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

Nguyen Lam; Guozhen Lu

doi:10.3934/dcds.2012.32.2187

Article Contents

2012, Volume 32, Issue 6: 2187-2205. Doi: 10.3934/dcds.2012.32.2187

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

Nguyen Lam^1, and
Guozhen Lu^1,

1.
Department of Mathematics, Wayne State University, Detroit, MI 48202

Received: March 31, 2011

Revised: May 31, 2011

Published: May 2012

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Abstract

The main purpose of this paper is to establish the existence of nontrivial solutions to semilinear polyharmonic equations with exponential growth at the subcritical or critical level. This growth condition is motivated by the Adams inequality [1] of Moser-Trudinger type. More precisely, we consider the semilinear elliptic equation \[ \left( -\Delta\right) ^{m}u=f(x,u), \] subject to the Dirichlet boundary condition $u=\nabla u=...=\nabla^{m-1}u=0$, on the 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 the well-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]).

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Mathematics Subject Classification: Primary: 35J91, 35J30.

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