Multiplicity of Positive Solutions for a p-q-Laplacian Type Equation with Critical Nonlinearities

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, $-{\mathrm{\Delta }}_{p}u-{\mathrm{\Delta }}_{q}u=f\left(x\right)\left|u{|}^{p^{*}-2}u+\lambda g\left(x\right)\right|u{|}^{r-2}u\text{in}\mathrm{\Omega }$, $u=0\text{on}\partial \mathrm{\Omega ,}$ where $\mathrm{\Omega }\subset {\mathbf{R}}^N$ is a bounded smooth domain, $N>p$, , $p^*=Np/(N-p)$ 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 $u$.