Solutions to nonlinear Neumann problems with an inverse square potential

Abstract

Let Ω be an open bounded domain in $\mathbb{R}^N (N\geq3)$ with smooth boundary $\partial\Omega, 0\in\partial\Omega$. We are concerned with the critical Neumann problem

$$\left\{ \begin{array}{ll} -\Delta{u}-\mu\frac{u}{|x|^2}+\lambda u=Q(x)|u|^{2^*-2}{u} \,\,& \quad \mbox{in}\,\,\Omega,\\ \frac{\partial u}{\partial \nu}=0\,\,&\quad \mbox{on}\,\,\partial\Omega, \end{array} \right. (*) $$

where $0 < \mu < \bar{\mu}=(\frac{N-2}{2})^2,\,\,2^*=\frac{2N}{N-2},\,\,\,\,\lambda > 0$ and Q(x) is a positive continuous function on $\overline{\Omega}$. Using Moser iteration, we give an asymptotic characterization of solutions for (*) at the origin. Under some conditions on Q,  μ, we, by means of a variational method, prove that there exists $\lambda_0=\lambda_0(\mu) > 0$ such that for every $\lambda > \lambda_0$, problem (*) has a positive solution and a pair of sign-changing solutions.