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Non existence of bounded-energy solutions for some semilinear elliptic equations with a large parameter. (English) Zbl 1121.35053
The authors prove that the energy of positive solutions to the Dirichlet problem \[ \begin{cases} -\Delta u+\lambda u=u^{{N+2}\over{N-2}} & \text{ in}\;\Omega\subset \mathbb R^N, \quad u>0 \text{ in } \Omega,\cr u=0 & \text{ on}\;\partial\Omega, \end{cases} \] where \(\Omega\) is a smooth bounded domain, \(N\geq 3\), tends to infinity as \(\lambda\to+\infty.\) Moreover it is proved that the bounded energy solutions of mixed (Dirichlet-Neumann) boundary value problems have at least one blow-up point on the Neumann component as \(\lambda\to+\infty.\)

35J60 Nonlinear elliptic equations
35B33 Critical exponents in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
Full Text: EuDML
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