Infinitely Many Solutions of Nonlinear Elliptic Systems

Abstract

In this paper we study elliptic systems of the form

$$ \left\{ {_{\Delta _v = H_{u(x,u,v)in\Omega } }^{ - \Delta _u = H_v (x,u,v)in\Omega } } \right. $$

((1.1))

where Ω ⊂ ℝ^N, N > 3, is a smooth bounded domain and H: Ω ℝ ℝ → ℝ C ¹-function. We shall also consider the case when Ω = ℝ^N and in this case the system takes the form

$$ \left\{ {_{\Delta _v + v = H_u (x,u,v)in\mathbb{R}^N }^{ - \Delta _u + u = H_v (x,u,v)in\mathbb{R}^N } } \right. $$

((1.2))