Multi-peak solutions for singularly perturbed nonlinear Dirichlet problems involving critical growth

Yi He; Juncheng Wei; Jianjun Zhang

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arXiv · 2207.04732

Multi-peak solutions for singularly perturbed nonlinear Dirichlet problems involving critical growth

Yi He·Juncheng Wei·Jianjun Zhang

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Abstract

We consider the following singularly perturbed elliptic problem \[ - {\varepsilon ^2}\Delta u + u = f(u){\text{ in }}\Omega ,{\text{ }}u > 0{\text{ in }}\Omega ,{\text{ }}u = 0{\text{ on }}\partial \Omega , \] where $\Omega$ is a domain in ${\mathbb{R}^N}(N \ge 3)$, not necessarily bounded, with boundary $\partial \Omega \in {C^2}$ and the nonlinearity $f$ is of critical growth. In this paper, we construct a family of multi-peak solutions to the equation given above which concentrate around any prescribed finite sets of local maxima of the distance function from the boundary $\partial \Omega$.

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Yi He, Juncheng Wei, Jianjun Zhang. 2022-07-11. Multi-peak solutions for singularly perturbed nonlinear Dirichlet problems involving critical growth. https://arxiv.org/abs/2207.04732

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