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The question of interior blow-up points for an elliptic Neumann problem: the critical case. (English) Zbl 1066.35033
The author considers, given a regular open bounded domain \(\Omega\subset {\mathbb R}^{3}\), the Neumann elliptic problem with critical nonlinearity \[ -\Delta u+\mu u= u^{5},\quad u>0\text{ in }\Omega;\quad{\partial u\over\partial\nu}=0 \text{ on }\partial\Omega.\eqno(P_{\mu}) \] In the paper a point \(y\in\bar\Omega\) is said to be a blow-up point for a family \((u_{\mu})_{\mu\geq\mu_{0}}\) of solutions of \((P_{\mu})\) when \[ \liminf_{r\to 0}\limsup_{\mu\to 0}\int_{\{x\in\Omega\,/\, \| x-y\| <r\}}| \nabla u_{\mu}| ^{2}+u_{\mu}^{6}\, dx\, >0. \] The main result proved in the paper is:
Theorem: Let \((u_{\mu})_{\mu\geq\mu_{0}}\) a sequence, bounded in \(H^{1}(\Omega)\) of solutions of \((P_{\mu})\). There exists at least one blow-up point which lies on the boundary of \(\Omega\).

MSC:
35J60 Nonlinear elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
35J25 Boundary value problems for second-order elliptic equations
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