Multiple solutions for some symmetric supercritical problems

TL;DR

This paper investigates the existence of multiple critical points for a class of supercritical symmetric problems involving nonlinear functionals, even under challenging growth conditions, using advanced variational methods.

Contribution

It establishes the existence of at least one or infinitely many critical points for a generalized supercritical problem with complex growth conditions, despite difficulties posed by variable coefficients.

Findings

Existence of at least one critical point under supercritical growth.

Infinitely many critical points when the functional is even.

Application of a novel intersection lemma and weak Cerami-Palais-Smale condition.

Abstract

The aim of this paper is investigating the existence of one or more critical points of a family of functionals which generalizes the model problem \[ \bar J(u)\ =\ \frac1p\ \int_\Omega \bar A(x,u)|\nabla u|^p dx - \int_\Omega G(x,u) dx \] in the Banach space $X = W^{1,p}_0(\Omega)\cap L^\infty(\Omega)$, where $\Omega \subset {\mathbb R}^N$ is an open bounded domain, $1 < p < N$ and the real terms $\bar A(x,t)$ and $G(x,t)$ are $C^1$ Carath\'eodory functions on $\Omega \times {\mathbb R}$. We prove that, even if the coefficient $\bar A(x,t)$ makes the variational approach more difficult, if it satisfies ``good'' growth assumptions then at least one critical point exists also when the nonlinear term $G(x,t)$ has a suitable supercritical growth. Moreover, if the functional is even, it has infinitely many critical levels. The proof, which exploits the interaction between two different norms on $X$, is based on a weak version of the Cerami-Palais-Smale condition and a suitable intersection lemma which allow us to use a Mountain Pass Theorem.