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

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
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### References:

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