Strongly Indefinite Functionals and Multiple Solutions of Elliptic Systems

Abstract

We study existence and multiplicity of solutions of the elliptic system $\left\{{\begin{array}{*{20}l} {- \Updelta u = H_{u} (x,u,v)} \hfill & {{\text{in}}\,\Upomega,} \hfill \\ {- \Updelta v = H_{v} (x,u,v)} \hfill & {{\text{in}}\,\Upomega,\quad u(x) = v(x) = 0\quad {\text{on}}\,\partial \Upomega,} \hfill \\ \end{array}} \right.$ where $\Upomega \subset {\mathbb{R}}^{N},\,N \ge 3,$ is a smooth bounded domain and $H \in {\mathcal{C}}^{1} (\overline{\Upomega} \times {\mathbb{R}}^{2},{\mathbb{R}}).$ We assume that the nonlinear term $H(x,\,u,\,v)\sim \left| u \right|^{p} + \left| v \right|^{q} + R(x,\,u,\,v)\,{\text{with}}\,\mathop {\lim}\limits_{{\left| {(u,v)} \right| \to \infty}} \frac{R(x,\,u,\,v)}{{\left| u \right|^{p} + \left| v \right|^{q}}} = 0,$ where $p \in (1,\,2^{*}),\,2^{*} : = 2N/(N - 2),\,{\text{and}}\,q \in (1,\,\infty).$ So some supercritical systems are included. Nontrivial solutions are obtained. When H(x, u, v) is even in (u, v), we show that the system possesses a sequence of solutions associated with a sequence of positive energies (resp. negative energies) going toward infinity (resp. zero) if p > 2 (resp. p < 2). All results are proved using variational methods. Some new critical point theorems for strongly indefinite functionals are proved.