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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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