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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].

35J65 Nonlinear boundary value problems for linear elliptic equations