On a Positive Solution for (p, q)-laplace Equation with Nonlinear Boundary Conditions and Indefinite Weights

Transcription

1 Bol. Soc. Paran. Mat. (3s.) v (0000):????. c SPM ISSN on line ISSN in ress SPM: doi: /bsm.v38i On a Positive Solution for (, )-Lalace Euation with Nonlinear Boundary Conditions and Indefinite Weights Abdellah Zerouali, Belhadj Karim, Omar Chakrone, Abdelmajid Boukhsas abstract: In the resent aer, we study the existence and non-existence results of a ositive solution for the Steklov eigenvalue roblem driven by nonhomogeneous oerator (, )-Lalacian with indefinite weights. We also rove, under aroriate conditions, that the results are comletely different from those for the usual Steklov eigenvalue roblem involving the -Lalacian with indefinite weight. Precisely, we show that there exists an interval of rincial eigenvalues for our Steklov eigenvalue roblem. Key Words:(,)-Lalacian, Nonlinear boundary conditions, Indefinite weight, Mountain ass theorem, Global minimizer. Contents 1 Introduction 1 2 Rayleigh uotient and non-existence results Rayleigh uotient for the roblem (P λ,µ ) Non-existence results Existence result with non-resonant case 7 (P λ,µ ) 1. Introduction Consider the (, )-Lalacian Steklov eigenvalue roblem { div[a (µ),( u)] = A (µ),(u) in, A (µ),( u),ν = λ[m (x) u 2 u+µm (x) u 2 u] on where is a bounded domain in R N (N 2) with smooth boundary, ν is the outward unit normal vector on,.,. is the scalar roduct of R N, λ R, µ 0 and 1 < < <. Let r =, and let N 1 r 1 < s r < if r < N and s r 1 if r N. A (µ),(s) = s 2 s+µ s 2 s and the function weight m r M r may be unbounded and change sign, where M r := {m r L sr ();m + r 0}. Theroblem(P λ,µ )comes,forexamle, fromageneralreactiondiffusionsystem u t = div(d(u) u)+c(x,u), (1.1) 2010 Mathematics Subject Classification: 35J20, 35J62, 35J70, 35P05, 35P3. Submitted Aril 11, Published December 26, Tyeset by B S P M style. c Soc. Paran. de Mat.

2 2 A. Zerouali, B. Karim, O. Chakrone, A. Boukhsas where D(u) = ( u 2 +µ u 2 ). This system has a wide range of alications in hysics and related sciences like chemical reaction design [2], biohysics [5] and lasma hysics [14]. In such alications, the function u describes a concentration, the first term on the right-hand side of (1.1) corresonds to the diffusion with a diffusion coefficient D(u); whereas the second one is the reaction and relates to source and loss rocesses. Tyically, in chemical and biological alications, the reaction term c(x; u) has a olynomial form with resect to the concentration. The nonhomogeneous oerator (, )-Lalacian have been the toic of many studies(see [6,7,13,17]). However, there are few results one the eigenvalue roblems for the (, )-Lalacian, we cite [3,9,10,15]. The classical eigenvalue roblem for the (, )-Lalacian { u µ u = λ[m (x) u 2 u+µm (x) u 2 u] in, (1.2) u = 0 on, where r u = div ( u r 2 u) indicate the r-lalacian, has attracted considerable attention. In [12], the authors study the roblem (1.2) for domains with boundary C 2 and bounded weights. They roved, in the case where µ > 0, the existence of an interval of eigenvalues and the existence of ositive solutions in nonresonant cases. A non-existence result is also given. In [18], A. Zerouali and B. Karim are roved the same results by assuming the singularities on the domain and the weights. Our urose in this article is to extend the results of the classical eigenvalue roblem involving the (, )-Lalacian (see for examle [11,12]) and generalize some results knouwn in the classical -Lalacian Steklov roblems (see [4]). We will write u r := ( u r dx ) 1/r for the L r () norm and W 1,r () will denote the usual Sobolev sace with usual norm u W 1,r () := ( u r r+ u r r) 1/r. We recall that a value λ R is an eigenvalue of roblem (P λ,µ ) if and only if there exists u W 1, ()\{0} such that A (µ), ( u) ϕdx+ A (µ), (u)ϕdx = λ [ ] (m (x) u 2 +µm (x) u 2 )uϕdσ (1.3) for all ϕ W 1, (), where dσ is the N 1 dimensional Hausdorff measure and u is then called an eigenfunction of λ. Lettingµ 0 +, ourroblem(p λ,µ )turnsintothe ( 1)-homogeneousroblem known as the usual weighted eigenvalue Steklov roblem for the -Lalacian with indefinite weight m : { (P λ,m ) u 2 u ν u = u 2 u in, = λm (x) u 2 u on Moreover, after multilying our euation (P λ,µ ) by 1/µ and then letting µ +, we obtain the ( 1)-homogeneous euations in: { (P λ,m ) u = u 2 u in, = λm (x) u 2 u on u 2 u ν

3 On a Positive Solution for (,)-Lalace Euation... 3 Nonlinear Steklov eigenvalue roblem (P λ,mr ), where r =, and with indefinite weight m r M r have been studied by several authors, for examle (see [4]). These works roved that there exists a first eigenvalue λ 1 (r,m r ) > 0, where { 1 λ 1 (r,m r ) := inf r u r W 1,r () ;u W1,r () and 1 } m r (x) u r dσ = 1, (1.4) r which is simle in the sense that two eigenfunctions corresonding to it are roortional. Moreover, the corresonding first eigenfunction φ 1 (r,m r ) can be assumed to be ositive. It was also shown in [4] that λ 1 (r,m r ) is isolated and monotone. This aer is divided into three sections, organized as follows. In Section 2, we study Rayleigh uotient for our roblem (P λ,µ ). In contrast to homogeneous case, we rove that if λ 1 (,m ) λ 1 (,m ) or φ 1 (,m ) kφ 1 (,m ) for every k > 0, then the infimum in Rayleigh uotient is not attained. We also show non-existence results for ositive solutions of the eigenvalue roblem (P λ,µ ) formulated as Theorem 2.5. Our existence results for ositive solutions of the eigenvalue roblem (P λ,µ ) are resented in Section 3. We study the non-resonant case (Theorem 3.1) which rove that when µ > 0 there exists an interval of ositive eigenvalues for the roblem (P λ,µ ). 2. Rayleigh uotient and non-existence results This section concerns the Rayleigh uotient and non-existence results for our eigenvalue Steklov roblem (P λ,µ ). It is insired from [11] and [15]. Remark 2.1. We start by ointing out to find a solution for the roblem (P λ,µ ) is euivalent to seek a solution in the case µ = 1, that is to solve the roblem (P λ,1 ). Indeed, if u is a solution of (P λ,1 ), then multilying euation (P λ,1 ) by s 1 for s > 0 we deduce that v = su is a solution for roblem (P λ,µ=s ). Conversely, let u be a solution of roblem (P λ,µ ). Then it follows that v = µ 1/ u is a solution of (P λ,1 ) Rayleigh uotient for the roblem (P λ,µ ) We introduce now the functionals A and B on W 1, () by for all u W 1, (). B(u) := 1 A(u) := 1 u W 1, () + 1 u W 1, () (2.1) m (x) u dσ + 1 m (x) u dσ (2.2) Proosition 2.2. (i) The functional A is well defined and seuently weakly lower semi-continuous. (ii) If m M and m M, then the functional B is also well defined and weakly continuous.

4 4 A. Zerouali, B. Karim, O. Chakrone, A. Boukhsas Proof. (i) The functional A is well defined. Indeed, since bounded and <, we have W 1, () W 1, (). Then for all u W 1, 1 (), u W 1, () < and 1 u W 1, () <. It follows that A(u) <. It is clear that A is seuently weakly lower semi-continuous. (ii) The functional B is also well defined. Indeed, for u W 1, (), by Hölder s ineuality, for r =, and s r = s r /(s r 1), we obtain 1 r (x) u r m r dσ 1 ( ) 1/sr ( ) 1 1/sr m r (x) sr dσ u rs r dσ r = 1 r m r sr, u r rs r, <, since m r M r and the trace embedding W 1,r () L rs r () is comact. Let us now show that B is weakly continuous. If u n u weakly in W 1,r (), u to a subseuence, u n u strongly in L rs r () and un r u r strongly in L s r () with r =,. Hence by Hölder s ineuality, we have B(u n ) B(u) 1 m (x)( u n u )dσ + 1 m (x)( u n u )dσ 1 m s, u n u s, + 1 m s, u n u s, 0. Thus the functional B is weakly continuous. Define now the Rayleigh uotient { } A(u) λ = inf B(u) ;u W1, (),B(u) > 0. (2.3) Proosition 2.3. One assumes that m M and m M. If λ 1 (,m ) λ 1 (,m ) or φ 1 (,m ) kφ 1 (,m ), for every k > 0. Then the infimum in (2.3) is not attained. For the roof of Proosition 2.3, we will need to use the following lemma. Lemma 2.4. The infimum in (2.3) verifies λ = min{λ 1 (,m ),λ 1 (,m )} Proof. For sufficiently large k > 0, using (2.1) and (2.2), we have ( B(kφ 1 (,m )) = k k + 1 ) m (x)φ 1 (,m )dσ > 0.

5 On a Positive Solution for (,)-Lalace Euation... 5 and A(kφ 1 (,m )) = k (λ 1 (,m )+ 1 k φ 1 (,m ) W 1, () ). By (2.3), we find λ A(kφ 1(,m )) B(kφ 1 (,m )) = λ 1(,m )+ 1 k φ 1 (,m ) W 1, () 1+ 1 k m (x)φ 1 (,m )dσ λ 1 (,m ) as k +, because <. It follows that λ λ 1 (,m ). On the other hand, we also have λ A(kφ 1 (,m )) B(kφ 1 (,m )) = λ 1(,m )+ 1 k φ 1 (,m ) W 1, () 1+ 1 k m (x)φ 1 (,m )dσ λ 1 (,m ) as k 0 +, because <. Thus, we obtain λ λ 1 (,m ), which imlies that λ min{λ 1 (,m ),λ 1 (,m )} Conversely, suose by contradiction that λ < min{λ 1 (,m ),λ 1 (,m )}. Then, by (2.3), there exists u W 1, () such that B(u) > 0 and A(u) B(u) < min{λ 1(,m ),λ 1 (,m )}. We distinguish three cases. Case (i): Suose that m u dσ > 0 and m u dσ 0. There hold B(u) m u dσ and A(u) u W 1, (). Using the definition of λ 1(,m ), we arrive at the contradiction. min{λ 1 (,m ),λ 1 (,m )} > A(u) B(u) u W 1, () m u dσ λ 1(,m ). (2.4) Case (ii): Suose that m u dσ 0 and m u dσ > 0. Using the definition of λ 1 (,m ), we also arrive at contradiction min{λ 1 (,m ),λ 1 (,m )} > A(u) B(u) u W 1, () m u dσ λ 1(,m ). (2.5) Case (iii): Suose now that m u dσ > 0 and m u dσ > 0. It follows from the definition of λ 1 (r,m r ), where r =, that u r W 1,r () λ 1(r,m r ) m r u r dσ.

6 6 A. Zerouali, B. Karim, O. Chakrone, A. Boukhsas Hence we get A(u) λ 1(,m ) m u dσ + λ 1(,m ) min{λ 1 (,m ),λ 1 (,m )}B(u). Against the assumtion in our reasoning by contradiction. m u dσ (2.6) Proof of Proosition 2.3. By contradiction, we suose that: there exists u W 1, () such that B(u) > 0 and A(u) B(u) = λ. Using Lemma 2.4, we give A(u) B(u) = λ = min{λ 1 (,m ),λ 1 (,m )}. (2.7) We argue by considering the three cases in the roof of Lemma 2.4. Case (i): By (2.4), (2.7) and m u dσ 0, we have λ = A(u) B(u) u W 1, () + u W 1, () m u dσ We deduce that u W 1, () = λ 1(,m ) u W 1, () m u dσ λ 1(,m ) λ. m u dσ and u W 1, () = 0. Thus u = 0. This contradicts the fact that u 0. Case (ii): similarly, By (2.5), (2.7) and m u dσ 0, we get u W 1, () = λ 1(,m ) m u dσ and u W 1, () = 0. Thus u = 0. Which contradicts u 0. Case (iii): In this case, using (2.6) and (2.7), we find A(u) = λ B(u) = λ 1(,m ) m u dσ + λ 1(,m ) It follows [λ 1 (,m ) λ ] m u dσ. m u dσ +[λ 1 (,m ) λ ] m u dσ = 0. Since m u dσ > 0, m u dσ > 0 and λ = min{λ 1 (,m ),λ 1 (,m )}, we have λ = λ 1 (,m ) = λ 1 (,m ).

7 On a Positive Solution for (,)-Lalace Euation... 7 We deduce that u W 1, () m u dσ = λ 1(,m ) = λ 1 (,m ) = u W 1, () m u dσ. Hence, the simlicity of eigenvalue λ 1 (r,m r ) (for r =,), guarantees that u = tφ 1 (,m ) = sφ 1 (,m ) for some t 0 and s 0. The hyotheses of roosition is thus contradicted Non-existence results The following theorem is the main result of this section. Theorem 2.5. One assumes that m M and m M. (a) If 0 < λ < λ, then the roblem (P λ,1 ) has no non-trivial solutions. (b) Moreover, if one of the following conditions holds (i) λ 1 (,m ) λ 1 (,m ); (ii) φ 1 (,m ) kφ 1 (,m ), for every k > 0, then the roblem (P λ,1 ), with λ = λ has no non-trivial solutions. Remark 2.6. It is easy to see that if λ 1 (,m ) = λ 1 (,m ) and φ 1 (,m ) = kφ 1 (,m ), for some k > 0, then φ 1 (,m ) and φ 1 (,m ) are ositive solutions of roblem (P λ,1 ), with λ = λ 1 (,m ) = λ 1 (,m ). Proof of Theorem 2.5. Assume by contradiction that there exists a non-trivial solution u of roblem (P λ,1 ). Then, for every s > 0, we have that v = su is a non-trivial solution of roblem (P λ,s ) (see Remark 2.1). Choose s = / and then act with su as test function on the roblem (P λ,s ). We arrive at 0 < A(su) = λb(su). (2.8) From the estimate (2.8) and according to Lemma 2.4, we obtain λ = A(su) B(su) λ = min{λ 1 (,m ),λ 1 (,m )}. This contradiction yields the first assertion of the theorem. The second art of the Theorem 2.5 follows by Proosition Existence result with non-resonant case Thefollowingtheoremisourmainexistenceresultforroblem(P λ,1 )(or(p λ,µ )) in the non-resonant case. This result rove that there exists an interval of ositive eigenvalues for the roblem (P λ,1 ) (or (P λ,µ ), with µ > 0).

8 8 A. Zerouali, B. Karim, O. Chakrone, A. Boukhsas Theorem 3.1. One suoses that m M, m M and λ 1 (,m ) λ 1 (,m ). If min{λ 1 (,m ),λ 1 (,m )} < λ < max{λ 1 (,m ),λ 1 (,m )}, then the roblem (P λ,1 ) has at least one ositive solution. Remark 3.2. The roof of Theorem 3.1 reduces to rovide a non-trivial critical oint of the functional I λ,m,m defined for all u W 1, () by I λ,m,m (u) := A(u) λb(u + ), whereu + = max{u,0}anda,b arethefunctionalsdefinedby(2.1)and(2.2). This non-trivial critical oint u of I λ,m,m is a non-negative solution of the roblem (P λ,1 ). Indeed, inserting u = max{ u,0} as test function leads to 0 = I λ,m,m (u), u = u W 1, () + u W 1, (), thus u = 0. We can check that u C 1,α () for some α (0,1) (see [1]). Then the maximum rincile of Vasuez [16] can be alied to ensure ositiveness of u. The argument will be searately develoed in two cases: (a) λ 1 (,m ) < λ < λ 1 (,m ). (b)λ 1 (,m ) < λ < λ 1 (,m ). In case (a), we aly the minimum rincile and in case (b), we use the mountain ass theorem. Proof of case (a). By Proosition 2.2, A is seuently weakly lower semicontinuousand B is weaklycontinuous. It followsthat I λ,m,m is seuently weakly lower semi-continuous. Moreover I λ,m,m is bounded from below. Indeed for all u W 1, (), we have I λ,m,m (u) λb(u + ) >. (3.1) It is remains to show that I λ,m,m is coercive in W 1, (). Fix ε > 0 such that (1 ε)λ 1 (,m ) > λ (3.2) which is ossible due to the assumtion in case (a). For every u W 1, () with m (u + ) dσ 0, through Holder s ineuality we obtain I λ,m,m (u) 1 u W 1, () + 1 u W 1, () λ m (u + ) dσ 1 u W 1, () + 1 u W 1, () λ m L s () (u + ) L s () (3.3) 1 u W 1, () λc m L s () u W 1, () for u W 1, () with m (u + ) dσ > 0, by (1.4) we have u + W 1, () λ 1(,m ) m (u + ) dσ.

9 On a Positive Solution for (,)-Lalace Euation... 9 Then, taking into account (3.2), we derive I λ,m,m (u) ε u W 1, () + (1 ε)λ 1(,m ) λ λ m L s () u + L s () m (u + ) dσ (3.4) ε u W 1, () Cλ m L s () u W 1, () Since <, it follows from (3.3) and (3.4) that I λ,m,m is coercive and bounded from below. Conseuently, by a standard result(see, e.g., [[8],Theorem 1.1]), there exists a global minimizer u 0 of I λ. In order to have u 0 0 it suffices to show that I λ (u 0 ) = min W 1, ()I λ < 0. Let ψ 1 be the eigenfunction corresonding to λ 1 (,m ) that satisfies m ψ 1 dσ = 1. Because λ > λ 1(,m ), for sufficiently small t > 0 it holds( ) I λ,m,m (tψ 1 ) = t t ψ 1 W 1, () λt m ψ λ1(,m) λ 1dσ + < 0, which comletes the roof. Proof of case (b). We organize the roof of this case in several lemmas. In the seuel, we design by o(1) a uantity tending to 0 as n. Lemma 3.3. Let m M, m M. If 0 λ λ 1 (,m ), Then the functional I λ,m,m satisfies the Palais-Smale condition. Proof. Let (u n ) W 1, () be a seuence such that I λ,m,m (u n ) c for c R and I λ,m,m (u n ) 0 in (W 1, ()) as n. Let us first show that the seuence (u n ) is bounded in W 1, (). It is sufficient only to rovethe boundedness of u n s, because using the Hölder s ineuality and the continuous embedding W 1, () L s (), we have u n W 1, () c+c λ m s u n s +c λ m s u n W 1, () (3.5) where c and c are the ositive constants. Suose by contradiction that u n s + and let v n := un u n. We claim s that the seuence v n bounded in W 1, (). Indeed, dividing (3.5) by u n s we have v n W 1, () c u n +C 1 +C 2 v n W 1, (), (3.6) s where the ositive contants C 1 and C 2 are defined by C 1 = c λ m s and C 2 = c λ m s. Since 1 < <, the ineuality (3.6) imlies the boundedness of v n in W 1, (). for a subseuence, v n v (weakly) in in W 1, (). By the comact embedding. W 1,r () L rs r (),(r =,) we have vn v strongly in L rs r () (r =,). First we observe that v 0 in. In fact, acting with u n as test function, we have o(1) u n s = I λ,m,m (u n ), u n = 1 u n W 1, () +1 u n W 1, () u n W 1, () (3.7)

10 10 A. Zerouali, B. Karim, O. Chakrone, A. Boukhsas theineuality(3.7)uarantinestheboundednessof v n W 1, () andso v n W 1, () = u n W 1, () u n s 0, thus v 0, holds, hence v 0 in. Now, by taking (v n v)/ u n 1 s as test function, we have o(1) = I λ,m,m (u n ), (v n v) u n 1 s = 1 v n 2 1 v n (v n v)dx+ u n v n 2 v n (v n v)dx s + 1 v n 2 1 v n (v n v)dx+ u n v n 2 v n (v n v)dx s λ m (v n) + 2 v n(v + n v)dσ λ u n m (v n + ) 2 v n + (v n v)dσ s = 1 v n 2 v n (v n v)dx+ 1 v n 2 v n (v n v)dx λ m (v n + ) 2 v n + (v n v)dσ +o(1) (3.8) because <, u n s +, v n is bounded in W 1, () and converge to v strongly in L s (). Thus by (3.8) and (S+ ) roerty of u + u 2 u in W 1, (), we deduce that v n v strongly in W 1, (). For any ϕ W 1, (), by taking as test function, we obtain ϕ u n 1 s o(1) = I λ,m ϕ,m (u n ), u n 1 s = 1 v n 2 1 v n ϕdx+ u n v n 2 v n ϕdx s + 1 v n 2 1 v n ϕdx+ u n v n 2 v n ϕdx s λ m (v n + ) 2 v n + ϕdσ λ u n m (v n + ) 2 v n + ϕdσ s (3.9) Passingto the limit in (3.9), we see that v is a non-negativeand non-trivialsolution of roblem (P λ,m ) (note v 0 and v W 1, () = 1). The eigenfunction v is C 1,α () for some α (0,1) (see [1]). According to maximum rincile of Vasuez, we have v > 0 in W 1, (). This imlies that λ = λ 1 (,m ) because any ositive eigenvalueotherthan λ 1 (,m ) has noositiveeigenfunction. Therefore, weobtain

11 On a Positive Solution for (,)-Lalace Euation a contradiction since we assumed λ λ 1 (,m ). Hence u n is bounded in W 1, (). Forasubseuence,u n u(weakly)inw 1, ()andu n u(strongly)inl s (). We claim now that u n u in W 1, (). As W 1, () is reflexive and uniformly convex, it suffices to rove that u n W 1, () u W 1, (). It is clear that o(1) = I λ,m,m (u n ),u n u = 1 u n 2 u n (u n u)dx+ 1 u n 2 u n (u n u)dx + 1 u n 2 u n (u n u)dx+ 1 u n 2 u n (u n u)dx+o(1). (3.10) Using Hölder s ineuality and for (r =,), we have u n r 2 u n (u n u)dx+ u n r 2 u n (u n u)dx = u n r dx+ u r dx u n r 2 u n udx u r 2 u u n dx = u n r dx+ u r dx u n r 2 u n udx u r 2 uu n dx + = 0 u n r dx+ u n r dx+ u r dx ( u r dx ( ( u n r 1 W 1,r () u r 1 W 1,r () u n r dx ) (r 1)/r ( ) (r 1)/r ( u n r dx )( u n W 1,r () u W 1,r () u r dx ) u r dx ) 1/r ) 1/r (3.11) Moreover, (3.10) and (3.11) imly that u n W 1, () u W 1, (). Thus u n u strongly in W 1, (). Lemma 3.4. Let m M, m M. If λ < λ 1 (,m ), then there exist δ > 0 and ρ > 0 such that I λ (u) δ whenever u s L = ρ. (3.12) () To rove the Lemma 3.4, we need the following lemma. Lemma 3.5. X(d) := { u W 1, (); u W 1, () d u L s () } (3.13) for d > 0. Then there exists C = C(d) > 0 such that u W 1, () C u s L () for all u X(d).

12 12 A. Zerouali, B. Karim, O. Chakrone, A. Boukhsas Proof. By way contradiction, we assume that 1 n N, u n X(d), n u n W 1, () > u n s L (3.14) () u Setv n = n u n ; hencev W 1, n boundedinw 1, (), thenthereexistsasubseuence () that we still denote v n such that v n v weakly in W 1, (). By the comact embedding W 1,r () L rs r ()(r =,) we have vn v strongly in L rs r (). By (3.14), we have v n s L () < 1 n, thus v n 0 in L s (). By uniueness of the limit we have v = 0, hence v n 0 in L s (). As un X(d) we have u 1 n d L s () u n = v n. W 1, L s () () Passing to the limit, we obtain a contradiction. Proof of Lemma 3.4. Let C r the constant from embedding W 1,r () L rs r (), where r =,. According to Lemma 3.5, there exists C(d) > 0 such that u W 1, () C(d) u L s () for all u X(d), (3.15) where d such that { } d > max 1,C λ m L s (),λ m L s (). (3.16) For any u X(d) satisfying m u +dσ 0 by (3.16) and (3.15) we have I λ (u) 1 d u W 1, () + 1 C u L s () + d u W 1, () λ m L s () u L s () 1 d u W 1, () + 1 C u L s () + d u W 1, () C λ m L s () u W 1, () 1 d u W 1, () + 1 u C + (d C λ m ()) Ls L s () (1 d)c (d) u + 1 L s u () C L s () u W 1, () (3.17) For any u / X(d) satisfying m u + dσ 0 thanks to (3.13) and (3.16) we find I λ (u) d λ m L s() u + 1 u L s () C L s () 1 u C L s () (3.18)

13 On a Positive Solution for (,)-Lalace Euation If u W 1, () fulfills m u + dσ > 0, we get u W 1, () u + W 1, () λ 1(,m ) m u +dσ (3.19) Our assumtion on λ enables us to fix (1 > ε > 0) with (1 ε)λ 1 (,m ) > λ. (3.20) If in addition u / X(d) then due to (3.16) and (3.20) we have the estimate I λ (u) d λ m L s() u + (1 ε)λ 1(,m ) λ m L s u + () dσ + ε u W 1, () ε u W 1, () ε u C L s () (3.21) Finally, if u X(d) and m u + dσ > 0, then (3.16), (3.19), (3.20) imly I λ (u) 1 d u W 1, () + ε u W 1, () + (1 ε)λ 1(,m ) λ m u + dσ + d λ m L s() u W 1, () u L s () (1 d)c (d) u + ε L s () u W 1, () + d u W 1, () C λ m L s() u W 1, () (1 d)c (d) u + ε u L s () C L s () + d C λ m L s() u W 1, () (1 d)c (d) u + ε L s u () C L s () (3.22) Using that <, the claim in (3.12) follows from (3.17), (3.18), (3.21), and (3.22). Lemma 3.6. Let m M, m M. If λ 1 (,m ) < λ, then there exist R > 0 such that Rϕ 1 L s () > ρ and I λ(rϕ 1 ) < 0, (3.23) where ρ > 0 is the constant in (3.12) and ϕ 1 is the ositive eigenfunction corresonding to λ 1 (,m ) satisfying m ϕ 1dσ = 1.

14 14 A. Zerouali, B. Karim, O. Chakrone, A. Boukhsas Proof. Taking into account that λ > λ 1 (,m ) and >, we claim that (3.23) is true because for a sufficiently large R > 0 we have I λ (Rϕ 1 ) R = λ 1(,m ) λ + 1 ) ( ϕ 1 R 1 W1,() m ϕ 1 dσ < 0. Recalling that I λ,m,m satisfies the Palais-Smale condition by virtue of Lemma 3.3, the roerties ointed out in (3.12) and (3.23) allow us to aly the mountain ass theorem, which guarantees the existence of a critical value c δ of I λ, with δ > 0 in (3.12), namely c := inf max I λ(γ(t)), γ Σt [0,1] Σ := { γ C([0,1],W 1, ());γ(0) = 0,γ(1) = Rϕ 1 }. This comletes the roof of Theorem 3.1. Acknowledgments The autors would like to thank the anonymous referee for valuable suggestions. References 1. Anane, A., Chakrone, O., Moradi, N., Regularity of the solutions to a nonlinear boundary roblem with indefinite weight, Bol. Soc. Paran. Mat. V. 29 1, 17 23, (2011). 2. Aris, R., Mathematical Modelling Techniues, Research Notes in Mathematics, Pitman, London, (1978). 3. Benouhiba, N., Belyacine, Z., On the solutions of (, )-Lalacian roblem at resonance, Nonlinear Anal. 77, 74 81, (2013). 4. Bonder, J.F., Rossi, J.D., A nonlinear eigenvalue roblem with indefinite weights related to the Sobolev trace embedding, Publ. Mat. 46, , (2002). 5. Fife, P.C., Mathematical Asects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics, 28, Sringer Verlag, Berlin-New York, (1979). 6. Li, G., Zhang, G., Multile solutions for the (, )-Lalacian roblem with critical exonent, Acta Mathematica Scientia, 29B, No.4, , (2009). 7. Marano, S.A., Paageorgiou, N.S., Constant-sign and nodal solutions of coercive (, )- Lalacian roblems, Nonlinear Anal. TMA, 77, , (2013). 8. Mawhin, J., Willem, M., Critical Point Theory and Hamiltonian System, Sringer-Verlag, New York, (1989). 9. Micheletti, A. M., Visetti, D., An eigenvalue roblem for a uasilinear ellitic field euation, J. Differential Euations, 184 2, , (2002). 10. Mihailescu, M., An eigenvalue roblem ossessing a continuous family of eigenvalues lus an isolated eigenvalue, Commun. Pure Al. Anal. 10, , (2011). 11. Motreanu, D., Tanaka, M., On a ositive solution for (, )-Lalace euation with indeffinite weight Minimax Theory Al. 1, 1 18, (2015). 12. Motreanu, D., M. Tanaka, M. Generalized eigenvalue roblems of nonho-mogeneous ellitic oerators and their alication, Pacific J. Math. 265, , (2013).

15 On a Positive Solution for (,)-Lalace Euation Sidirooulos, N.E., Existence of solutions to indefinite uasilinear ellitic roblems of (, )- Lalacian tye, Elect. J. Diff. Eu., 162, 1 23, (2010). 14. Struwe, M., Variational Methods, Alications to Nonlinear Partial Differential Euations and Hamiltonian Systems, Sringer-Verlag, Berlin, Heidelberg, New York, (1996). 15. Tanaka, M., Generalized eigenvalue roblems for (, )-Lalace euation with indefinite weight, J. Math. Anal. Al., Vol. 419(2), , (2014). 16. Vazuez, J.L., A strong maximum rincile for some uasi-linear ellitic euations, Al.Math. Otim., , (1984). 17. Yin, H., Yang, Z., A class of (, )-Lalacian tye euation with noncaveconvex nonlinearities in bounded domain, J. Math. Anal. Al. 382, , (2011). 18. Zerouali, A., Karim, B., Existence and non-existence of a ositive solution for (, )-Lalacian with singular weights Bol. Soc. Paran. Mat. (3s.) v. 34 2, , (2016). Abdellah Zerouali Centre Régional des Métiers de l Éducation et de la Formation, Oujda, Maroc. address: abdellahzerouali@yahoo.fr and Belhadj Karim Faculté des Sciences et Téchniues, Errachidia, Maroc. address: karembelf@gmail.com and Omar Chakrone Faculté des Sciences Oujda, Maroc. address: chakrone@yahoo.fr and Abdelmajid Boukhsas Faculté des Sciences, Oujda, Maroc. address: abdelmajidboukhsas@gmail.com