Existence of multiple positive weak solutions and estimates for extremal values for a class of concave-convex elliptic problems with an inverse-square potential

Yaoping Chen; Jianqing Chen

doi:10.3934/cpaa.2017073

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2017, Volume 16, Issue 5: 1531-1552. Doi: 10.3934/cpaa.2017073

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Existence of multiple positive weak solutions and estimates for extremal values for a class of concave-convex elliptic problems with an inverse-square potential

Yaoping Chen^1,2, and
Jianqing Chen^2,

1.
Concord University College, Fujian Normal University, Fuzhou, 350117, China
2.
College of Mathematics and Computer Science, Fujian Normal University, Fuzhou, 350108, China

Received: September 30, 2014

Revised: January 31, 2017

Published: October 2017

This work is supported by NSF of China (No. 11371091) and the innovation group of `Nonlinear analysis and its applications' (No. 021337120).

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Abstract

In this paper, variational methods are used to establish some existence and multiplicity results and provide uniform estimates of extremal values for a class of elliptic equations of the form:

$-Δ u - {{λ}\over{|x|^2}}u = h(x) u^q + μ W(x) u^p,\ \ x∈Ω\backslash\{0\}$

with Dirichlet boundary conditions, where $0∈ Ω\subset\mathbb{R}^N $($N≥q 3 $) be a bounded domain with smooth boundary $\partial Ω $, $μ>0 $ is a parameter, $0 < λ < Λ={{(N-2)^2}\over{4}}$, $0 < q < 1 < p < 2^*-1 $, $h(x)>0 $ and $W(x) $ is a given function with the set $\{x∈ Ω: W(x)>0\} $ of positive measure.

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Mathematics Subject Classification: Primary: 35J20, 35J60.

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References

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