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Singular elliptic problems with unbalanced growth and critical exponent
Nonlinearity ( IF 1.7 ) Pub Date : 2020-05-27 , DOI: 10.1088/1361-6544/ab81ed
Deepak Kumar 1 , Vicenţiu D Rădulescu 2, 3, 4 , K Sreenadh 1
Affiliation  

In this article, we study the existence and multiplicity of solutions of the following $(p,q)$-Laplace equation with singular nonlinearity: \begin{equation*} \left\{\begin{array}{rllll} -\Delta_{p}u-\ba\Delta_{q}u & = \la u^{-\de}+ u^{r-1}, \ u>0, \ \text{ in } \Om \\ u&=0 \quad \text{ on } \pa\Om, \end{array} \right. \end{equation*} where $\Om$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary, $1 p$ and $\la,\, \ba>0$ are parameters. We prove existence, multiplicity and regularity of weak solutions of $(P_\la)$ for suitable range of $\la$. We also prove the global existence result for problem $(P_\la)$.

中文翻译:

具有不平衡增长和临界指数的奇异椭圆问题

在本文中,我们研究以下具有奇异非线性的 $(p,q)$-Laplace 方程解的存在性和多重性: \begin{equation*} \left\{\begin{array}{rllll} -\Delta_ {p}u-\ba\Delta_{q}u & = \la u^{-\de}+ u^{r-1}, \ u>0, \ \text{ in } \Om \\ u&= 0 \quad \text{ on } \pa\Om, \end{array} \right。\end{equation*} 其中 $\Om$ 是 $\mathbb{R}^n$ 中边界平滑的有界域,$1 p$ 和 $\la,\, \ba>0$ 是参数。我们证明了$(P_\la)$ 弱解在$\la$ 的合适范围内的存在性、多重性和规律性。我们也证明了问题$(P_\la)$的全局存在结果。
更新日期:2020-05-27
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