Fractional elliptic problems with critical growth in the whole of $\R^n$

By Serena Dipierro, María Medina, Enrico Valdinoci. 2015

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We study the following nonlinear and nonlocal elliptic equation in~$\R^n$ $$ (-\Delta)^s u = \epsilon\,h\,u^q + u^p \ {\mbox{ in }}\R^n, $$ where~$s\in(0,1)$, $n>2s$, $\epsilon>0$ is a small parameter, $p=\frac{n+2s}{n-2s}$, $q\in(0,1)$, and~$h\in L^1(\R^n)\cap L^\infty(\R^n)$. The problem has a variational structure, and this allows us to find a positive solution by looking at critical points of a suitable energy functional. In particular, in this paper, we find a local minimum and a mountain pass solution of this functional. One of the crucial ingredient is a Concentration-Compactness principle. Some difficulties arise from the nonlocal structure of the problem and from the fact that we deal with an equation in the whole of~$\R^n$ (and this causes lack of compactness of some embeddings). We overcome these difficulties by looking at an equivalent extended problem.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d6627f11fe5ab50fc4c17556f6363a5c6f66d664

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In Applied Mathematics

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Citations in the corpus

Kirchhoff–Hardy Fractional Problems with Lack of Compactness (2017, background): “These models arise in a quite natural way in many different applications, and we refer to the recent monograph [27], to the extensive paper [10] and the references cited therein for further details.”
The concentration behavior of ground state solutions for a fractional Schrödinger–Poisson system (2017, unknown): “6) Let T > 0, we follow [16] and define”
On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent (2017, methodology): “We also refer to the recent monographs [15, 5, 10] for a thorough variational approach of nonlocal problems.”