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Positive solutions of some nonlinear elliptic problems in exterior domains. (English) Zbl 0635.35036
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. 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.
Reviewer: P.W.Schaefer

35J65Nonlinear boundary value problems for linear elliptic equations
35J20Second order elliptic equations, variational methods
35A05General existence and uniqueness theorems (PDE) (MSC2000)
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
Full Text: DOI
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