## 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

Zbl 1204.35090
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### References:

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