A biharmonic equation in $\mathbb{R}^4$ involving nonlinearities with critical exponential growth

Federica Sani

doi:10.3934/cpaa.2013.12.405

Article Contents

2013, Volume 12, Issue 1: 405-428. Doi: 10.3934/cpaa.2013.12.405

This issue Previous Article Global existence and nonexistence for the viscoelastic wave equation with nonlinear boundary damping-source interaction Next Article Existence and multiplicity of semiclassical states for a quasilinear Schrödinger equation in $\mathbb{R}^N$

A biharmonic equation in $\mathbb{R}^4$ involving nonlinearities with critical exponential growth

Federica Sani^1,

1.
Dipartimento di Matematica Federigo Enriques, Università degli Studi di Milano

Received: May 2011

Revised: December 2011

Published: January 2013

Abstract / Introduction Related Papers Cited by

Abstract

In this paper we give sufficient conditions for the existence of solutions of a biharmonic equation of the form

$ \Delta^2 u + V(x)u = f(u)$ in $\mathbb{R}^4$

where $V$ is a continuous positive potential bounded away from zero and the nonlinearity $f(s)$ behaves like $e^{\alpha_0 s^2}$ at infinity for some $\alpha_0>0$.
In order to overcome the lack of compactness due to the unboundedness of the domain $\mathbb{R}^4$, we require some additional assumptions on $V$. In the case when the potential $V$ is large at infinity we obtain the existence of a nontrivial solution, while requiring the potential $V$ to be spherically symmetric we obtain the existence of a nontrivial radial solution. In both cases, the main difficulty is the loss of compactness due to the critical exponential growth of the nonlinear term $f$.

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Mathematics Subject Classification: Primary: 35J30.

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