# On quasi-linear equation problems involving critical and singular nonlinearities

Volume 10, Issue 9, pp 4966--4982
Publication Date: September 23, 2017 Submission Date: June 25, 2017
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### Authors

Yanbin Sang - Department of Mathematics, School of Science, North University of China, Taiyuan, Shanxi 030051, China. Xiaorong Luo - Department of Mathematics, School of Science, North University of China, Taiyuan, Shanxi 030051, China.

### Abstract

We consider the singular boundary value problem $\left\{\begin{array}{l} -\text{div}(|x|^{-ap}|\nabla u|^{p-2}\nabla u)=h(x)\frac{u^{-\gamma}}{|x|^{b(1-\gamma)}}+\mu \frac{u^{p^*-1}}{|x|^{bp^*}} \ \ \ \text{in} \ \Omega\backslash\{0\}, \cr u>0\ \ \ \ \ \text{in}\ \Omega\backslash\{0\}, \cr u=0\ \ \ \ \ \text{on} \ \partial\Omega, \end{array}\right.$ where $\Omega\subset \mathbb{R}^N(N\geq 3)$ is a bounded domain such that $0\in \Omega$, $0<\gamma<1$, $0\leq a<\frac{N-p}{p}$, $a\leq b<a+1$, $p^* :=p^* (a,b)=\frac{Np}{N-(1+a-b)p}$, and $h(x)$ is a given function. Based on different assumptions, using variational methods and Ekeland's principle, we admit that this problem possesses two positive solutions. keywords

### Share and Cite

##### ISRP Style

Yanbin Sang, Xiaorong Luo, On quasi-linear equation problems involving critical and singular nonlinearities, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 9, 4966--4982

##### AMA Style

Sang Yanbin, Luo Xiaorong, On quasi-linear equation problems involving critical and singular nonlinearities. J. Nonlinear Sci. Appl. (2017); 10(9):4966--4982

##### Chicago/Turabian Style

Sang, Yanbin, Luo, Xiaorong. "On quasi-linear equation problems involving critical and singular nonlinearities." Journal of Nonlinear Sciences and Applications, 10, no. 9 (2017): 4966--4982

### Keywords

• Critical exponent
• $p$-Laplacian operator
• extremal value
• singular nonlinearity

•  5A15
•  35B33
•  35J62

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