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Zbl 0635.35036
Benci, Vieri; Cerami, Giovanna
Positive solutions of some nonlinear elliptic problems in exterior domains.
(English)
[J] Arch. Ration. Mech. Anal. 99, 283-300 (1987). ISSN 0003-9527; ISSN 1432-0673/e

Consider the question of the existence of positive solutions u in $H\sp 1\sb 0(\Omega)$ of (*): $-\Delta u+\lambda u=\vert u\vert\sp{p-2} u$ in $\Omega$, where $\Omega \subset R\sp N$ is an unbounded domain, $\partial \Omega \ne \phi$ is bounded, $\lambda \in R\sb+$, $N\ge 3$, and $2\le p<2N/(N-2).$ After a nice discussion of recent results and the difficulty encountered when $\Omega$ is unbounded - a lack of compactness of the embedding $J: H\sp 1\sb 0(\Omega)\to L\sp p(\Omega)$- the authors examine the obstruction to the compactness and obtain some estimates of the energy levels where the Palais-Smale condition can fail. This enables them to prove the following two theorems. \par Theorem A: If $p<(2N-2)/(N-2)$ for $N=3,4$ and $p=1+8/N$ for $4<N<8$, then there exists a $\lambda\sb c(\Omega)$ such that for $\lambda \in (0,\lambda\sb c)$, problem (*) has at least one positive solution. Theorem B: Let N and p be as in Theorem A and $x\sb 0\in R\sp n-\Omega$. Then for all $\lambda$ there is a $\rho$ ($\lambda)$ such that if $R\sp N-\Omega \subset B\sb{\rho}(x\sb 0)=\{x\in R\sp N:\vert x-x\sb 0\vert \le \rho \}$, problem (*) has at least one positive solution. The conditions on N and p are a consequence of {\it K. McLeod} and {\it J. Serrin} [Proc. Natl. Acad. Sci. USA 78, 6592-6595 (1981; Zbl 0474.35047)] and Theorem B shows that the geometry of $\Omega$ plays a role in the existence question.
[P.W.Schaefer]
MSC 2000:
*35J65 (Nonlinear) BVP for (non)linear elliptic equations
35J20 Second order elliptic equations, variational methods
35A05 General existence and uniqueness theorems (PDE)
35B05 General behavior of solutions of PDE

Keywords: exterior domains; existence; positive solutions; unbounded domain; compactness; Palais-Smale condition

Citations: Zbl 0474.35047

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