Multiple positive solutions for a quasilinear nonhomogeneous Neumann problems with critical Hardy exponents. (English) Zbl 1131.35030

Deng, Yinbin; Jin, Lingyu

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 67, No. 12, 3261-3275 (2007).

Summary: We study the existence, nonexistence and multiplicity of positive solutions for nonhomogeneous Neumann boundary value problems of the type
\[ \begin{aligned} -\Delta_pu+\lambda u^{p-1}= \frac{u^q}{|x|^s}, &\quad x\in\Omega,\\ u>0, &\quad x\in\Omega,\\ |\nabla u|^{p-2}D_\gamma u=\varphi, &\quad x\in\partial\Omega \setminus\{0\}, \end{aligned} \]
where \(\Omega\) is a bounded domain in \(\mathbb R^n\) with smooth boundary, \(0\in\partial\Omega\), \(2\leq p<n\). \(\Delta_pu= \text{div}(|\nabla u|^{p-2}\nabla u)\) is the \(p\)-Laplacian operator, \(p-1<q\leq p^*(s)-1\), \(0\leq s<p-1\), \(p^*(s)= \frac{(n-s)p}{n-p}\), \(\varphi\in C^\alpha (\overline{\Omega})\), \(0<\alpha<1\), \(\varphi(x)\geq 0\), \(\varphi(x)\not\equiv 0\) and \(\lambda\) is a real constant.

MSC:

35J65	Nonlinear boundary value problems for linear elliptic equations
35B50	Maximum principles in context of PDEs

Keywords:

Neumann boundary value problems; positive solutions; \(p\)-Laplacian; Hardy equality; quasilinear elliptic problems

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