Existence and multiplicity results for a class of coupled quasilinear elliptic systems of gradient type

TL;DR

This paper establishes the existence and multiplicity of weak solutions for a class of coupled quasilinear elliptic systems of gradient type using variational methods and critical point theory.

Contribution

It introduces new conditions under which the associated functional admits critical points, even with complex coefficients, and extends multiplicity results to these systems.

Findings

Existence of at least one weak solution under certain hypotheses.

Infinitely many solutions if the functional is even.

Application of a weak Cerami-Palais-Smale condition and advanced variational techniques.

Abstract

The aim of this paper is investigating the existence of one or more weak solutions of the coupled quasilinear elliptic system of gradient type \[ (P)\qquad \left\{ \begin{array}{ll} - {\rm div} (A(x, u)\vert\nabla u\vert^{p_1 -2} \nabla u) + \frac{1}{p_1}A_u (x, u)\vert\nabla u\vert^{p_1} = G_u(x, u, v) &\hbox{ in $\Omega$,}\\[5pt] - {\rm div} (B(x, v)\vert\nabla v\vert^{p_2 -2} \nabla v) +\frac{1}{p_2}B_v(x, v)\vert\nabla v\vert^{p_2} = G_v\left(x, u, v\right) &\hbox{ in $\Omega$,}\\[5pt] u = v = 0 &\hbox{ on $\partial\Omega$,} \end{array} \right. \] where $\Omega \subset \mathbb{R}^N$ is an open bounded domain, $p_1$, $p_2 > 1$ and $A(x,u)$, $B(x,v)$ are $\mathcal{C}^1$-Carath\'eodory functions on $\Omega \times \mathbb{R}$ with partial derivatives $A_u(x,u)$, respectively $B_v(x,v)$, while $G_u(x,u,v)$, $G_v(x,u,v)$ are given Carath\'eodory maps defined on $\Omega \times \mathbb{R}\times \mathbb{R}$ which are partial derivatives of a function $G(x,u,v)$. We prove that, even if the coefficients make the variational approach more difficult, under suitable hypotheses functional $\cal{J}$, related to problem $(P)$, admits at least one critical point in the ''right'' Banach space $X$. Moreover, if $\cal{J}$ is even, then $(P)$ has infinitely many weak bounded solutions. The proof, which exploits the interaction between two different norms, is based on a weak version of the Cerami-Palais-Smale condition, a ''good'' decomposition of the Banach space $X$ and suitable generalizations of the Ambrosetti-Rabinowitz Mountain Pass Theorems.