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2014 Multiplicity of Positive Solutions for a $p$-$q$-Laplacian Type Equation with Critical Nonlinearities
Tsing-San Hsu, Huei-Li Lin
Abstr. Appl. Anal. 2014(none): 1-9 (2014). DOI: 10.1155/2014/829069

## Abstract

We study the effect of the coefficient $f(x)$ of the critical nonlinearity on the number of positive solutions for a $p$-$q$-Laplacian equation. Under suitable assumptions for $f(x)$ and $g(x)$, we should prove that for sufficiently small $\lambda >0$, there exist at least $k$ positive solutions of the following $p$-$q$-Laplacian equation, $-{\mathrm{\Delta }}_{p}u-{\mathrm{\Delta }}_{q}u=f(x)|u{|}^{{p}^{\ast}-2}u+\lambda g(x)|u{|}^{r-2}u\text{\hspace\{0.17em\}\hspace\{0.17em\}in\hspace\{0.17em\}\hspace\{0.17em\}}\mathrm{\Omega }$, $u=0\text{\hspace\{0.17em\}\hspace\{0.17em\}on\hspace\{0.17em\}\hspace\{0.17em\}}\partial \mathrm{\Omega ,}$ where $\mathrm{\Omega }\subset {\mathbf{R}}^{N}$ is a bounded smooth domain, $N>p$, $1, $p^{\ast}=Np/(N-p)$ is the critical Sobolev exponent, and ${\Delta }_{s}u=\text{d}\text{i}\text{v}(|\nabla u{|}^{s-2}\nabla u$ is the $s$-Laplacian of $u$.

## Citation

Tsing-San Hsu. Huei-Li Lin. "Multiplicity of Positive Solutions for a $p$-$q$-Laplacian Type Equation with Critical Nonlinearities." Abstr. Appl. Anal. 2014 1 - 9, 2014. https://doi.org/10.1155/2014/829069

## Information

Published: 2014
First available in Project Euclid: 2 October 2014

zbMATH: 07023153
MathSciNet: MR3193552
Digital Object Identifier: 10.1155/2014/829069  