A class of Schrödinger elliptic equations involving supercritical exponential growth

Abstract

This paper studies the existence of nontrivial solutions to the following class of Schrödinger equations: $$ \textstyle\begin{cases} -\operatorname{div}(w(x)\nabla u) = f(x,u),&\ x \in B_{1}(0), \\ u = 0,&\ x \in \partial B_{1}(0), \end{cases} $$ { − div ( w ( x ) ∇ u ) = f ( x , u ) , x ∈ B 1 ( 0 ) , u = 0 , x ∈ ∂ B 1 ( 0 ) , where $w(x)= (\ln (1/|x|) )^{\beta}$ w ( x ) = ( ln ( 1 / | x | ) ) β for some $\beta \in [0,1)$ β ∈ [ 0 , 1 ) , the nonlinearity $f(x,s)$ f ( x , s ) behaves like ${\exp} (|s|^{\frac{2}{1-\beta}+h(|x|)} )$ exp ( | s | 2 1 − β + h ( | x | ) ) , and h is a continuous radial function such that $h(r)$ h ( r ) can be unbounded as r tends to 1. Our approach is based on a new Trudinger–Moser-type inequality for weighted Sobolev spaces and variational methods.