Semilinear elliptic equations with Hardy potential and subcritical source term

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Abstract

Let \(\Omega \) be a smooth bounded domain in \({\mathbb {R}}^N\) (\(N>2\)) and \(\delta (x):=\text {dist}\,(x,\partial \Omega )\). Assume \(\mu \in {\mathbb {R}}_+, \nu \) is a nonnegative finite measure on \(\partial \Omega \) and \(g \in C(\Omega \times {\mathbb {R}}_+)\). We study positive solutions of
$$\begin{aligned} -\Delta u - \frac{\mu }{\delta ^2} u = g(x,u) \text { in } \Omega , \qquad \text {tr}^*(u)=\nu . \end{aligned}$$
(P)
Here \(\text {tr}^*(u)\) denotes the normalized boundary trace of u which was recently introduced by Marcus and Nguyen (Ann Inst H Poincaré Anal Non Linéaire, 34, 69–88, 2017). We focus on the case \(0<\mu < C_H(\Omega )\) (the Hardy constant for \(\Omega \)) and provide qualitative properties of positive solutions of (P). When \(g(x,u)=u^q\) with \(q>0\), we prove that there is a critical value \(q^*\) (depending only on \(N, \mu \)) for (P) in the sense that if \(q<q^*\) then (P) possesses a solution under a smallness assumption on \(\nu \), but if \(q \ge q^*\) this problem admits no solution with isolated boundary singularity. Existence result is then extended to a more general setting where g is subcritical [see (1.28)]. We also investigate the case where g is linear or sublinear and give an existence result for (P).

Mathematics Subject Classification

35J60 35J75 35J10 
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Notes

Acknowledgements

This research was supported by Fondecyt Grant 3160207. The author is grateful to Dr. Q. H. Nguyen for many useful discussions. The author would like to thank the anonymous referee for a careful reading of the manuscript and helpful comments.

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© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Departamento de MatemáticasPontificia Universidad Católica de ChileSantiagoChile

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