Strongly Indefinite Functionals and Multiple Solutions of Elliptic Systems

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.