Some results for a class of quasilinear elliptic equations with singular nonlinearity

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• 作者：Joã ; o Marcos do Ó ; ^a ; ^{jmbo@pq.cnpq.br" class="auth_mail" title="E-mail the corresponding author} ; Esteban da Silva^b ; ^{esteban@dmat.ufpe.br" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35J62 ; 35J70 ; 34B15 ; 35B35
• 刊名：Nonlinear Analysis
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：148
• 期：Complete
• 页码：1-29
• 全文大小：750 K

文摘

In this paper, motivated by recent works on the study of the equations which model the electrostatic MEMS devices, we study the quasilinear elliptic equation involving a singular nonlinearity According to the choice of the parameters $\alpha ,\beta$ and γ$\gamma$, the differential operator which we are dealing with corresponds to the radial form of the Laplacian, the p$p$-Laplacian and the k$k$-Hessian. In this work we present conditions over which we can assert regularity for solutions, including the case λ=λ^∗$\lambda ={\lambda }^{\ast }$, where λ^∗${\lambda }^{\ast }$ is a critical value for the existence of solutions. Moreover, we prove that whenever the critical solution is regular, there exists another solution of mountain pass type for λ$\lambda$ close to the critical one. In addition, we use the Shooting Method to prove uniqueness of solutions for λ$\lambda$ in a neighborhood of 0$0$.