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

TL;DR

This paper establishes the existence of nontrivial solutions for a class of gradient-type quasilinear elliptic systems using variational methods and critical point theory, under suitable hypotheses on the involved functions.

Contribution

It introduces new conditions ensuring the functional is well-behaved and applies generalized Mountain Pass Theorems to prove existence and multiplicity of solutions.

Findings

Existence of at least one weak bounded solution under certain conditions.

Infinitely many solutions if the functional is even.

The functional satisfies the Cerami-Palais-Smale condition.

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.