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Semilinear elliptic problems with mixed Dirichlet-Neumann boundary conditions. (English) Zbl 1034.35041
The authors study the problem \begin{aligned} -\Delta u &= \lambda u^q + u^r \quad\text{in } \Omega, \cr u &> 0 \quad\text{in } \Omega, \cr B(u) &= 0 \quad\text{on } \partial\Omega, \end{aligned} \eqno(*) where $$\Omega\subset{\mathbb R}^N$$, $$N\geq 3$$, is a smooth bounded domain, $$1<r<2^*-1=(N+2)/(N-2)$$, $$0<q<r$$, and $$B$$ is a mixed boundary operator of the form $B(u)= u \chi_{\Sigma_1} + {{\partial u}\over{\partial \nu}}\chi_{\Sigma_2},$ where $$\Sigma_1$$ and $$\Sigma_2$$ are smooth $$(N-1)$$-dimensional submanifolds of $$\partial\Omega$$ such that $$\Sigma_1\cap\Sigma_2=\emptyset$$, $$\overline\Sigma_1\cup\overline\Sigma_2=\partial\Omega$$ and $$\overline\Sigma_1\cup\overline\Sigma_2=\Gamma$$ is a smooth $$(N-2)$$-dimensional submanifold of $$\partial\Omega$$.
The main results are existence, nonexistence and multiplicity results, and a priori $$L^\infty$$ estimates for solutions of $$(*)$$. These are proved by a blow-up argument that relies on some refinements of the global Hölder estimates for solutions of mixed boundary value problems established by G. Stampacchia [Ann. Mat. Pura Appl. (4) 51, 1–37 (1960; Zbl 0204.42001)].

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35J20 Variational methods for second-order elliptic equations 35J25 Boundary value problems for second-order elliptic equations 35J60 Nonlinear elliptic equations 35B45 A priori estimates in context of PDEs
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