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

Abstract

The aim of this paper is to investigate the existence of weak bounded solutions of the gradient-type quasilinear elliptic system\begin{equation}\tag{P}\begin{cases}- {\rm div} ( a_i(x, u_i, \nabla u_i) )+ A_{i, t} (x, u_i, \nabla u_i) = G_i(x, \mathbf{u}) \quad \hbox{in $\Omega$ for $i\in\{1,\dots,m\}$,}\\ {\mathbf{u}} = 0\quad \hbox{on $\partial\Omega$,}\end{cases}\end{equation}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 $A_i\colon \Omega\times{\mathbb R}\times{\mathbb R}^N\rightarrow{\mathbb R}$,$i\in\{1,\dots,m\}$, and $G\colon \Omega\times{\mathbb R}^m\rightarrow{\mathbb R}$ 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 additionally ${\mathcal J}$ is even, of infinitely many critical points.