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Concentrating solutions for an anisotropic elliptic problem with large exponent

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  • We consider the following anisotropic boundary value problem $$\nabla (a(x)\nabla u) + a(x)u^p = 0, \;\; u>0 \ \ \mbox{in} \ \Omega, \quad u = 0 \ \ \mbox{on} \ \partial\Omega,$$ where $\Omega \subset \mathbb{R}^2$ is a bounded smooth domain, $p$ is a large exponent and $a(x)$ is a positive smooth function. We investigate the effect of anisotropic coefficient $a(x)$ on the existence of concentrating solutions. We show that at a given strict local maximum point of $a(x)$, there exist arbitrarily many concentrating solutions.
    Mathematics Subject Classification: Primary: 35B40, 35B45; Secondary: 35J40.


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