(ε 0 ε)y n (z) p 1 χ Eε,n (z) u n (z) x n p 1 χ E ε,n (z)(c + ε)y n (z) p 1 for a.e. on {y >0}. Taking weak limits in L 1 ({y >0}), we obtain (ε 0 ε)y(z) p 1 h(z) (c + ε)y(z) p 1 a.e. on {y >0}.
15 NON-RESONANCE, RESONANCE AND NEAR RESONANCE 2541 Because ε>0 was arbitrary, we conclude (4.6) ε 0 y(z) p 1 h(z) cy(z) p 1 a.e. on {y >0}. Also from (4.5) it is clear that (4.7) h(z) =0 a.e. on{y =0}. Taking into account that = {y =0} {y>0}, from (4.6) and (4.7) we infer (4.8) h(z) =g(z)y(z) p 1 with ε 0 g(z) c a.e. on. Thechoiceofthesequence{x n } W 1,p 0 () implies A(y n),v λ y n p 2 u n y n vdz vdz x n p 1 ε n v, v W 1,p 0 (), with ε n 0. Using y n y in W 1,p 0 () and (4.8), we obtain that in W 1,p () there holds (4.9) div ( Dy(z) p 2 Dy(z)) = (λ + g(z)) y(z) p 2 y(z). Exploiting (4.8), the assumption λ>λ 1 ε 0 and the decreasing monotonicity of the principal eigenvalue λ 1 (ĝ) > 0 of the weighted eigenvalue problem p v(z) = λĝ(z) v(z) p 2 v(z) a.e. on, v = 0, on the weight function ĝ L () +,we deduce (4.10) λ 1 (g + λ) <λ 1 (λ 1 )=1. By (4.9) in conjunction with y 0,wegetthaty is an eigenfunction whose corresponding eigenvalue is 1 for the problem with weight g + λ L () +. According to (4.10), y cannot be a principal eigenfunction that is associated to the first eigenvalue λ 1 (g + λ). Knowing that only the principal eigenfunctions have constant sign (see Anane [2]), we deduce that y must change sign, thus achieving a contradiction with y 0. This proves that {x n } W 1,p 0 () is bounded, from which as in Claim 1 of the proof of Theorem 3.1 we conclude that ϕ 1,λ satisfies the nonsmooth PS-condition. Claim 2. For every λ (λ 1 ε 0,λ 1 ), we can find v λ W 1,p 0 () of arbitrarily large norm such that ϕ 1,λ (v λ ) < 0=ϕ 1,λ (0). Fix λ (λ 1 ε 0,λ 1 ). For a.a. z and all s R \ D(z) with D(z) 1 =0there exists d ds j(z, s) and d ds j(z, s) j(z, s). So by H(j) 4(iv) for ε (0,ε 0 ), there is M = M(ε) > 0 such that d (4.11) ds j(z, s) (ε 0 ε)s p 1 for a.a. z, s M, s D(z). By H(j) 4 (iii) and (4.11), it is allowed to write j(z, x) = x 0 d j(z, s) ds = ds M 0 d j(z, s) ds + ds x M d j(z, s) ds ds ξ ε + 1 p (ε 0 ε)(x p M p ) for a.a. z, x M, for some constant ξ ε > 0. Thus there is c ε > 0 such that (4.12) j 1 (z, x) =j(z, x) 1 p (ε 0 ε) x p c ε for a.a. z, x 0.
16 2542 D. MOTREANU, V. V. MOTREANU, AND N. S. PAPAGEORGIOU Then, from (3.3), (2.2) and (4.12), for all t>0 we get with some constant ĉ ε > 0 that ( ϕ 1,λ (tu 1 ) tp 1 λ + ε ) 0 ε Du 1 p p +ĉ ε. p λ 1 Choosing ε<λ λ 1 + ε 0, we obtain that ϕ 1,λ (tu 1 ) as t +, which establishes Claim 2. Using H(j) 1 (v), as in Claim 2 of the proof of Theorem 3.1 we find ρ>0with (4.13) inf{ϕ 1,λ (x) : x = ρ} = β>0. Claims 1, 2 and assertion (4.13) enable us to make use of Theorem 2.1, which gives for all λ (λ 1 ε 0,λ 1 ) the existence of an x = x(λ) W 1,p 0 () such that ϕ 1,λ (x) β>0=ϕ 1,λ (0) (thus x 0)and0 ϕ 1,λ (x). The reasoning in the final part of the proof of Theorem 3.1, based on hypothesis H(j) 1 (vi), allows us to conclude that x is a positive solution of problem (1.1). Example 4.2. Consider problem (1.1) with the locally Lipschitz potential j(z, x) = j(x) givenby { } 1 1 j(x) =min p x p, r x r, with 1 <p<r<p. A direct verification shows that the assumptions of Theorem 4.1 are fulfilled. We obtain the existence of a positive solution to problem (1.1) with j(z, x) =j(x) as above and λ (λ 1 1,λ 1 ). 5. Multiple solutions with near resonance from the right We now examine the eigenvalue problem (1.1) near resonance from the right of λ 1. The hypotheses on the nonsmooth potential j(z, x) are the following: H(j) 5 j : R R is a function such that j(z, 0) = 0 a.e. on, itsatisfies H(j) 1 (i) (iii), and (iv) there exists ˆη L () such that ˆη(z) 0a.e.on with strict inequality on a set of positive measure and pj(z, x) lim sup x + x p ˆη(z) uniformly for almost all z ; (v) there exists δ>0such that for almost all z and all 0 x δ or δ x 0, we have j(z, x) 0. Theorem 5.1. If hypotheses H(j) 5 hold, then there exists ˆε >0 such that for all λ (λ 1,λ 1 +ˆε) problem (1.1) has a nontrivial solution x C 1 0(). Proof. Suppose that λ = λ 1 + ε with ε>0. From hypotheses H(j) 1 (iii), H(j) 5 (iv) and the mean value theorem for locally Lipschitz functions, we can find a constant c ε > 0 such that (5.1) j(z, x) 1 p (ˆη(z)+ε) x p + c ε for a.a. z, x R. Consider the locally Lipschitz functional ϕ λ : W 1,p 0 () R defined by (5.2) ϕ λ (x) = 1 p Dx p p λ p x p p j(z, x(z)) dz, x W 1,p 0 ().
17 NON-RESONANCE, RESONANCE AND NEAR RESONANCE 2543 By (5.2), (5.1) and Lemma 3.4 we derive the estimate ϕ λ (x) 1 p Dx p p 1 p 1 ( ξ 0 2ε p λ 1 (λ 1 +ˆη(z)) x(z) p dz 2ε p x p p ĉ ε ) Dx p p ĉ ε, x W 1,p 0 (), with a constant ĉ ε > 0andwhereξ 0 > 0 is independent of ε>0. We see that if ε (0, ˆε) withˆε = 1 2 ξ 0λ 1,thenϕ λ is coercive, and being weakly lower semicontinuous, we obtain x = x(λ) W 1,p 0 () such that ϕ λ (x) = inf W 1,p 0 () ϕ λ. Therefore x = x(λ) solves problem (1.1). We claim ϕ λ (x) < 0=ϕ λ (0). Suppose that the first option in hypothesis H(j) 5 (v) is valid (the reasoning is similar in the other case). Since u 1 C0(), 1 we can find t>0 such that tu 1 δ. Then, by (5.2) and H(j) 5 (v),wehave ϕ λ (tu 1 ) tp p (1 λλ1 ) Du 1 p p < 0. The claim is verified and so x 0. From nonlinear regularity theory we obtain that x C0(), 1 thus completing the proof. Example 5.2. Consider problem (1.1) with { x if x 1, j(z, x) = x +ˆη(z)( x 1) p if x 1 where ˆη L () is such that ˆη(z) 0a.e.on with strict inequality on a set of positive measure. One can easily check that hypotheses H(j) 5 are satisfied and so Theorem 5.1 applies to problem (1.1) with j(z, x) as defined above and λ near λ 1 from the right. Finally, strengthening hypothesis H(j) 5 (v) we prove a theorem on the existence of multiple nontrivial solutions for problem (1.1) under near resonance at λ 1 > 0 from the right. Consider the direct sum decomposition (5.3) W 1,p 0 () =Ru 1 V with V = {v W 1,p 0 () : up 1 1 vdz =0}. Since λ 1 > 0 is isolated, there is ˆλ 2 >λ 1 such that (5.4) ˆλ2 v p p Dv p p, v V. We formulate the assumptions H(j) 6 j : R R is a function such that j(z, 0) = 0 a.e. on, itsatisfies H(j) 1 (i) (iii), H(j) 5 (iv) and there exists δ>0such that for almost all z and all x [ δ, δ], one has (v) 0 j(z, x) β p x p with β<ˆλ 2 λ 1. Theorem 5.3. If hypotheses H(j) 6 hold, then there exists ˆε 0 > 0 such that for all λ (λ 1,λ 1 +ˆε 0 ) problem (1.1) has at least two nontrivial solutions x 1,x 2 C 1 0().
18 2544 D. MOTREANU, V. V. MOTREANU, AND N. S. PAPAGEORGIOU Proof. In the proof of Theorem 5.1 we obtained an ˆε >0 such that for all ε (0, ˆε), the functional ϕ λ in (5.2) is coercive with λ = λ 1 +ε. Therefore it is bounded below and satisfies the nonsmooth PS-condition. Moreover, again in the proof of Theorem 5.1, it was established that inf W 1,p 0 () ϕ λ <ϕ λ (0) = 0. Since u 1 C0(), 1 we can find t 0 > 0 such that if t t 0,then tu 1 (z) δ for all z, withδ>0determined in hypothesis H(j) 6 (v). By (5.2), H(j) 6 (v) and since λ>λ 1,weget ) (5.5) ϕ λ (tu 1 ) (1 tp λλ1 Du 1 p p 0 for t t 0. p On the other hand, from hypotheses H(j) 1 (iii) and H(j) 6 (v) we have (5.6) j(z, x) β p x p +ĉ x r for a.a. z, x R, with a constant ĉ>0. Then from (5.2) (5.4), (5.6) we obtain (5.7) ϕ λ (v) 1 ( 1 λ ) 1 + ε + β Dv p ˆλ p p ĉ 1 Dv r p, v V, λ = λ 1 +ε, ε>0, 2 for a new constant ĉ 1 > 0. Since ˆλ 2 λ 1 β>0(cf. hypothesish(j) 6 (v) ), we may take ε (0, ˆλ 2 λ 1 β). Hence by (5.7), recalling r>p, we can find ˆδ >0 such that (5.8) ϕ λ (v) 0 for all v V with v ˆδ. So if ˆε 0 =min{ˆε, ˆλ 2 λ 1 β}, then due to (5.5), (5.8), we can apply Theorem 2.2 that provides two nontrivial critical points of ϕ λ. This amounts to saying that there exist two nontrivial solutions of problem (1.1). Through the nonlinear regularity theory, they belong to C0 1 (). Example 5.4. Consider problem (1.1) with the locally Lipschitz potential j(z, x) = j(x) introduced as follows: β p ln( x p +1) if x 1, j(x) = (2 x ) β p ln( x p +1)+( x 1) ˆη(z) p x p if 1 < x < 2, ˆη(z) p x p if x 2. Here ˆη L () is such that ˆη(z) 0a.e. on with strict inequality on a set of positive measure and 0 β<ˆλ 2 λ 1. It is easily seen that hypotheses H(j) 6 are verified. Hence Theorem 5.3 can be applied to problem (1.1) with the above potential j(z, x) =j(x) in the case of near resonance from the right at λ 1. References 1. A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), MR (51:6412) 2. A. Anane, Simplicité et isolation de la première valeur propre du p-laplacien avec poids, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), MR (89e:35124) 3. G. Barletta and S.A. Marano, Some remarks on critical point theory for locally Lipschitz functions, Glasg. Math. J. 45 (2003), MR (2004e:58016)
19 NON-RESONANCE, RESONANCE AND NEAR RESONANCE K.-C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), MR (82h:35025) 5. F.H. Clarke, Optimization and Nonsmooth Analysis, John Wiley and Sons, New York, MR (85m:49002) 6. L. Gasiński and N.S. Papageorgiou, Multiple solutions for semilinear hemivariational inequalities at resonance, Publ. Math. Debrecen 59 (2001), MR (2003d:35202) 7. L. Gasiński and N.S. Papageorgiou, Existence of solutions and of multiple solutions for eigenvalue problems of hemivariational inequalities, Adv. Math. Sci. Appl. 11 (2001), MR (2002c:49014) 8. L. Gasiński and N.S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems, Chapman & Hall/CRC, Boca Raton, FL, MR (2006f:58013) 9. D. Goeleven, D. Motreanu, and P.D. Panagiotopoulos, Multiple solutions for a class of eigenvalue problems in hemivariational inequalities, Nonlinear Anal. 29 (1997), MR (98f:47072) 10. D. Goeleven, D. Motreanu, and P.D. Panagiotopoulos, Eigenvalue problems for variationalhemivariational inequalities at resonance, Nonlinear Anal. 33 (1998), MR (99b:47094) 11. G. Li and H.-S. hou, Asymptotically linear Dirichlet problem for the p-laplacian, Nonlinear Anal. 43 (2001), MR (2001m:35113) 12. R. Livrea, S.A. Marano, and D. Motreanu, Critical points for nondifferentiable functions in presence of splitting, J.DifferentialEquations226 (2006), MR D. Motreanu and P.D. Panagiotopoulos, A minimax approach to the eigenvalue problem of hemivariational inequalities and applications, Appl. Anal. 58 (1995), MR (97h:47064) 14. D. Motreanu and P.D. Panagiotopoulos, On the eigenvalue problem for hemivariational inequalities: existence and multiplicity of solutions, J. Math. Anal. Appl. 197 (1996), MR (96k:47113) 15. D. Motreanu and P.D. Panagiotopoulos, Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities, Kluwer Academic Publishers, Dordrecht, MR (2000a:49015) 16. D. Motreanu and N.S. Papageorgiou, Multiple solutions for nonlinear elliptic equations at resonance with a nonsmooth potential, Nonlinear Anal. 56 (2004), MR (2005b:35087) 17.. Naniewicz and P.D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Marcel Dekker, Inc., New York, MR (96d:47067) 18.J.L.Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optimization 12 (1984), MR (86m:35018) 19. X. Wu, A new critical point theorem for locally Lipschitz functionals with applications to differential equations, Nonlinear Anal. 66 (2007), MR (2007g:35068) Département de Mathématiques, Université de Perpignan, Perpignan, France address: motreanu@univ-perp.fr Département de Mathématiques, Université de Perpignan, Perpignan, France address: viorica@univ-perp.fr Department of Mathematics, National Technical University, ografou Campus, Athens 15780, Greece address: npapg@math.ntua.gr
EXISTENCE RESULTS FOR QUASILINEAR HEMIVARIATIONAL INEQUALITIES AT RESONANCE. Leszek Gasiński
DISCRETE AND CONTINUOUS Website: www.aimsciences.org DYNAMICAL SYSTEMS SUPPLEMENT 2007 pp. 409 418 EXISTENCE RESULTS FOR QUASILINEAR HEMIVARIATIONAL INEQUALITIES AT RESONANCE Leszek Gasiński Jagiellonian
More informationElliptic problems with discontinuities
J. Math. Anal. Appl. 276 (2002) 13 27 www.elsevier.com/locate/jmaa Elliptic problems with discontinuities Nikolaos Halidias Department of Statistics and Actuarial Science, University of the Aegean, 83200
More informationON THE EXISTENCE OF THREE SOLUTIONS FOR QUASILINEAR ELLIPTIC PROBLEM. Paweł Goncerz
Opuscula Mathematica Vol. 32 No. 3 2012 http://dx.doi.org/10.7494/opmath.2012.32.3.473 ON THE EXISTENCE OF THREE SOLUTIONS FOR QUASILINEAR ELLIPTIC PROBLEM Paweł Goncerz Abstract. We consider a quasilinear
More informationNONTRIVIAL SOLUTIONS FOR SUPERQUADRATIC NONAUTONOMOUS PERIODIC SYSTEMS. Shouchuan Hu Nikolas S. Papageorgiou. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 34, 29, 327 338 NONTRIVIAL SOLUTIONS FOR SUPERQUADRATIC NONAUTONOMOUS PERIODIC SYSTEMS Shouchuan Hu Nikolas S. Papageorgiou
More informationEXISTENCE OF FIVE NONZERO SOLUTIONS WITH EXACT. Michael Filippakis. Alexandru Kristály. Nikolaos S. Papageorgiou
DISCRETE AND CONTINUOUS doi:10.3934/dcds.2009.24.405 DYNAMICAL SYSTEMS Volume 24, Number 2, June 2009 pp. 405 440 EXISTENCE OF FIVE NONERO SOLUTIONS WITH EXACT SIGN FOR A p-laplacian EQUATION Michael Filippakis
More informationMultiple Solutions for Parametric Neumann Problems with Indefinite and Unbounded Potential
Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 8, Number 2, pp. 281 293 (2013) http://campus.mst.edu/adsa Multiple Solutions for Parametric Neumann Problems with Indefinite and Unbounded
More informationarxiv: v2 [math.ap] 3 Nov 2014
arxiv:22.3688v2 [math.ap] 3 Nov 204 Existence result for differential inclusion with p(x)-laplacian Sylwia Barnaś email: Sylwia.Barnas@im.uj.edu.pl Cracow University of Technology Institute of Mathematics
More informationNonlinear Analysis 72 (2010) Contents lists available at ScienceDirect. Nonlinear Analysis. journal homepage:
Nonlinear Analysis 72 (2010) 4298 4303 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Local C 1 ()-minimizers versus local W 1,p ()-minimizers
More informationNonlinear elliptic systems with exponential nonlinearities
22-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 9, 22, pp 139 147. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu
More informationResonant nonlinear Neumann problems with indefinite weight
Resonant nonlinear Neumann problems with indefinite weight Dimitri Mugnai Dipartimento di Matematica e Informatica Università di Perugia Via Vanvitelli 1, 06123 Perugia - Italy tel. +39 075 5855043, fax.
More informationJournal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied Mathematics http://jipam.vu.edu.au/ Volume 2, Issue 1, Article 12, 2001 ON KY FAN S MINIMAX INEQUALITIES, MIXED EQUILIBRIUM PROBLEMS AND HEMIVARIATIONAL INEQUALITIES
More informationMultiple positive solutions for a class of quasilinear elliptic boundary-value problems
Electronic Journal of Differential Equations, Vol. 20032003), No. 07, pp. 1 5. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu login: ftp) Multiple positive
More informationFROM VARIATIONAL TO HEMIVARIATIONAL INEQUALITIES
An. Şt. Univ. Ovidius Constanţa Vol. 12(2), 2004, 41 50 FROM VARIATIONAL TO HEMIVARIATIONAL INEQUALITIES Panait Anghel and Florenta Scurla To Professor Dan Pascali, at his 70 s anniversary Abstract A general
More informationEXISTENCE RESULTS FOR OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS VIA THE MOUNTAIN PASS THEOREM
EXISTENCE RESULTS FOR OPERATOR EQUATIONS INVOLVING DUALITY MAPPINGS VIA THE MOUNTAIN PASS THEOREM JENICĂ CRÎNGANU We derive existence results for operator equations having the form J ϕu = N f u, by using
More informationResearch Article Dirichlet Problems with an Indefinite and Unbounded Potential and Concave-Convex Nonlinearities
Abstract and Applied Analysis Volume 2012, Article ID 492025, 36 pages doi:10.1155/2012/492025 Research Article Dirichlet Problems with an Indefinite and Unbounded Potential and Concave-Convex Nonlinearities
More informationVariational eigenvalues of degenerate eigenvalue problems for the weighted p-laplacian
Variational eigenvalues of degenerate eigenvalue problems for the weighted p-laplacian An Lê Mathematics Sciences Research Institute, 17 Gauss Way, Berkeley, California 94720 e-mail: anle@msri.org Klaus
More informationWeak solvability of quasilinear elliptic inclusions with mixed boundary conditions
Weak solvability of quasilinear elliptic inclusions with mixed boundary conditions Nicuşor Costea a, and Felician Dumitru Preda b a Department of Mathematics and its Applications, Central European University,
More informationEXISTENCE OF SOLUTIONS FOR A RESONANT PROBLEM UNDER LANDESMAN-LAZER CONDITIONS
Electronic Journal of Differential Equations, Vol. 2008(2008), No. 98, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) EXISTENCE
More informationEXISTENCE OF THREE WEAK SOLUTIONS FOR A QUASILINEAR DIRICHLET PROBLEM. Saeid Shokooh and Ghasem A. Afrouzi. 1. Introduction
MATEMATIČKI VESNIK MATEMATIQKI VESNIK 69 4 (217 271 28 December 217 research paper originalni nauqni rad EXISTENCE OF THREE WEAK SOLUTIONS FOR A QUASILINEAR DIRICHLET PROBLEM Saeid Shokooh and Ghasem A.
More informationSemilinear Robin problems resonant at both zero and infinity
Forum Math. 2018; 30(1): 237 251 Research Article Nikolaos S. Papageorgiou and Vicențiu D. Rădulescu* Semilinear Robin problems resonant at both zero and infinity DOI: 10.1515/forum-2016-0264 Received
More informationBIFURCATION OF POSITIVE SOLUTIONS FOR NONLINEAR NONHOMOGENEOUS ROBIN AND NEUMANN PROBLEMS WITH COMPETING NONLINEARITIES. Nikolaos S.
DISCRETE AND CONTINUOUS doi:10.3934/dcds.2015.35.5003 DYNAMICAL SYSTEMS Volume 35, Number 10, October 2015 pp. 5003 5036 BIFURCATION OF POSITIVE SOLUTIONS FOR NONLINEAR NONHOMOGENEOUS ROBIN AND NEUMANN
More informationCOMBINED EFFECTS FOR A STATIONARY PROBLEM WITH INDEFINITE NONLINEARITIES AND LACK OF COMPACTNESS
Dynamic Systems and Applications 22 (203) 37-384 COMBINED EFFECTS FOR A STATIONARY PROBLEM WITH INDEFINITE NONLINEARITIES AND LACK OF COMPACTNESS VICENŢIU D. RĂDULESCU Simion Stoilow Mathematics Institute
More informationNONHOMOGENEOUS ELLIPTIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENT AND WEIGHT
Electronic Journal of Differential Equations, Vol. 016 (016), No. 08, pp. 1 1. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu NONHOMOGENEOUS ELLIPTIC
More informationSYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS. Massimo Grosi Filomena Pacella S. L. Yadava. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 21, 2003, 211 226 SYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS Massimo Grosi Filomena Pacella S.
More informationarxiv: v1 [math.ap] 16 Jan 2015
Three positive solutions of a nonlinear Dirichlet problem with competing power nonlinearities Vladimir Lubyshev January 19, 2015 arxiv:1501.03870v1 [math.ap] 16 Jan 2015 Abstract This paper studies a nonlinear
More informationMULTIPLE POSITIVE SOLUTIONS FOR P-LAPLACIAN EQUATION WITH WEAK ALLEE EFFECT GROWTH RATE
Differential and Integral Equations Volume xx, Number xxx,, Pages xx xx MULTIPLE POSITIVE SOLUTIONS FOR P-LAPLACIAN EQUATION WITH WEAK ALLEE EFFECT GROWTH RATE Chan-Gyun Kim 1 and Junping Shi 2 Department
More informationVARIATIONAL AND TOPOLOGICAL METHODS FOR DIRICHLET PROBLEMS. The aim of this paper is to obtain existence results for the Dirichlet problem
PORTUGALIAE MATHEMATICA Vol. 58 Fasc. 3 2001 Nova Série VARIATIONAL AND TOPOLOGICAL METHODS FOR DIRICHLET PROBLEMS WITH p-laplacian G. Dinca, P. Jebelean and J. Mawhin Presented by L. Sanchez 0 Introduction
More informationEXISTENCE OF NONTRIVIAL SOLUTIONS FOR A QUASILINEAR SCHRÖDINGER EQUATIONS WITH SIGN-CHANGING POTENTIAL
Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 05, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE OF NONTRIVIAL
More informationThe Brézis-Nirenberg Result for the Fractional Elliptic Problem with Singular Potential
arxiv:1705.08387v1 [math.ap] 23 May 2017 The Brézis-Nirenberg Result for the Fractional Elliptic Problem with Singular Potential Lingyu Jin, Lang Li and Shaomei Fang Department of Mathematics, South China
More informationAsymptotic Analysis 93 (2015) DOI /ASY IOS Press
Asymptotic Analysis 93 215 259 279 259 DOI 1.3233/ASY-151292 IOS Press Combined effects of concave convex nonlinearities and indefinite potential in some elliptic problems Nikolaos S. Papageorgiou a and
More informationNONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian UNDER NONHOMOGENEOUS NEUMANN BOUNDARY CONDITION
Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 210, pp. 1 7. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu NONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian
More informationNONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS
Fixed Point Theory, Volume 9, No. 1, 28, 3-16 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html NONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS GIOVANNI ANELLO Department of Mathematics University
More informationMartin Luther Universität Halle Wittenberg Institut für Mathematik
Martin Luther Universität Halle Wittenberg Institut für Mathematik Constant-Sign and Sign-Changing Solutions for Nonlinear Elliptic Equations with Neumann Boundary Values Patrick Winkert Report No. 20
More informationThe local equicontinuity of a maximal monotone operator
arxiv:1410.3328v2 [math.fa] 3 Nov 2014 The local equicontinuity of a maximal monotone operator M.D. Voisei Abstract The local equicontinuity of an operator T : X X with proper Fitzpatrick function ϕ T
More informationON THE EIGENVALUE PROBLEM FOR A GENERALIZED HEMIVARIATIONAL INEQUALITY
STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume XLVII, Number 1, March 2002 ON THE EIGENVALUE PROBLEM FOR A GENERALIZED HEMIVARIATIONAL INEQUALITY ANA-MARIA CROICU Abstract. In this paper the eigenvalue
More informationNOTE ON THE NODAL LINE OF THE P-LAPLACIAN. 1. Introduction In this paper we consider the nonlinear elliptic boundary-value problem
2005-Oujda International Conference on Nonlinear Analysis. Electronic Journal of Differential Equations, Conference 14, 2006, pp. 155 162. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
More informationPseudo-monotonicity and degenerate elliptic operators of second order
2002-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 09, 2002, pp 9 24. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu
More informationSYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS. M. Grossi S. Kesavan F. Pacella M. Ramaswamy. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 12, 1998, 47 59 SYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS M. Grossi S. Kesavan F. Pacella M. Ramaswamy
More informationarxiv: v1 [math.ca] 5 Mar 2015
arxiv:1503.01809v1 [math.ca] 5 Mar 2015 A note on a global invertibility of mappings on R n Marek Galewski July 18, 2017 Abstract We provide sufficient conditions for a mapping f : R n R n to be a global
More informationSubdifferential representation of convex functions: refinements and applications
Subdifferential representation of convex functions: refinements and applications Joël Benoist & Aris Daniilidis Abstract Every lower semicontinuous convex function can be represented through its subdifferential
More informationASYMPTOTIC BEHAVIOUR OF NONLINEAR ELLIPTIC SYSTEMS ON VARYING DOMAINS
ASYMPTOTIC BEHAVIOUR OF NONLINEAR ELLIPTIC SYSTEMS ON VARYING DOMAINS Juan CASADO DIAZ ( 1 ) Adriana GARRONI ( 2 ) Abstract We consider a monotone operator of the form Au = div(a(x, Du)), with R N and
More informationCritical Groups in Saddle Point Theorems without a Finite Dimensional Closed Loop
Math. Nachr. 43 00), 56 64 Critical Groups in Saddle Point Theorems without a Finite Dimensional Closed Loop By Kanishka Perera ) of Florida and Martin Schechter of Irvine Received November 0, 000; accepted
More informationEXISTENCE OF SOLUTIONS TO NONHOMOGENEOUS DIRICHLET PROBLEMS WITH DEPENDENCE ON THE GRADIENT
Electronic Journal of Differential Equations, Vol. 2018 (2018), No. 101, pp. 1 18. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS TO NONHOMOGENEOUS
More informationEXISTENCE AND REGULARITY RESULTS FOR SOME NONLINEAR PARABOLIC EQUATIONS
EXISTECE AD REGULARITY RESULTS FOR SOME OLIEAR PARABOLIC EUATIOS Lucio BOCCARDO 1 Andrea DALL AGLIO 2 Thierry GALLOUËT3 Luigi ORSIA 1 Abstract We prove summability results for the solutions of nonlinear
More informationContents: 1. Minimization. 2. The theorem of Lions-Stampacchia for variational inequalities. 3. Γ -Convergence. 4. Duality mapping.
Minimization Contents: 1. Minimization. 2. The theorem of Lions-Stampacchia for variational inequalities. 3. Γ -Convergence. 4. Duality mapping. 1 Minimization A Topological Result. Let S be a topological
More informationExistence and multiple solutions for a second-order difference boundary value problem via critical point theory
J. Math. Anal. Appl. 36 (7) 511 5 www.elsevier.com/locate/jmaa Existence and multiple solutions for a second-order difference boundary value problem via critical point theory Haihua Liang a,b,, Peixuan
More informationOn pseudomonotone variational inequalities
An. Şt. Univ. Ovidius Constanţa Vol. 14(1), 2006, 83 90 On pseudomonotone variational inequalities Silvia Fulina Abstract Abstract. There are mainly two definitions of pseudomonotone mappings. First, introduced
More informationHistory-dependent hemivariational inequalities with applications to Contact Mechanics
Annals of the University of Bucharest (mathematical series) 4 (LXII) (213), 193 212 History-dependent hemivariational inequalities with applications to Contact Mechanics Stanislaw Migórski, Anna Ochal
More informationON WEAK SOLUTION OF A HYPERBOLIC DIFFERENTIAL INCLUSION WITH NONMONOTONE DISCONTINUOUS NONLINEAR TERM
Internat. J. Math. & Math. Sci. Vol. 22, No. 3 (999 587 595 S 6-72 9922587-2 Electronic Publishing House ON WEAK SOLUTION OF A HYPERBOLIC DIFFERENTIAL INCLUSION WITH NONMONOTONE DISCONTINUOUS NONLINEAR
More informationApplications of the periodic unfolding method to multi-scale problems
Applications of the periodic unfolding method to multi-scale problems Doina Cioranescu Université Paris VI Santiago de Compostela December 14, 2009 Periodic unfolding method and multi-scale problems 1/56
More informationMULTI-VALUED BOUNDARY VALUE PROBLEMS INVOLVING LERAY-LIONS OPERATORS AND DISCONTINUOUS NONLINEARITIES
MULTI-VALUED BOUNDARY VALUE PROBLEMS INVOLVING LERAY-LIONS OPERATORS,... 1 RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO Serie II, Tomo L (21), pp.??? MULTI-VALUED BOUNDARY VALUE PROBLEMS INVOLVING LERAY-LIONS
More informationCRITICAL POINT METHODS IN DEGENERATE ANISOTROPIC PROBLEMS WITH VARIABLE EXPONENT. We are interested in discussing the problem:
STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume LV, Number 4, December 2010 CRITICAL POINT METHODS IN DEGENERATE ANISOTROPIC PROBLEMS WITH VARIABLE EXPONENT MARIA-MAGDALENA BOUREANU Abstract. We work on
More informationHAIYUN ZHOU, RAVI P. AGARWAL, YEOL JE CHO, AND YONG SOO KIM
Georgian Mathematical Journal Volume 9 (2002), Number 3, 591 600 NONEXPANSIVE MAPPINGS AND ITERATIVE METHODS IN UNIFORMLY CONVEX BANACH SPACES HAIYUN ZHOU, RAVI P. AGARWAL, YEOL JE CHO, AND YONG SOO KIM
More informationMULTIPLE SOLUTIONS FOR AN INDEFINITE KIRCHHOFF-TYPE EQUATION WITH SIGN-CHANGING POTENTIAL
Electronic Journal of Differential Equations, Vol. 2015 (2015), o. 274, pp. 1 9. ISS: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu MULTIPLE SOLUTIOS
More informationLERAY LIONS DEGENERATED PROBLEM WITH GENERAL GROWTH CONDITION
2005-Oujda International Conference on Nonlinear Analysis. Electronic Journal of Differential Equations, Conference 14, 2006, pp. 73 81. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
More informationEXISTENCE OF WEAK SOLUTIONS FOR A NONUNIFORMLY ELLIPTIC NONLINEAR SYSTEM IN R N. 1. Introduction We study the nonuniformly elliptic, nonlinear system
Electronic Journal of Differential Equations, Vol. 20082008), No. 119, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu login: ftp) EXISTENCE
More informationON NONHOMOGENEOUS BIHARMONIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENT
PORTUGALIAE MATHEMATICA Vol. 56 Fasc. 3 1999 ON NONHOMOGENEOUS BIHARMONIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENT M. Guedda Abstract: In this paper we consider the problem u = λ u u + f in, u = u
More informationON GAP FUNCTIONS OF VARIATIONAL INEQUALITY IN A BANACH SPACE. Sangho Kum and Gue Myung Lee. 1. Introduction
J. Korean Math. Soc. 38 (2001), No. 3, pp. 683 695 ON GAP FUNCTIONS OF VARIATIONAL INEQUALITY IN A BANACH SPACE Sangho Kum and Gue Myung Lee Abstract. In this paper we are concerned with theoretical properties
More informationA NOTE ON THE EXISTENCE OF TWO NONTRIVIAL SOLUTIONS OF A RESONANCE PROBLEM
PORTUGALIAE MATHEMATICA Vol. 51 Fasc. 4 1994 A NOTE ON THE EXISTENCE OF TWO NONTRIVIAL SOLUTIONS OF A RESONANCE PROBLEM To Fu Ma* Abstract: We study the existence of two nontrivial solutions for an elliptic
More informationON THE RANGE OF THE SUM OF MONOTONE OPERATORS IN GENERAL BANACH SPACES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 11, November 1996 ON THE RANGE OF THE SUM OF MONOTONE OPERATORS IN GENERAL BANACH SPACES HASSAN RIAHI (Communicated by Palle E. T. Jorgensen)
More informationExistence and Multiplicity of Solutions for a Class of Semilinear Elliptic Equations 1
Journal of Mathematical Analysis and Applications 257, 321 331 (2001) doi:10.1006/jmaa.2000.7347, available online at http://www.idealibrary.com on Existence and Multiplicity of Solutions for a Class of
More informationSobolev Spaces. Chapter Hölder spaces
Chapter 2 Sobolev Spaces Sobolev spaces turn out often to be the proper setting in which to apply ideas of functional analysis to get information concerning partial differential equations. Here, we collect
More informationEXISTENCE OF SOLUTIONS TO ASYMPTOTICALLY PERIODIC SCHRÖDINGER EQUATIONS
Electronic Journal of Differential Equations, Vol. 017 (017), No. 15, pp. 1 7. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS TO ASYMPTOTICALLY PERIODIC
More informationFIRST CURVE OF FUČIK SPECTRUM FOR THE p-fractional LAPLACIAN OPERATOR WITH NONLOCAL NORMAL BOUNDARY CONDITIONS
Electronic Journal of Differential Equations, Vol. 208 (208), No. 74, pp. 2. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu FIRST CURVE OF FUČIK SPECTRUM FOR THE p-fractional
More informationASYMMETRIC SUPERLINEAR PROBLEMS UNDER STRONG RESONANCE CONDITIONS
Electronic Journal of Differential Equations, Vol. 07 (07), No. 49, pp. 7. ISSN: 07-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ASYMMETRIC SUPERLINEAR PROBLEMS UNDER STRONG RESONANCE
More informationOPTIMALITY CONDITIONS AND ERROR ANALYSIS OF SEMILINEAR ELLIPTIC CONTROL PROBLEMS WITH L 1 COST FUNCTIONAL
OPTIMALITY CONDITIONS AND ERROR ANALYSIS OF SEMILINEAR ELLIPTIC CONTROL PROBLEMS WITH L 1 COST FUNCTIONAL EDUARDO CASAS, ROLAND HERZOG, AND GERD WACHSMUTH Abstract. Semilinear elliptic optimal control
More informationPositive eigenfunctions for the p-laplace operator revisited
Positive eigenfunctions for the p-laplace operator revisited B. Kawohl & P. Lindqvist Sept. 2006 Abstract: We give a short proof that positive eigenfunctions for the p-laplacian are necessarily associated
More informationMULTIPLE SOLUTIONS FOR A KIRCHHOFF EQUATION WITH NONLINEARITY HAVING ARBITRARY GROWTH
MULTIPLE SOLUTIONS FOR A KIRCHHOFF EQUATION WITH NONLINEARITY HAVING ARBITRARY GROWTH MARCELO F. FURTADO AND HENRIQUE R. ZANATA Abstract. We prove the existence of infinitely many solutions for the Kirchhoff
More informationLaplace s Equation. Chapter Mean Value Formulas
Chapter 1 Laplace s Equation Let be an open set in R n. A function u C 2 () is called harmonic in if it satisfies Laplace s equation n (1.1) u := D ii u = 0 in. i=1 A function u C 2 () is called subharmonic
More informationJournal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied Mathematics http://jipam.vu.edu.au/ Volume 4, Issue 4, Article 67, 2003 ON GENERALIZED MONOTONE MULTIFUNCTIONS WITH APPLICATIONS TO OPTIMALITY CONDITIONS IN
More informationCONDITIONS FOR HAVING A DIFFEOMORPHISM BETWEEN TWO BANACH SPACES
Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 99, pp. 1 6. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu CONDITIONS FOR
More informationA CHARACTERIZATION OF STRICT LOCAL MINIMIZERS OF ORDER ONE FOR STATIC MINMAX PROBLEMS IN THE PARAMETRIC CONSTRAINT CASE
Journal of Applied Analysis Vol. 6, No. 1 (2000), pp. 139 148 A CHARACTERIZATION OF STRICT LOCAL MINIMIZERS OF ORDER ONE FOR STATIC MINMAX PROBLEMS IN THE PARAMETRIC CONSTRAINT CASE A. W. A. TAHA Received
More informationOPTIMAL CONTROL AND STRANGE TERM FOR A STOKES PROBLEM IN PERFORATED DOMAINS
PORTUGALIAE MATHEMATICA Vol. 59 Fasc. 2 2002 Nova Série OPTIMAL CONTROL AND STRANGE TERM FOR A STOKES PROBLEM IN PERFORATED DOMAINS J. Saint Jean Paulin and H. Zoubairi Abstract: We study a problem of
More informationSTOKES PROBLEM WITH SEVERAL TYPES OF BOUNDARY CONDITIONS IN AN EXTERIOR DOMAIN
Electronic Journal of Differential Equations, Vol. 2013 2013, No. 196, pp. 1 28. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu STOKES PROBLEM
More informationNecessary and Sufficient Conditions for the Existence of a Global Maximum for Convex Functions in Reflexive Banach Spaces
Laboratoire d Arithmétique, Calcul formel et d Optimisation UMR CNRS 6090 Necessary and Sufficient Conditions for the Existence of a Global Maximum for Convex Functions in Reflexive Banach Spaces Emil
More informationPERTURBATION THEORY FOR NONLINEAR DIRICHLET PROBLEMS
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, 207 222 PERTURBATION THEORY FOR NONLINEAR DIRICHLET PROBLEMS Fumi-Yuki Maeda and Takayori Ono Hiroshima Institute of Technology, Miyake,
More informationTechnische Universität Dresden
Technische Universität Dresden Als Manuskript gedruckt Herausgeber: Der Rektor Existence of a sequence of eigensolutions for the 1-Laplace operator Z. Milbers and F. Schuricht Institut für Analysis MATH-AN-04-2008
More informationHESSIAN VALUATIONS ANDREA COLESANTI, MONIKA LUDWIG & FABIAN MUSSNIG
HESSIAN VALUATIONS ANDREA COLESANTI, MONIKA LUDWIG & FABIAN MUSSNIG ABSTRACT. A new class of continuous valuations on the space of convex functions on R n is introduced. On smooth convex functions, they
More informationEXISTENCE OF STRONG SOLUTIONS OF FULLY NONLINEAR ELLIPTIC EQUATIONS
EXISTENCE OF STRONG SOLUTIONS OF FULLY NONLINEAR ELLIPTIC EQUATIONS Adriana Buică Department of Applied Mathematics Babeş-Bolyai University of Cluj-Napoca, 1 Kogalniceanu str., RO-3400 Romania abuica@math.ubbcluj.ro
More informationMULTIPLE SOLUTIONS FOR CRITICAL ELLIPTIC PROBLEMS WITH FRACTIONAL LAPLACIAN
Electronic Journal of Differential Equations, Vol. 016 (016), No. 97, pp. 1 11. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu MULTIPLE SOLUTIONS
More informationGlobal Maximum of a Convex Function: Necessary and Sufficient Conditions
Journal of Convex Analysis Volume 13 2006), No. 3+4, 687 694 Global Maximum of a Convex Function: Necessary and Sufficient Conditions Emil Ernst Laboratoire de Modélisation en Mécaniue et Thermodynamiue,
More informationON A HYBRID PROXIMAL POINT ALGORITHM IN BANACH SPACES
U.P.B. Sci. Bull., Series A, Vol. 80, Iss. 3, 2018 ISSN 1223-7027 ON A HYBRID PROXIMAL POINT ALGORITHM IN BANACH SPACES Vahid Dadashi 1 In this paper, we introduce a hybrid projection algorithm for a countable
More informationCHARACTERIZATION OF (QUASI)CONVEX SET-VALUED MAPS
CHARACTERIZATION OF (QUASI)CONVEX SET-VALUED MAPS Abstract. The aim of this paper is to characterize in terms of classical (quasi)convexity of extended real-valued functions the set-valued maps which are
More informationSUPER-QUADRATIC CONDITIONS FOR PERIODIC ELLIPTIC SYSTEM ON R N
Electronic Journal of Differential Equations, Vol. 015 015), No. 17, pp. 1 11. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SUPER-QUADRATIC CONDITIONS
More informationTiziana Cardinali Francesco Portigiani Paola Rubbioni. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 32, 2008, 247 259 LOCAL MILD SOLUTIONS AND IMPULSIVE MILD SOLUTIONS FOR SEMILINEAR CAUCHY PROBLEMS INVOLVING LOWER
More informationInternal Stabilizability of Some Diffusive Models
Journal of Mathematical Analysis and Applications 265, 91 12 (22) doi:1.16/jmaa.21.7694, available online at http://www.idealibrary.com on Internal Stabilizability of Some Diffusive Models Bedr Eddine
More informationEXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS WITH UNBOUNDED POTENTIAL. 1. Introduction In this article, we consider the Kirchhoff type problem
Electronic Journal of Differential Equations, Vol. 207 (207), No. 84, pp. 2. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS
More informationSOME PROPERTIES ON THE CLOSED SUBSETS IN BANACH SPACES
ARCHIVUM MATHEMATICUM (BRNO) Tomus 42 (2006), 167 174 SOME PROPERTIES ON THE CLOSED SUBSETS IN BANACH SPACES ABDELHAKIM MAADEN AND ABDELKADER STOUTI Abstract. It is shown that under natural assumptions,
More informationOn the Schrödinger Equation in R N under the Effect of a General Nonlinear Term
On the Schrödinger Equation in under the Effect of a General Nonlinear Term A. AZZOLLINI & A. POMPONIO ABSTRACT. In this paper we prove the existence of a positive solution to the equation u + V(x)u =
More informationA CONVEX-CONCAVE ELLIPTIC PROBLEM WITH A PARAMETER ON THE BOUNDARY CONDITION
A CONVEX-CONCAVE ELLIPTIC PROBLEM WITH A PARAMETER ON THE BOUNDARY CONDITION JORGE GARCÍA-MELIÁN, JULIO D. ROSSI AND JOSÉ C. SABINA DE LIS Abstract. In this paper we study existence and multiplicity of
More informationREGULARIZED PENALTY METHOD FOR NON-STATIONARY SET VALUED EQUILIBRIUM PROBLEMS IN BANACH SPACES. Salahuddin
Korean J. Math. 25 (2017), No. 2, pp. 147 162 https://doi.org/10.11568/kjm.2017.25.2.147 REGULARIZED PENALTY METHOD FOR NON-STATIONARY SET VALUED EQUILIBRIUM PROBLEMS IN BANACH SPACES Salahuddin Abstract.
More informationPARTIAL REGULARITY OF BRENIER SOLUTIONS OF THE MONGE-AMPÈRE EQUATION
PARTIAL REGULARITY OF BRENIER SOLUTIONS OF THE MONGE-AMPÈRE EQUATION ALESSIO FIGALLI AND YOUNG-HEON KIM Abstract. Given Ω, Λ R n two bounded open sets, and f and g two probability densities concentrated
More informationSPECTRAL PROPERTIES OF THE LAPLACIAN ON BOUNDED DOMAINS
SPECTRAL PROPERTIES OF THE LAPLACIAN ON BOUNDED DOMAINS TSOGTGEREL GANTUMUR Abstract. After establishing discrete spectra for a large class of elliptic operators, we present some fundamental spectral properties
More informationOn the distributional divergence of vector fields vanishing at infinity
Proceedings of the Royal Society of Edinburgh, 141A, 65 76, 2011 On the distributional divergence of vector fields vanishing at infinity Thierry De Pauw Institut de Recherches en Mathématiques et Physique,
More informationOn a weighted total variation minimization problem
On a weighted total variation minimization problem Guillaume Carlier CEREMADE Université Paris Dauphine carlier@ceremade.dauphine.fr Myriam Comte Laboratoire Jacques-Louis Lions, Université Pierre et Marie
More informationResearch Article Hybrid Algorithm of Fixed Point for Weak Relatively Nonexpansive Multivalued Mappings and Applications
Abstract and Applied Analysis Volume 2012, Article ID 479438, 13 pages doi:10.1155/2012/479438 Research Article Hybrid Algorithm of Fixed Point for Weak Relatively Nonexpansive Multivalued Mappings and
More informationA nodal solution of the scalar field equation at the second minimax level
Bull. London Math. Soc. 46 (2014) 1218 1225 C 2014 London Mathematical Society doi:10.1112/blms/bdu075 A nodal solution of the scalar field equation at the second minimax level Kanishka Perera and Cyril
More informationOn nonexpansive and accretive operators in Banach spaces
Available online at www.isr-publications.com/jnsa J. Nonlinear Sci. Appl., 10 (2017), 3437 3446 Research Article Journal Homepage: www.tjnsa.com - www.isr-publications.com/jnsa On nonexpansive and accretive
More informationWEAK CONVERGENCE OF RESOLVENTS OF MAXIMAL MONOTONE OPERATORS AND MOSCO CONVERGENCE
Fixed Point Theory, Volume 6, No. 1, 2005, 59-69 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.htm WEAK CONVERGENCE OF RESOLVENTS OF MAXIMAL MONOTONE OPERATORS AND MOSCO CONVERGENCE YASUNORI KIMURA Department
More informationRemarks on Multiple Nontrivial Solutions for Quasi-Linear Resonant Problems
Journal of Mathematical Analysis and Applications 258, 209 222 200) doi:0.006/jmaa.2000.7374, available online at http://www.idealibrary.com on Remarks on Multiple Nontrivial Solutions for Quasi-Linear
More information