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

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)