Advances in Differential Equations

Multiplicity of positive solutions for a class of nonlinear Schrödinger equations

Claudianor O. Alves and Sérgio H. M. Soares

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This paper proves the multiplicity of positive solutions for the following class of quasilinear problems: $$ \begin{cases} -\epsilon^{p}\Delta_{p}{u}+(\lambda A(x)+1)|u|^{p-2}u=f(u), \,\,\, \mathbb{R}^{N}\\ u(x)>0 \,\,\, \mbox{in} \,\, \mathbb{R}^{N}, \end{cases} $$ where $\Delta_{p}$ is the p-Laplacian operator, $ N >p \geq 2$, $\lambda$ and $\epsilon$ are positive parameters, $A$ is a nonnegative continuous function and $f$ is a continuous function with subcritical growth. Here, we use variational methods to get multiplicity of positive solutions involving the Lusternick-Schnirelman category of ${\rm{int}}A^{-1}(0)$ for all sufficiently large $\lambda$ and small~$\epsilon$.

Article information

Adv. Differential Equations Volume 15, Number 11/12 (2010), 1083-1102.

First available in Project Euclid: 18 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Primary: 35A15, 35H30, 35Q55


Alves, Claudianor O.; Soares, Sérgio H. M. Multiplicity of positive solutions for a class of nonlinear Schrödinger equations. Adv. Differential Equations 15 (2010), no. 11/12, 1083--1102.

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