Solutions to nonlinear Neumann problems with an inverse square potential

Han, Pigong; Liu, Zhaoxia
November 2007
Calculus of Variations & Partial Differential Equations;Nov2007, Vol. 30 Issue 3, p315
Academic Journal
Let O 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 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, �, 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.


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