Existence and nonexistence of solutions for a singular elliptic problem with a nonlinear boundary condition
Volume 109 / 2013
Abstract
We consider the existence and nonexistence of solutions for the following singular quasi-linear elliptic problem with concave and convex nonlinearities: $$ \left\{ \begin{array}{@{}l} -\,\mathrm{div}(|x|^{-ap}|\nabla u|^{p-2}\nabla u)+h(x)|u|^{p-2}u=g(x)|u|^{r-2}u,\quad x\in\varOmega,\\ |x|^{-ap}|\nabla u|^{p-2}\frac{\partial u}{\partial \nu}=\lambda f(x)|u|^{q-2}u, \quad x \in \partial\varOmega, \end{array} \right. $$ where $\varOmega$ is an exterior domain in $\mathbb{R}^N$, that is, $\varOmega={\mathbb {R}^N} \setminus D $, where $D$ is a bounded domain in $ \mathbb {R}^N $ with smooth boundary $\partial D$ $(=\partial\varOmega)$, and $0\in \varOmega.$ Here $\lambda>0$, $0\le a < (N-p)/p$, $1< p< N $, ${\partial}/{\partial\nu}$ is the outward normal derivative on $\partial\varOmega$. By the variational method, we prove the existence of multiple solutions. By the test function method, we give a sufficient condition under which the problem has no nontrivial nonnegative solutions.
