Existence results for problems involving non local operator with an asymmetric weight and with a critical nonlinearity
Abstract
Recently, a great attention has been focused on the study of fractional and nonlocal operators of elliptic type, both for the pure mathematical research and in view of concrete realworld applications. We consider the following non local problem on $\mathbb{H}_0^s(\Omega) \subset L^{q_s}(\Omega)$, with $q_s :=\frac{2n}{n2s}$, $s\in ]0, 1[$ and $n\geq 3$ \begin{equation}\int_{\mathbb{R}^n}p(x) \bigg(\int_{\mathbb{R}^n}\frac{u(x)u(y)^2}{xy^{n+2s}}dy\bigg)dx\lambda \int_\Omega u(x)^q dx, \qquad(1)\end{equation} where $\Omega$ is a bounded domain in $\mathbb{R}^n, p :\mathbb{R}^n \to \mathbb{R}$ is a given positive weight presenting a global minimum $p_0 >0$ at $a \in \Omega$ and $\lambda$ is a real constant. In this work we show that for $q=2$ the infimum of (1) over the set $\{u\in \mathbb{H}_0^s(\Omega), u_{L^{q_s}(\Omega)}=1\}$ does exist for some $k, s, \lambda$ and $n$ and for $q\geq 2$ we study non ground state solutions using the Mountain Pass Theorem.
 Publication:

arXiv eprints
 Pub Date:
 April 2024
 DOI:
 10.48550/arXiv.2404.07531
 arXiv:
 arXiv:2404.07531
 Bibcode:
 2024arXiv240407531B
 Keywords:

 Mathematics  Analysis of PDEs