On Some Quasilinear Elliptic Systems with Singular and Sign-Changing Potentials

Abstract

Using variational methods, we study the existence and nonexistence of nontrivial weak solutions for the quasilinear elliptic system

$$\left\{\begin{array}{ll}- {\rm div}(h_1(|\nabla u|^2)\nabla u) = \frac{\mu}{|x|^2}u + \lambda F_u(x, u, \upsilon)\quad {\rm in}\,\Omega,\\- {\rm div}(h_2(|\nabla \upsilon|^2)\nabla \upsilon) = \frac{\mu}{|x|^2}\upsilon + \lambda F_\upsilon(x,u,\upsilon)\quad {\rm in}\,\Omega,\\u = \upsilon = 0 \qquad \qquad \qquad \qquad \qquad \qquad {\rm in}\, \partial\Omega, \end{array}\right.$$

where ${\Omega \subset \mathbb{R}^N,N \geq 3}$ , is a bounded domain containing the origin with smooth boundary ${\partial \Omega ; h_i, i = 1, 2}$ , are nonhomogeneous potentials; ${(F_u, F_v) = \nabla F}$ stands for the gradient of a sign-changing C ¹-function ${F : \Omega \times \mathbb{R}^2 \to \mathbb{R}}$ in the variable ${{w = (u, v) \in \mathbb{R}^2}}$ ; and λ and μ are parameters.