Existence of positive solutions for a semilinear elliptic problem with critical Sobolev and Hardy terms

Hirano, Norimichi; Shioji, Naoki

doi:10.1090/S0002-9939-06-08405-X

Existence of positive solutions for a semilinear elliptic problem with critical Sobolev and Hardy terms
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by Norimichi Hirano and Naoki Shioji PDF

Proc. Amer. Math. Soc. 134 (2006), 2585-2592 Request permission

Abstract:

Let $N\geq 4$, let $2^{\ast }=2N/(N-2)$ and let $\Omega \subset \mathbb {R}^{N}$ be a bounded domain with a smooth boundary $\partial \Omega$. Our purpose in this paper is to consider the existence of solutions of the problem: \[ \left \{ \begin {aligned} -\Delta u - \mu \frac {u}{\vert x\vert ^2} &= \vert u\vert ^{2^\ast -1} && \text {in $\Omega $}, \\ u & > 0 && \text {in $\Omega $},\\ u & = 0 && \text {on $\partial \Omega $}, \end {aligned} \right . \] where $0<\mu <(\frac {N-2}{2})^{2}.$

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