On a Nonlocal Elliptic System of p-kirchhoff-type Under Neumann Boundary Condition

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1 On a Nonlocal Elliptic System of p-kirchhoff-type Under Neumann Boundary Condition Francisco Julio S.A Corrêa,, UFCG - Unidade Acadêmica de Matemática e Estatística, Campina Grande - PB - Brazil julio@dme.ufcg.edu.br, Rúbia G. Nascimento UFPA - Campus Universitário de Abaetetuba Abatetuba - PA - Brazil. rubia@ufpa.br Abstract: In this paper we investigate questions of existence of solution for the system [M u p ] p p u =f(u, v+ρ (x, ] p v p p v =g(u, v+ρ 2 (x, =,. vector on, p is the p-laplacian operator p u = div( u p2 u and the involved functions in the problem satisfy: (M M, M 2 : IR + IR are continuous functions and there is a positive constant m > such that M (t, M 2 (t m >, t ; Motivated by a problem in Costa [3], who studies a single local equation, we study the above problem by using variational methods. Since we wiil work in the space W,p ( the functional associated to the above problem will not be coercive. So, we have to consider the Poincaré-Wirtinger s inequality in the subspace of W,p ( formed by the functions with null mean in. In this way, and motivated by physical motivations related with the wave equation we consider the conditions (F (F 2. Introduction In this paper we deal with the nonlocal elliptic system of the p-kirchhoff type given by ] p [M u p p u =f(u, v+ρ (x, ] (P p v p p v =g(u, v+ρ 2 (x, =,. where IR N, N is a bounded smooth domain, < p < N, η is the unit exterior (F There is a C function F : IR 2 IR such that F u (u, v = f(u, v and F v (u, v = g(u, v (u, v IR 2 ; There is k > such that (F 2 F (u + k, v + k = F (u, v, (u, v IR 2 ; (ρ 33 ρ, ρ 2 L q (, p + q =, < q < p and ρ = ρ 2 =. Where p = { Np Np if < p < N, if p N is the critical Sobolev exponent. Here h means h(xdx. Using a variational method we will establish the following result: Theorem.. Under assumptions (M, (F, (F 2 and (ρ, problem (P possesses a weak solution (u, v W,p ( W,p (.

2 By a weak solution of (P we mean a function (u, v W,p ( W,p ( such that ] p [M u p u p2 u ϕ = [f(u, v+ρ ]ϕ and ] p v p v p2 v ψ = [g(u, v+ρ 2 ]ψ for all ϕ, ψ W,p (. System (P is a generalization of a model introduced by Kirchhoff [5]. More precisely, Kirchhoff proposed a model given by the equation ρ 2 u t 2 ( P h + E 2L L x 2 dx 2 u =, ( x2 where ρ, P, h, E, L are constants, which extends the classical D Alembert wave equation, by considering the effects of the changes in the length of the strings during the vibrations. The equation 2 u t 2 M( u 2 u = f(u in (2 generalizes equation (, where M : IR + IR is a given function and u 2 = u 2 is the usual norm of H (. The stationary counterpart of (2 is M( u 2 u = f(x, u in (3 We now set X = W,p ( W,p ( and Y = L p ( L p ( which are Banach spaces equipped, respectively, by the norms ( (u, v X = ( u p + v p /p + ( u p + v p = and (u, v Y = ( u p + v p ( u p + v p /p /p = ( u p p + v p p /p. We remark that W,p ( may be split in the following way. Let W c =, that is, the subspace of W,p ( { spanned by the constant } function and W = z W,p (; z = which is called the space of functions of W,p ( with null mean in. Thus W,p ( = W W c, i.e., every function u W,p ( is of the form where u = u + α, u = and α is a constant. Consequently, if (u, v X, then which, in the most of the papers, has been studied under Dirichlet boundary condition. In the present work, besides the Neumann boundary condition, we consider the case of a system and, instead of the Laplacian, we work with the p-laplacian. This work is organized as follows: in section 2 we present some preliminary results and in the section 3 we prove the main result. (u, v = (u + α, v + β = (u, v + (α, β, where u = v = and α, β are constants. As it is well known the Poincaré s inequality does not hold in the space W,p (. However, it is true in W as shows the next lemma for which we will give an alternative proof by using minimization on a certain manifold. 2 Preliminaries We work essentially in the space W,p ( endowed with the norm u p = ( u p + u p. The norm in L p ( will be denoted by u p p = u p. Lemma 2.. (Poincaré-Wirtinger s Inequality There is a positive constant C such that z p C z p, for all z W. 34 Proof. Let ψ : W IR be the functional given by ψ(z = z p, for all z W,

3 and M be the manifold { M = z W ; } z p =. Since ψ is bounded from below on M, it follows that there is a minimizing sequence (z n M, that is, ψ(z n inf ψ = ψ. M Consequently, z n p = and there is a positive constant C such that z n p C, for all n IN. From these facts we infer that the sequence (z n is bounded in W,p ( and in view of this, the real sequence ( z n p possesses a convergent subsequence. So, z n z in W,p ( and by virtue of compactness of the Sobolev embedding we have, perhaps for a subsequence, that z n z in L r (, r < p, with p the critical Sobolev exponent. In particular, = z n z = and = z n p z p and thus z M. Let us show that ψ >. Suppose, on the contrary, that ψ =. In this case, up to subsequences, we have = lim z n p = lim z n p + z n p z n p = lim ( z n p z n p p = lim z n p lim z n p p z p z p p = z p which yields z p. Therefore, z(x = C a.e. in, with C a real constant. Because z M W, one has z = C = and we conclude that C =, which is impossible because z p =. Consequently, ψ >. Thus, ψ = inf ψ M = lim Hence, z n p = lim( z n p z n p p = lim z n p lim z n p p z p z p p = z p ψ. ψ = z p = ψ(z which shows that the infimum of ψ is attained on M. Consequently, ψ z p, for all z W with z p =. If z W, ( ψ z p z p from which z p p ψ = z p z p p z p, for all z W which shows the Poincaré-Wirtinger s inequality in W. We will use the following version of the Ekeland Variational Principle [4] that will play a key role in the proof of our main result. Proposition 2.2. Let E be a Banach space and let ψ : E IR be a C function which is bounded from below. Then, for any ɛ > there exists z E such that ψ(z inf E ψ + ɛ and ψ (z E ɛ. 3 Proofs Let us start by considering the functional I : X IR given by I(u, v = p M u p + p M 2 v p F (u, v for all (u, v X, where M i (t = t ρ u ρ 2 v, [M i (s] p ds, i =, 2. 35

4 Proposition 3.. The functional I is bounded from below. Proof. Firstly, we note that I is well defined. For, it is enough to show that J : X IR (u, v J(u, v = F (u, v is well defined. Since F is continuous on [, k] [, k] and F (u + k, v + k = F (u, v for all (u, v IR 2, it follows that F (u, v C, for all (u, v IR 2, and so J(u, v F (u, v C, for all (u, v X, where is the Lebesgue measure of. Let us show that I is bounded from below. If (u, v X, u and v may be written as u = u + α and v = v + β, where α, β IR and u = v =. Thus I(u, v = p M u p + p M 2 v p F (u + α, v + β ρ (u + α ρ 2 (v + β p M u p + p M 2 v p C ρ (u + α p M C ρ u α ρ ρ 2 v β ρ 2. u p + p M 2 ρ 2 (v + β v p Since ρ, ρ 2 L q (, u, v L p ( and using the Hölder inequality ρ u ρ q u p and ρ 2 v ρ 2 q v p we obtain I(u, v p M u p + p M 2 ρ q u p ρ 2 q v p C v p and using assumption (M I(u, v p mp u p + p mp ρ q u p ρ 2 q v p C. v p By means of Poincaré-Wirtinger s inequality I(u, v p mp v p ψ /p ψ /p Because the function ρ q ρ 2 q u p + u p /p (s, t p mp (s + t ψ /p v p /p C ψ /p ρ 2 q t /p, s, t, ρ q s /p is bounded from below, we conclude that I is also bounded from below. Remark 3.2. By this last proposition, the main difference between the Dirichlet and Neumann problem is related with the noncoerciveness of the energy functional associated to Neumann problem. At this point the Poincaré-Wirtinger s inequality plays a fundamental role. The reader may consult [], [6], [7] where some Kirchhoff equations are considered, from the variational point of view, under Dirichlet boundary conditions in the scalar case. Some classes of M and nonlinearities f are considered in problems of the type { [ ] p2 M( u p,p p u = f(u in, where u p,p = W,p (. u = on, u p is the usual norm in Proof of Theorem. We will find a critical point of the functional I. As I is C functional bounded from below, it follows from the Ekeland Variational Principle that there exists ((u n, v n X such that 36 I(u n, v n inf X I and I (u n, v n. (4

5 For each n IN, Recalling that u n = u n + α n and v n = vn + β n, where α n and β n are real constants and u n = vn =. From (4 we have I(u n, v n C, for some positive constant C and for all n IN. We now use the preceding proposition to obtain C 2 p mp ψ /p ψ /p ρ q ρ 2 q u n p + u n p /p v n p /p v n p I(u n, v n C which implies that the sequences u n p and vn p are bounded. By virtue of the Poincaré inequality u n p and vn p are bounded too. Consequently, (u n and (v n are bounded sequences in W,p (. Remarking that α n, β n may be taken in the interval [, k], u n = u n + α n and v n = v n + β n, for all n IN, the sequences (u n and (v n are bounded in W,p (. Hence, u n u and v n v in W,p (, (5 perhaps for subsequences, and so ρ u n ρ u, ρ 2 v n and thanks to Sobolev embeddings ρ 2 v u n u and v n v in L s (, s < p, where p is the critical Sobolev exponent, and up to subsequences, u n (x u(x and v n (x v(x a.e. in. Due to the continuity of F F (u n (x, v n (x F (u(x, v(x a.e. in and because F (u n (x, v n (x C for all n IN a.e. in, we may use the Lebesgue dominated convergence theorem to conclude that F (u n, v n F (u, v. (6 inf I = lim I(u n, v n X [ ( = lim p M u n p + ( ] p M 2 v n p F (u n, v n ρ u n ρ 2 v n it rests to analyze the convergence of ( M u n p and ( M 2 v n. p As u n u in W,p ( one has that u n p u p and u p lim inf u n p + Since u n p. (7 u n p is bounded, it is convergent, at least for a subsequence. Hence, from (7, u p lim which implies u p + Thus, u n p + lim u n p u p lim u n p + u p. u p lim u n p. Noticing that M and M 2 are continuous and increasing, we get M u p lim M u n p. Analogously, M 2 v p lim M 2 Consequently, inf I X p M 37 F (u, v v n p. u p + p M 2 v p ρ u ρ 2 v = I(u, v

6 that implies that I(u, v = inf I. Since (u, v X W,p ( and is a weak solution of problem (P we conclude that such a function satisfies the Neumann boundary condition in the trace sense. This finishes the proof of the theorem. Remark 3.3. If one considers the Hamiltonian system [M u p ] p p u = f(v + ρ (x, ] p v p p v = g(u + ρ 2 (x, =,. (8 where f, g : IR IR are P-periodic and continuous functions satisfying P f(sds = P g(sds = we can prove, reasoning as in the theorem., the existence of a weak solution of the problem (P, since M, ρ, ρ 2 satisfy assumptions (M and (ρ. The situation described in the problem (8 is the counterpart, in nonlocal partial differential equations setting, of that considered in Willem [8] for the forced pendulum equation. Also related with this kind of problem the reader may consult Willem [9] and Brézis [2]. Note that these assumptions imply that the involved functions f, g, ρ, ρ 2 change sign. Thus we can not expect to obtain positive solutions. Referências [] C.O. Alves,F.J.S.A. Corrêa and T.F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type,comput. Math. Appl., 49( [2] H. Brézis, Periodic solutions of nonlinear vibrations strings and duality principles, Bull. Amer. Math. Soc., 8( [3] D. G. Costa, Tópicos em análise funcional não-linear e aplicações às equações diferenciais,viii Esola Latino-Americana de Matemática, Rio de Janeiro-Brazil(986. [4] I. Ekeland, On the variational principle,j. Math. Anal. Appl., 47( [5] G. Kirchhoff, Mechanik, Teubner, Leipzig,,(883. [6] T.F. Ma,Remarks on an elliptic equation of Kirchhoff type,nonlinear Anal., Vol. 63( [7] R.G. Nascimento, Problemas elíticos nãolocais do tipo p-kirchhoff, Doctoral Dissertation, IMECC-UNICAMP, January(28. [8] M. Willem, Lectures on critical point theory, Trabalho de Matemática 99, Universidade de Brasília, Brasília-Brazil(983. [9] M. Willem, Periodic oscilations of odd second order Hamiltonian systems, Bolletino U.M.I., 6( Remark 3.4. As it is easy to see the condition (ρ is necessary in the (M, M 2 linear case, that is, [M u p ] p p u = ρ (x, ] p v p p v = ρ 2 (x, =,. (9 Thus, it is natural to consider this assumption in the semilinear case. 38