Existence of nontrivial solutions to Polyharmonic equations with subcritical and critical exponential growth

Abstract

The main purpose of this paper is to establishthe existence of nontrivial solutions to semilinear polyharmonicequations with exponential growth at the subcritical or critical level. This growth condition is motivated by the Adamsinequality [1] of Moser-Trudinger type. More precisely, we consider thesemilinear elliptic equation\[\left( -\Delta\right) ^{m}u=f(x,u),\]subject to the Dirichlet boundary condition $u=\nabla u=...=\nabla^{m-1}u=0$, onthe bounded domains $\Omega\subset\mathbb{R}^{2m}$ when the nonlinear term $f$ satisfies exponential growth condition. We will study the above problem both in the case when $f$ satisfies thewell-known Ambrosetti-Rabinowitz condition and in the case without the Ambrosetti-Rabinowitz condition.This is one of a series of works by the authors on nonlinear equations of Laplacian in $\mathbb{R}^2$ and $N-$Laplacian in $\mathbb{R}^N$ when the nonlinear term has the exponential growth and with a possible lack of the Ambrosetti-Rabinowitz condition (see [23], [24]).