Nontrivial solutions for a class of gradient-type quasilinear elliptic systems

Abstract

The aim of this paper is investigating the existence of weak bounded solutions of the gradient-type quasilinear elliptic system $$(P)\qquad \left\{ \begin{array}{ll} - {\rm div} ( a_i(x, u_i, \nabla u_i) ) + A_{i, t} (x, u_i, \nabla u_i) = G_i(x, \mathbf{u}) &\hbox{ in $\Omega$}\\ \quad\qquad\qquad\qquad\qquad \mbox{ for }\; i\in\{1,\dots,m\},\\ \mathbf{u} = 0 &\hbox{ on $\partial\Omega$,} \end{array} \right.$$ with $m\geq 2$ and $\mathbf{u}=(u_1,\dots, u_{m})$, where $\Omega\subset\mathbb{R}^N$ is an open bounded domain and some functions $A_i:\Omega\times\mathbb{R} \times \mathbb{R}^N\rightarrow\mathbb{R}$, $i\in\{1,\dots,m\}$, and $G:\Omega\times\mathbb{R}^m\rightarrow\mathbb{R}$ exist such that $a_i(x,t,\xi) = \nabla_{\xi} A_i(x,t,\xi)$, $A_{i, t} (x,t,\xi) = \frac{\partial A_i}{\partial t} (x,t,\xi)$ and $G_{i}(x,\mathbf{u}) = \frac{\partial G}{\partial u_i}(x,\mathbf{u})$. We prove that, under suitable hypotheses, the functional $\mathcal{J}$ related to problem $(P)$ is $\mathcal{C}^1$ on a "good" Banach space $X$ and satisfies the weak Cerami-Palais-Smale condition. Then, generalized versions of the Mountain Pass Theorems allow us to prove the existence of at least one critical point and, if $\mathcal{J}$ is even, of infinitely many ones, too.