On a Bi-Nonlocal p(x)-kirchhoff Equation via Krasnoselskii s Genus

Transcription

1 On a Bi-Nonlocal p(x-kirchhoff Equation via Krasnoselskii s Genus Francisco Julio S.A. Corrêa Universidade Federal de Campina Grande Centro de Ciências e Tecnologia Unidade Acadêmica de Matemática e Estatística CEP: , Campina Grande - PB - Brazil fjsacorrea@gmail.com & Augusto César dos Reis Costa Universidade Federal do Pará Instituto de Ciências Exatas e Naturais Faculdade de Matemática CEP: , Belém - PA - Brazil aug@ufpa.br November 27, 203 Abstract We study existence and multiplicity of solutions to the following bi-nonlocal p(x- Kirchhoff equation via Krasnoselskii s Genus, on the Sobolev space with variable exponent, ( M p(x u p(x [ r p(x u = f(x, u F (x, u] in, u = 0 on, where is a bounded smooth domain of IR N, < p(x < N, M and f are continuous functions, f is an odd function, F (x, u = real parameter. u 0 f(x, ξdξ and r > 0 is a MSC: 35J60; 35J70; 58E05 Keywords: p(x-laplace operator; Sobolev space with variable exponent; Krasnoselskii s genus Partially supported by CNPq-Brazil under Grant 30056/200-5.

2 Introduction In this paper we are going to study existence and multiplicity of solutions of the p(x- Kirchhoff equation, with an additional nonlocal term, M ( [ r p(x u p(x p(x u = f(x, u F (x, u] in, u = 0 on, (. where IR N is a bounded smooth domain, f : IR IR and M : IR + IR + are continuous functions that satisfy conditions which will be stated later, F (x, u = u f(x, ξdξ, r > 0 is a real parameter and p(x is the p(x-laplacian operator, that 0 is, p(x u = ( N p(x 2 u u, < p(x < N. x i x i i= We assume the following hypotheses on M and f: there are positive constants A 0, A, B 0, B, Q, Q 2 and functions α(x, β(x, γ(x, q(x C + (, see section 2, such that A 0 + At α(x M(t B 0 + Bt β(x, (.2 and Q t γ(x f(x, t Q 2 t q(x, (.3 for all t 0 and for all x. Furthermore, for all x, α(x β(x and γ(x q(x < p = Np(x N p(x, with p > q + (r +, (.4 where and h + = max h(x and h = min h(x, f(x, t = f(x, t (.5 for all t IR and for all[ x. r Note that the term F (x, u] makes sense because, in view of assumption (.3, F (x, u 0 for all u IR. We use the genus theory, introduced by Krasnoselskii [4], to prove our main result, as follows: Theorem. Assume (.2, (.3, (.4 and (.5. Then (. has infinitely many solutions. 2

3 Problem (. was initially motivated by Corrêa-Figueiredo [8]. However, after concluding this paper we have learned that Avci-Cekic-Mashyiev [2] have obtained a similar result using Genus Theory. Because of this we should point[ out that the novelty r in the present paper is the appearance of another nonlocal term, F (x, u] and the variation of the function M. This paper is organized as follows: In section 2 we present some preliminaries on the variable exponent spaces. In section 3, we give some basic notions on the Krasnoselskii s genus that we will be used in the proof of our main result in section 4. Acknowledgement. This work was done while Francisco Julio S.A. Corrêa was visiting the Post-Graduate Programme in Mathematics of the Universidade Federal do Pará - Brazil. I would like to thank Profs. Giovany M. Figueiredo and Rúbia G. Nascimento for their warm hospitality. 2 Preliminaries on variable exponent spaces First of all, we set C + ( = { h; h C(, h(x > for all x } and for each h C + ( we define h + = max h(x and h = min h(x. We denote by M( the set of real measurable functions defined on. For each p C + (, we define the generalized Lebesgue space by L p(x ( = { } u M(; u(x p(x dx <. We consider L p(x ( equipped with the Luxemburg norm u p(x = inf µ > 0; u(x p(x dx µ. The generalized Lebesgue - Sobolev space W,p(x ( is defined by W,p(x ( = { u L p(x (; u L p(x ( } with the norm u,p(x = u p(x + u p(x. 3

4 We define W,p(x 0 ( as being the closure of Cc ( in W,p(x ( with respect the norm u,p(x. Accordingly to Fan - Zhao [3], the spaces L p(x (, W,p(x ( and W,p(x 0 ( are separable and reflexive Banach spaces. Furthermore, if the Lebesgue measure of is finite, p, p 2 C( and p (x p 2 (x, for all x, then we have the continuous embedding L p2(x ( L p(x (. The proof of the following propositions may be found in Fan [0], Fan-Shen-Zhao [], Fan-Zhang [2] and [3]. Proposition 2. Suppose that is a bounded smooth domain in IR N and p C( with p(x < N for all x. If p C( and p (x p (x ( p (x < p (x for x, then there is a continuous (compact embedding W,p(x ( L p(x (, where p (x = Np(x N p(x. Proposition 2.2 Set ρ(u =. For u 0, u p(x = λ ρ( u λ = ; 2. u p(x < (= ; > ρ(u < (= ; > ; 3. If u p(x >, then u p p(x ρ(u u p+ p(x ; 4. If u p(x <, then u p+ p(x ρ(u u p p(x ; 5. lim k + u k p(x = 0 6. lim k + u k p(x = + u(x p(x dx. For all u L p(x (, we have: lim ρ(u k = 0; k + lim ρ(u k = +. k + Proposition 2.3 (Poincaré Inequality If u W,p(x 0 (, then u p(x C u p(x, where C is a constant that does not depend on u. Note that, in view of Poincaré inequality, the norms,p(x and u = u p(x are equivalent in W,p(x 0 (. From now on we work on W,p(x 0 ( with the norm u = u p(x. We denote by L p (x ( the conjugate space of L p(x (, where p(x + p (x =, for all x. 4

5 Proposition 2.4 (Hölder Inequality If u L p(x ( and v L p (x (, then ( uvdx p + u p p(x v p (x. Proposition 2.5 (Fan-Zhang [2] (i L p(x : W,p(x 0 ( (W,p(x 0 ( is a continuous, bounded and strictly monotone operator; (ii L p(x is a mapping of type S +, i.e., if u n u in W,p(x 0 ( and lim sup(l p(x (u n L p(x (u, u n u 0, then u n u in W,p(x 0 (; (iii L p(x, N : W,p(x 0 ( (W,p(x 0 ( is a homeomorphism. 3 Preliminaries on Krasnoselskii s genus In this section we present some basic notions on the Krasnoselskii s genus that we will use in the proof of our main result. Let X be a real Banach space. Let us denote by U the class of all closed subsets A X \ {0} that are symmetric with respect to the origin, that is, u A implies u A. Definition 3. Let A U. The genus γ(a of A is defined as being the least positive integer k such that there is an odd mapping ϕ C(A, IR k such that ϕ(x 0 for all x A. If such a k does not exist we set γ(a =. Furthermore, by definition γ( = 0. In the sequel we will establish only the properties of genus that will be used through this work. More information on this subject may be found in the references [],[5],[9] and [4]. Theorem 3. Let X = IR N and be the boundary of an open, symmetric and bounded subset IR N with 0. Then γ( = N. Corollary 3. γ(s N = N. As a consequence of this, if X is of infinite dimension and separable and S is the unit sphere in X, then γ(s =. We now establish a result due to Clark [6]. Theorem 3.2 Let J C (X, IR be a functional satisfying the Palais-Smale condition. Furthermore, let us suppose that: (i J is bounded from below and even; (ii There is a compact set K U such that γ(k = k and sup x K J(x < J(0. Then J possesses at least k pairs of distinct critical points and their corresponding critical values are less than J(0. 5

6 We now return to the problem (.. For this, we consider the functional J : W,p(x 0 ( IR defined by where M(t = and J (uv = M t 0 J(u = M ( ( M(sds and F (x, t = p(x u p(x r + t 0 [ ] r+ F (x, u f(x, sds. Plainly, J C (Wo,p(x (, IR [ ] r p(x u p(x u p(x 2 u v F (x, u f(x, uv, for all u, v W,p(x 0 (. Next we present two important lemmas for the proof of the main result. Lemma 3. J is bounded from below. Proof. Using (.2 and (.3 we get ( p(x u p(x J(u [As α(x + A 0 ]ds [ u ] r+ (Q 2 u q(x ds 0 ( r + 0 A α(x+ α(x + p(x u p(x + (A 0 p(x u p(x ( r+ ( Q2 r+ u dx q(x. r + q Taking u > we obtain from Proposition 2.2 and Sobolev immersions A J(u (p + α+ + (α + + u p (α + + A 0 p + u p C u (r+q±. So J is bounded from below, since p (α + + > p > q + (r + and the lemma is proved. Lemma 3.2 J satisfies the (PS condition. Proof. Let (u n W,p(x 0 ( be a sequence such that J(u n C and J (u n 0. Arguing as in the above lemma, we obtain a positive constant C such that C J(u n A (p + α+ + (α + + u n p (α + C u n (r+q±. 6

7 Since p (α + + > q + (r +, we conclude that ( u n is bounded. Thus, there exists a subsequence, still denoted by (u n, such that u n u in W,p(x 0 (. From we have J (u n 0, that is, J (u n (u n u 0, M ( [ p(x u n p(x u n p(x 2 u n (u n u By the Hölder inequality we obtain ] r F (x, u n f(x, u n (u n u 0. (3.6 f(x, u n (u n udx Q 2 u n q(x (u n udx u n q(x (u n udx u n q(x (u n u dx C u q(x/q(x u n u q(x. Since q(x < p (x para todo x we deduce that W,p(x 0 ( is compactly embedded in L q(x (, hence (u n converges strongly to u in L q(x (. Then [ ] r F (x, u n f(x, u n (u n udx 0. Note that there exist nonnegative constants c and c 2 such that c [Q ] r [ ] r ] r γ(x u n γ(x F (x, u n [Q 2 q(x u n q(x c 2. (3.7 Since (u n is bounded in W,p(x 0 (, we may assume that p(x u n p(x dx t 0 0. If t 0 = 0 then u n 0 in W,p(x 0 ( and the proof is finished. If t 0 > 0 then from (3.6 we obtain L p(x (u n (u n u = u n p(x 2 u n (u n u 0, 7

8 because there exist positive constants c 3 and c 4 such that c 3 M c 4. We also have Consequently, L p(x (u(u n u = From Proposition 2.5 we have u n condition. ( u p(x 2 u (u n u 0. (L p(x (u n L p(x (u, u n u 0. p(x u n p(x u em W,p(x 0 (. Hence J satisfies the (PS 4 Proof of Theorem. The following result, whose proof can be seen Brezis [3], will play a key role in the proof of our main result Theorem 4. Let X be a separable and reflexive Banach space, then there exists (e n X and (e n X such that < e n, e m >= δ n,m = { if n = m 0 if n m, X = span{e n ;, 2, } and X = span{e n;, 2, }. Proof of Theorem.. For each k IN consider X k = span{e, e 2,, e k }, the subspace of W,p(x 0 ( spanned by the vectors e, e 2,, e k. Note that X k L γ(x (, < γ(x < p with continuous immersions. Thus, the norm W,p(x 0 ( and L q(x ( are equivalent on X k. Note that using (.2 and (.3 we obtain ( ( B β(x+ J(u B 0 p(x u p(x + (β(x + p(x u p(x ( r+ ( Q r+ u dx γ(x. r + γ + If u is small enough, from Proposition 2.2 we obtain u p(x u p and u γ+ γ(x u γ(x. By equivalent norms on X k we have C(k u γ+ u γ(x where C(k is a positive constant. Hence 8

9 or also J(u B 0 B p u p + (p α + (α + u p (α + (r+γ+ C(K u, [( ] B0 J(u u (r+γ+ p + B u p (r+γ + C(K. (p α + (α + Let R be a positive constant such that ( B0 p + B R p (r+γ + < C(K. (p α + (α + Thus, for all 0 < r 0 < R, and [( considering K = {u X k : u = r 0 }, we get ] J(u r (r+γ+ B0 0 p + B r p (r+γ + (p α + (α 0 C(K [( + ] B0 < R (r+γ+ p + B R p (r+γ + C(K < 0 = J(0, (p α + (α + which implies sup J(u < 0 = J(0. K Since X k and IR k are isomorphic and K and S k are homeomorphic, we conclude that γ(k = k. Moreover, from (.5, J is even. By the Clark theorem, J has at least k pair of different critical points. Since k is arbitrary, we obtain infinitely many critical points of J. Remark 4. Following the same steps in this article, we obtain the main result, changing the condition (.2 by: A 0 M(t B 0, At α(x M(t Bt β(x, A 0 M(t Bt β(x, A 0 M(t B 0 + Bt β(x, At α(x M(t B 0 + Bt β(x, A 0 + At α(x M(t Bt β(x, where A 0, B 0, A and B are positive constants. Note that the result obtained in this paper is more general than the result obtained in [2] and [8], because we introduced the additional nonlocal term, we varied the exponents of functions limits, we did translation of these functions and we conclude that the result also holds true for bounded functions M. Remark 4.2 This work partially complements a previous paper by the authors. See Corrêa-Costa [7]. In this paper the authors focus their attention on the problem M ( [ ] r p(x u p(x p(x u = λ u q(x 2 u q(x u q(x in, u = 0 on. 9 (4.8

10 Depending on some relations between the the various functions contained in the problem, the authors use the Mountain Pass Theorem or the Ekeland Variational Principle to prove the existence of a positive solution. The results are summarized in the table below: Table with results M(t hypotheses MPT EVP m p + m 0 M(t m < (q r+ (r + m 0 (q + r yes not used and q (r + > p + for all λ > 0 m 0 M(t m q (r + < p no yes for λ (0, λ M(t = a + bt max {p +, 2(p+ 2 } (p 2 < (r + (q r+ (q + r yes not used and q (r + > 2p + for all λ > 0 M(t = a + bt q (r + < p no yes for λ (0, λ M(t = t α α(p + α (q α < (q r+ (r + (q + r yes not used and q (r + > p + for all λ > 0 M(t = t α q (r + < αp no yes for λ (0, λ MPT - Existence of solution via The Mountain Pass Theorem EVP - Existence of solution via Ekeland s Var. Principle In particular, the case M(t = t α is related with the present paper. Remark 4.3 In the present paper and in [7] the authors deal with subcritical growth. However, in a forthcoming article they will study the critical case by using the Concentration Compactness Principle in L p(x ( spaces due to Yongqiang [6]. In this last work it is proved a generalization of the well known Lions s Concentration Compactness Principle due to Lions [5] in the context of the generalized Lebesgue space L p(x (. Remark 4.4 In the problem (., as well as those cited in [7] and in the above remark, the authors treat of variational problems. However, in a paper in preparation, we prove a result involving the problem M ( [ r p(x u p(x p(x u = f(x, u g(x, u] + h(x in, u = 0 on (4.9 in bounded domains and h 0 is a given nonzero function and f, g are nondecreasing functions in which it is allowed critical growth. To attack problem (4.9 we invoke a fixed point theorem due to Carl-Heikkilä [4]. In this theorem the authors provide a new fixed point theorem for increasing self-mappings G : B B of a closed ball B X, where X is a Banach semilattice which is reflexive or has a weakly fully regular order cone X +. 0

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