Ground state solutions for a semilinear elliptic problem with critical-subcritical growth

TL;DR

This paper investigates positive ground state solutions for a semilinear elliptic problem with variable exponent growth, focusing on existence and properties when the nonlinearity exhibits critical and subcritical behavior.

Contribution

It establishes existence results for ground state solutions in unbounded and bounded domains with variable exponent nonlinearities.

Findings

Existence of ground state solutions in unbounded domains.

Existence of ground state solutions in bounded domains.

Analysis of solutions with critical and subcritical growth conditions.

Abstract

In this work, we study the of positive ground state solution for the semilinear elliptic problem $$ \left\{ \begin{array} [c]{ll}% -\Delta u=u^{p(x)-1},\quad u>0 & \mathrm{in}\,G\subseteq\mathbb{R}^{N}% ,\,N\geq3\\ u\in D_{0}^{1,2}(G), & \end{array} \right. $$ where $G$ is either $\mathbb{R}^{N}$ or a bounded domain, and $p:G\rightarrow \mathbb{R}$ is a continuous function assuming critical and subcritical values.