Existence of two positive solutions for a class of semilinear elliptic equations with singularity and critical exponent
Volume 116 / 2016
Annales Polonici Mathematici 116 (2016), 273-292
MSC: Primary 35J15; Secondary 35A15, 35B09.
DOI: 10.4064/ap3606-10-2015
Published online: 16 March 2016
Abstract
We study the following singular elliptic equation with critical exponent $$ \begin{cases} -\varDelta u=Q(x)u^{2^{*}-1}+\lambda u^{-\gamma}&\text{in } \varOmega, \\ u \gt 0 &\text{in } \varOmega, \\ u=0 &\text{on } \partial\varOmega, \end{cases} $$ where $\varOmega\subset\mathbb{R}^{N}$ $(N\geq3)$ is a smooth bounded domain, and $\lambda \gt 0$, $\gamma\in(0,1)$ are real parameters. Under appropriate assumptions on $Q,$ by the constrained minimizer and perturbation methods, we obtain two positive solutions for all $\lambda \gt 0$ small enough.
