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2017 On nonhomogeneous elliptic problems involving the Hardy potential and critical Sobolev exponent
Jing Zhang, Shiwang Ma
Rocky Mountain J. Math. 47(2): 687-710 (2017). DOI: 10.1216/RMJ-2017-47-2-687

Abstract

In this paper, we are concerned with elliptic equations with Hardy potential and critical Sobolev exponents where $2^{*}={2N}/({N-2})$ is the critical Sobolev exponent, $N\geq 3$, $0\leq \mu \lt \overline {\mu }={(N-2)^2}/{4}$, $\mathbf {\Omega }\subset \mathbb {R}^{N}$ an open bounded set. For $\lambda \in [0,\lambda _{1})$ with $\lambda _{1}$ being the first eigenvalue of the operator $-\Delta -{\mu }/{|x|^{2}}$ with zero Dirichlet boundary condition, and for $f\in H_{0}^{1}(\mathbf {\Omega })^{-1}=H^{-1}$, $f\neq 0$, we show that (\ref {eq1}) admits at least two distinct nontrivial solutions $u_{0}$ and $u_{1}$ in $H_{0}^{1}(\mathbf {\Omega })$. Furthermore, $u_{0}\geq 0$ and $u_{1}\geq 0$ whenever $f\geq 0$.

Citation

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Jing Zhang. Shiwang Ma. "On nonhomogeneous elliptic problems involving the Hardy potential and critical Sobolev exponent." Rocky Mountain J. Math. 47 (2) 687 - 710, 2017. https://doi.org/10.1216/RMJ-2017-47-2-687

Information

Published: 2017
First available in Project Euclid: 18 April 2017

zbMATH: 1376.35061
MathSciNet: MR3635381
Digital Object Identifier: 10.1216/RMJ-2017-47-2-687

Subjects:
Primary: 35B33
Secondary: 35B09, 35B20

Rights: Copyright © 2017 Rocky Mountain Mathematics Consortium

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Vol.47 • No. 2 • 2017
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