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Solutions to nonlinear Neumann problems with an inverse square potential

Authors:

Institution:

(1) Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100080, People’s Republic of China;(2) School of Mathematics and Computer Science, Central University for Nationalities, Beijing, 100081, People’s Republic of China

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.

Keywords:

Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) 35J65 35J25 47J30

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