Multiplicity solutions for fully nonlinear equationinvolving nonlinearity with zeros

Xiaohui Yu

doi:10.3934/cpaa.2013.12.451

Article Contents

2013, Volume 12, Issue 1: 451-459. Doi: 10.3934/cpaa.2013.12.451

This issue Previous Article Existence and multiplicity of semiclassical states for a quasilinear Schrödinger equation in $\mathbb{R}^N$ Next Article Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\mathbb{R}^3$

Multiplicity solutions for fully nonlinear equation involving nonlinearity with zeros

Xiaohui Yu^1,

1.
Institute for Advanced Study, Shenzhen University, Shenzhen Guangdong, 518060

Received: April 30, 2011

Revised: November 30, 2011

Published: December 2012

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Abstract

We study the multiplicity solutions for the nonlinear elliptic equation

$ -\mathcal{M}_{\lambda,\Lambda}^+ (D^2u)=f(u) $ in $\Omega$,

$ u=0 $ on $\partial \Omega$

and a more general fully nonlinear elliptic equation

$ F(D^2u)=f(u) $ in $\Omega$,

$ u=0 $ on $\partial \Omega$,

where $\Omega$ is a bounded domain in $\mathbb{R}^N, N\geq 3$, $f$ is a locally Lipschitz continuous function with superlinear growth at infinity. We will show that the equation has at least two positive solutions under some assumptions.

Keywords:

Mathematics Subject Classification: Primary: 35J60, 35J65; Secondary: 35B45.

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References

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