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A Palais-Smale approach to problems in Esteban-Lions domains with holes. (English) Zbl 0951.35043
In this paper, the author considers the problem of finding positive solutions \(u>0\) to the equation (1) \(-\delta u+u= u^{p-1}\) in a domain \(\Omega \subset {\mathbb R}^N\) where \(2<p<2N/(N-1)\). It is known that (1) admits positive solutions when \(\Omega\) is bounded or when \(\Omega={\mathbb R}^N\). When \(\Omega\) is an unbounded proper domain, a non-existence result in (what the author calls) “Esteban-Lions domains” was proved by Esteban and Lions in [Proc. R. Soc. Edinb., Sect. A 93, 1-14 (1982; Zbl 0506.35035)]. Similarly to the famous non-existence result of Pohozaev in starshaped domains for a class of elliptic problems, both the topology and the geometry of the domain seem to influence strongly the existence of solutions of (1). For example, in [Differ. Integral Equ. 6, No. 6, 1281-1298 (1993; Zbl 0837.35051)], it is proved that when adding a ball to the Esteban-Lions domain, the existence of a ground state solution may be obtained. In this paper, the author asserts that, although (1) admits no ground state solution, a higher energy solution indeed exists in various Esteban-Lions domains with a hole, i.e. with a ball taken out from. This is done by a close study of the asymptotic behaviour of possible positive solutions to (1).
Dynamic systems of solutions of (1) are also studied and multiple solutions of (1) in the presence of a forcing term are examined.

35J60 Nonlinear elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
35A15 Variational methods applied to PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35J20 Variational methods for second-order elliptic equations
Full Text: DOI
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