## 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

### MSC:

 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
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