Nontrivial solutions for semilinear elliptic systems via Orlicz-Sobolev theory

TL;DR

This paper establishes the existence of nontrivial solutions for semilinear elliptic systems with N-function growth by extending subcritical growth concepts to Orlicz-Sobolev spaces, without requiring the Ambrosetti-Rabinowitz condition.

Contribution

It introduces a novel approach using Orlicz-Sobolev spaces to handle superlinear elliptic systems with N-function growth, relaxing traditional growth conditions.

Findings

Existence of nontrivial solutions under N-function growth

Extension of subcritical growth to Orlicz-Sobolev spaces

Solutions obtained without Ambrosetti-Rabinowitz condition

Abstract

In this paper, the semilinear elliptic systems with Dirichlet boundary value are considered \begin{align} \left\{\begin{array}{ll} -\Delta v=f(u) & \mathrm{in}\ \Omega, -\Delta u=g(v) & \mathrm{in}\ \Omega, u=0, \ v=0 & \mathrm{on}\ \partial\Omega, \end{array} \right. \end{align} We extend the notion of subcritical growth from polynomial growth to N-function growth. Under N-function growth, nontrivial solutions are obtained via Orlicz-Sobolev spaces and variational methods. It's also noteworthy that the nonlinear term $g(v)$ does not have to satisfy the usual Ambrosetti-Rabinowitz condition. So, in a sense, we enrich recent results of D. ~G. de Figueiredo, J. ~M. do {\'O} and B. ~Ruf [D. ~G. de Figueiredo, J.M. do {\'O}, B. ~Ruf, An {O}rlicz-space approach to superlinear elliptic systems, J. Funct. Anal. 224 (2005) 471--496].