## The Neumann problem for elliptic equations with critical nonlinearity.(English)Zbl 0836.35048

Ambrosetti, A. (ed.) et al., Nonlinear analysis. A tribute in honour of Giovanni Prodi. Pisa: Scuola Normale Superiore, Quaderni. Universitá di Pisa. 9-25 (1991).
The authors consider the semilinear elliptic boundary value problem in a bounded domain $$\Omega \subset \mathbb{R}^n$$, $$n \geq 3$$: $\Delta u + \alpha (x)u = u^p,\;u > 0 \text{ in } \Omega, \quad u = 0 \text{ on }\Gamma_0,\;\partial u/ \partial n = 0 \text{ on } \Gamma_1,\;\partial \Omega = \Gamma_0 \cup \Gamma_1,$ where $$p = (n + 2)/(n - 2)$$ is the critical exponent. They prove existence of at least one nontrivial solution in the following cases:
1) $$\alpha (x) \equiv \lambda > \lambda_1$$, where $$\lambda_1$$ is a sufficiently large constant, and $$\Gamma_0 = \emptyset$$, $$\Gamma_1 = \partial \Omega$$;
2) $$\alpha \in L^\infty$$ such that $$- \Delta + \alpha$$ is a positive operator on $$\{u \in H^1 (\Omega) \mid u = 0$$ on $$\Gamma_0\}$$ and $$\Gamma_1$$ satisfies a certain geometric property.
A similar result holds also in the case, where the Neumann condition on $$\Gamma_1$$ is replaced by the boundary condition $$\partial u/ \partial n + \beta u = 0$$ on $$\Gamma_1$$.
For the entire collection see [Zbl 0830.00011].

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations

### Keywords:

critical exponent; existence