一个椭圆系统的无穷多个解

Infinitely many solutions for an elliptic system Lihong Huang, Ronghui Hu
College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, P. R. China ( E-mail addresses: lhhuang@ (L. Huang);
|u|→∞
H (x, u) ≤ b < ∞ for all x ∈ Ω. |u|q
(2) Comparing Theorem 3.1 in [1], Condition (H4 ) differs from the one in [1]; (3) (H5 ) guarantees that H (x, u) grows nonquadratic in u for |u| → ∞ for all x ∈ Ω. Now we can state our main results: Theorem 1.1 Suppose H satisfies (H1 ) − (H5 ) and (H6 ) H (x, v, −w) = H (x, v, w) for all x ∈ Ω, v, w ∈ R. Then there exist infinitely many solutions of (1.1). The paper is organized as follows: In section 2, we shall give some preliminaries. In section 3, by using the generalized symmetry mountain theorem, we shall give the proof of Theorem 1.1. 2. Preliminaries Let X be a Hilbert space and ϕ ∈ C 1 (X, R) be of the form ϕ(x) =
∗
huronghui 2004@ (R. Hu)) ¯
Abstract. By using the generalized symmetric mountain pass theorem developed by Bartsch and Clapp through the limit relative category of Fournier et al, we prove the existence of infinitely many solutions for an elliptic system with Dirichlet boundary conditions. Some existing results are extended. Keywords. elliptic system; Dirichlet boundary condition; multiple solutions; generalized mountain pass theorem AMS(2000) Subject Classification. 39A11 1. Introduction This paper is devoted to the study of multiple solutions of the following elliptic system
1 2
Ax, x + ψ (x) with
A : X → X a bounded self adjoint linear operator and ψ a nonlinear operator whose gradient ∇ψ ∈ C 0 (X, X ) be completely continuous. Such a functional is said to be strongly indefinite if the quadratic form negative eigenvalues. Let G be a compact Lie group. By a G− space X we mean a topological space together with a continuous map G × X → X, (g, x) → gx, satisfying ex = x, and (gh)x = g (hx) where e is the identity element of G and g, h ∈ G, x ∈ X are arbitrary. A subset Y of X is said to be G-invariant, or a G-subset, if gx ∈ Y for every g ∈ G, x ∈ X. A function ϕ : X → R is invariant if ϕ(gx) = ϕ(x) for every g ∈ G, x ∈ X . For G− space X, Y, a map f : X → Y is equivariant if f (gx) = gf (x) for every g ∈ G, x ∈ X . For a subgroup H of G X H := {x ∈ X : gx = x for all g ∈ H }, denotes the H-fixed point set. Let X be a Hilbert space with an orthogonal action of a compact Lie group G and consider a G-invariant C 1 -function J : X → R. Assume T be a maximal torus of G, N be its
1

1 (Ω) × H 1 (Ω). As a norm in X we take for u = (v, w ) defined for u = (v, w) ∈ X = H0 0
u =
Ω
|∇u|2 dx =
Ω
(|∇v |2 + |∇w|2 )dx.
The functional J is strongly indefinite. The study of strongly indefinite functionals is motivated by a number of problems from mathematical physics. They appear for example in the search for periodic solutions of Hamiltonian systems or the one-dimensional wave equations, or (weak) solutions of boundary value problems for systems of elliptic equations. In applications the original problem is symmetric with respect to the action of some Lie group, such as Z2 , S 1 . One expects that symmetry forces the existence of multiple solutions. There are a lot of papers concerned with this topic. By Lusternik-Schnirelmann category theory, saddle-point reduction, index and pseudoindex theory, relative category cat∞ , Morse theory et., Rabinowitz and Ambrosetti [3], Ammann and Zehnder [4, 5], Benci [6, 7], Szulkin [8, 9] studied the strongly indefinite functional with symmetry through restricting the functional to finite-dimensional subspace respectively. Recently by limit index theory Huang and Li in [10] studied a noncooperative p-Laplacian elliptic system in RN . However, Bartsch and Clapp [1] developed another approach, the limit relative category of Fournier et al [2] which used a Galerkin type approximation ( reducing the study of strongly indefinite functionals to a semidefinite situation, but not to a finitedimensional one). Inspired by [1] we study the elliptic system (1.1) and extend some results in them. Our fundamental assumptions are as follows: (H1 ) H : Ω × R2 → R is of class C 1 ; (H2 ) There exist c > 0 and 1 ≤ p < (N + 2)/(N − 2) such that for all x ∈ Ω and u ∈ R2 |Hu (x, u)| ≤ c(1 + |u|p ); (H3 ) H (x, v, 0) ≥ 0 for all x ∈ Ω, v ∈ R; (H4 ) There exists an open subset Ω0 of Ω such that H (x, v, w) → +∞, as |w| → ∞ for all x ∈ Ω0 . (H5 ) There exists µ > N (p − 1)/2 if N ≥ 2(resp. µ > p − 1 if N = 1.) such that for all x∈Ω lim inf