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 \(H^ 1_ 0(\Omega)\) of (*): \(-\Delta u+\lambda u=| u|^{p-2} u\) in \(\Omega\), where \(\Omega \subset R^ N\) is an unbounded domain, \(\partial \Omega \neq \phi\) is bounded, \(\lambda \in R_+\), \(N\geq 3\), and \(2\leq p<2N/(N-2).\) After a nice discussion of recent results and the difficulty encountered when \(\Omega\) is unbounded - a lack of compactness of the embedding \(J: H^ 1_ 0(\Omega)\to L^ p(\Omega)\)- 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 \(\lambda_ c(\Omega)\) such that for \(\lambda \in (0,\lambda_ c)\), problem (*) has at least one positive solution. Theorem B: Let N and p be as in Theorem A and \(x_ 0\in R^ n-\Omega\). Then for all \(\lambda\) there is a \(\rho\) (\(\lambda)\) such that if \(R^ N-\Omega \subset B_{\rho}(x_ 0)=\{x\in R^ N:| x-x_ 0| \leq \rho \}\), problem (*) has at least one positive solution. The conditions on N and p are a consequence of K. McLeod and 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


35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs


Zbl 0474.35047
Full Text: DOI


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