Existence of minimizers for a quasilinear elliptic system of gradient type

Federica Mennuni; Addolorata Salvatore

doi:10.3934/dcdss.2022013

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Existence of minimizers for a quasilinear elliptic system of gradient type

Federica Mennuni ^, and Addolorata Salvatore

Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, Via E. Orabona 4, 70125 Bari, Italy

^* Corresponding author: Federica Mennuni

Received August 2021 Early access January 2022

Fund Project: The research that led to the present paper was partially supported by MIUR-PRIN project "Qualitative and quantitative aspects of non linear PDEs"(2017 JPCAPN-005), Fondi di Ricerca di Ateneo 2017/18 "Problemi differenziali non lineari"

The aim of this paper is to investigate the existence of weak solutions for the coupled quasilinear elliptic system of gradient type

$ \left\{ \begin{array}{ll} - {\rm div} (a(x, u, \nabla u)) + A_t (x, u,\nabla u) = g_1(x, u, v) &{\rm{ in}} \; \Omega ,\\ - {\rm div} (B(x, v, \nabla v)) + B_t (x, v,\nabla v) = g_2(x, u, v) &{\rm{ in}}\; \Omega ,\\ \quad u = v = 0 &{\rm{ on}}\;\partial\Omega , \end{array} \right. $

where

$ \Omega \subset \mathbb R^N $

is an open bounded domain,

$ N \geq 2 $

and

$ A(x,t,\xi) $

$ B(x,t, {\xi}) $

are

$ \mathcal{C}^1 $

–Carathéodory functions on

$ \Omega \times \mathbb R \times { \mathbb R}^{N} $

with partial derivatives

$ A_t = \frac{\partial A}{\partial t} $

$ a = {\nabla}_{\xi}A $

, respectively

$ B_t = \frac{\partial B}{\partial t} $

$ b = {\nabla}_{{\xi}}B $

, while

$ g_1(x,t,s) $

$ g_2(x,t,s) $

are given Carathéodory maps defined on

$ \Omega \times \mathbb R\times \mathbb R $

which are partial derivatives with respect to

$ t $

and

$ s $

of a function

$ G(x,t,s) $

We prove that, even if the general form of the terms

$ A $

and

$ B $

makes the variational approach more difficult, under suitable hypotheses, the functional related to the problem is bounded from below and attains its minimum in a "right" Banach space

$ X $

The proof, which exploits the interaction between two different norms, is based on a weak version of the Cerami–Palais–Smale condition and a suitable generalization of the Weierstrass Theorem.

Keywords: Quasilinear elliptic system, weak Cerami–Palais–Smale condition, Minimum Principle, "sublinear" growth.

Mathematics Subject Classification: Primary: 35J20, 35J92; Secondary: 35J25, 58E05.

Citation: Federica Mennuni, Addolorata Salvatore. Existence of minimizers for a quasilinear elliptic system of gradient type. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022013