On quasi-linear equation problems involving critical and singular nonlinearities
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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.
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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
MSC
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