Existence of two positive solutions for a class of semilinear elliptic equations with singularity and critical exponent

Jia-Feng Liao¹ ; Jiu Liu² ; Peng Zhang³ ; Chun-Lei Tang²

¹ School of Mathematics and Statistics Southwest University 400715 Chongqing People’s Republic of China and School of Mathematics and Computational Science Zunyi Normal College 563002 Zunyi, Guizhou People’s Republic of China
² School of Mathematics and Statistics Southwest University 400715 Chongqing People’s Republic of China
³ School of Mathematics and Computational Science Zunyi Normal College 563002 Zunyi, Guizhou People’s Republic of China

Annales Polonici Mathematici, Tome 116 (2016) no. 3, pp. 273-292.

Résumé

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.

Détail

DOI : 10.4064/ap3606-10-2015

Keywords: study following singular elliptic equation critical exponent begin cases vardelta u * lambda gamma text varomega text varomega text partial varomega end cases where varomega subset mathbb geq smooth bounded domain lambda gamma real parameters under appropriate assumptions constrained minimizer perturbation methods obtain positive solutions lambda small enough