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Boundary clustered layers near the higher critical exponents. (English) Zbl 1288.35240

In this paper, the authors consider the following supercritical problem \[ \begin{cases} -\Delta u = |u|^{p-2}u & \text{in} \;\Omega \\ u=0 & \text{on} \;\partial \Omega \end{cases}\tag{1} \] with \(\Omega\) a bounded smooth domain in \(\mathbb{R}^n\) and \(2<p<2_{N,k}^*\). Here \(2_{N,k}^*={2(N-k)\over N-k-2}\) is the critical exponent for the Sobolev embedding of \(H^1(\mathbb{R}^{N-k})\) in \(L^q(\mathbb{R}^{N-k})\), \(1\leq k\leq N-3\).
The main results of the paper consist of providing affirmative answers to the following two questions proposed in the paper [M. del Pino et al., J. Eur. Math. Soc. (JEMS) 12, No. 6, 1553–1605 (2010; Zbl 1204.35090)]:
1. Are there domains \(\Omega\) in which equation (1) has a positive solution for each \(p\) with the property that these solutions concentrate along a \(k\)-dimensional submanifold of the boundary \(\partial\Omega\) as \(p\rightarrow 2_{N,k}^*\)?
2. Are there domains \(\Omega\) in which equation (1) has a sign changing solution for each \(p\) with the property that these solutions concentrate along a \(k\)-dimensional submanifold of the boundary \(\partial\Omega\) as \(p\rightarrow 2_{N,k}^*\)?

MSC:

35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
35B33 Critical exponents in context of PDEs
35B09 Positive solutions to PDEs

Citations:

Zbl 1204.35090
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References:

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