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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 $$\lambda\to+\infty.$$

##### MSC:
 35J60 Nonlinear elliptic equations 35B33 Critical exponents in context of PDEs 35J25 Boundary value problems for second-order elliptic equations
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##### References:
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