Supercritical elliptic problems on domains of \(\mathbb{R}^ N\). (English) Zbl 0806.35036

Isselkou, O. A.

Chipot, M. (ed.), Progress in partial differential equations: the Metz surveys 2. Proceedings of the conferences given at the University of Metz (France) during the 1992 ”Metz Days”. Harlow: Longman Scientific & Technical. Pitman Res. Notes Math. Ser. 296, 172-184 (1993).

The paper concerns semilinear second order elliptic equations with supercritical growth on domains \(\Omega\) in \(\mathbb{R}^ N\) \((N \geq 3)\). The boundary conditions are \(u = \varphi>0\) on \(\partial \Omega\) (if \(\Omega \neq \mathbb{R}^ N)\) resp. \(u(x) \to \ell\) for \(\| x \| \to \infty\) (if \(\Omega = \mathbb{R}^ N)\).
If \(\Omega\) is bounded, various existence results are proved for small and for bounded boundary data \(\varphi\). If \(\Omega\) is an exterior domain, then for small \(\varphi\) there exists a solution which behaves as \(\| x \|^{N - 2}\) at infinity. Finally, in the case \(\Omega = \mathbb{R}^ N\) the author shows: If \(u\) is a supersolution then \(\liminf_{\| x \| \to \infty} u(x) = 0\), and if \(u\) is a radial supersolution then \(\lim_{r \to \infty} u(r) = 0\). Hence, \(\ell > 0\) implies nonexistence.
For the entire collection see [Zbl 0785.00025].

Reviewer: L.Recke (Berlin)

MSC:

35J65	Nonlinear boundary value problems for linear elliptic equations
35B40	Asymptotic behavior of solutions to PDEs
35A05	General existence and uniqueness theorems (PDE) (MSC2000)

Keywords:

semilinear second order elliptic equations; supercritical growth; exterior domain; nonexistence