Nontrivial solutions for a class of gradient-type quasilinear elliptic systems - Candela, Anna Maria; Sportelli, Caterina (Topological Methods in Nonlinear Analysis, 2022)
The aim of this paper is to investigate the existence of weak bounded solutionsof the gradient-type quasilinear elliptic system\begin{equation}\label{aP}\tag{P}\begin{cases}- {\rm div} ( a_i(x, u_i, abla u_i) )+ A_{i, t} (x, u_i, abla u_i) = G_i(x, \bu)\hidewidth \\&\hskip3cm \hbox{in $\Omega$ for $i\in\{1,\dots,m\}$,}\\\bu = 0 &\hskip 3cm \hbox{on $\partial\Omega$,}\end{cases}\end{equation}with $m\geq 2$ and $\bu=(u_1,\dots, u_{m})$, where$\Omega\subset{\mathbb R}^N$ is an open bounded domain andsome functions $A_i\colon \Omega\times{\mathbb R}\times{\mathbb R}^N\rightarrow{\mathbb R}$,$i\in\{1,\dots,m\}$,and $G\colon \Omega\times{\mathbb R}^m\rightarrow{\mathbb R}$ existsuch that$a_i(x,t,\xi) = abla_{\xi} A_i(x,t,\xi)$,$A_{i, t} (x,t,\xi) = \frac{\partial A_i}{\partial t} (x,t,\xi)$, and$G_{i}(x,\bu) = \frac{\partial G}{\partial u_i}(x,\bu)$.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 provethe existence of at least one critical point and, if additionally ${\mathcal J}$ is even, of infinitely many critical points.