1. Existence results for problems involving non local operator with an asymmetric weight and with a critical nonlinearity
- Author
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Benhafsia, Sana and Hadiji, Rejeb
- Subjects
Mathematics - Analysis of PDEs - 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 real-world applications. We consider the following nonlocal problem on $\mathbb{H}_0^s(\Omega) \subset L^{q_s}(\Omega)$, with $q_s :=\frac{2n}{n-2s}$, $s\in ]0, 1[$ and $n\geq 3$ \begin{equation} \int_{\mathbb{R}^n}p(x) \bigg(\int_{\mathbb{R}^n}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}dy\bigg)dx-\lambda \int_\Omega |u(x)|^q dx, \qquad (1.1) \end{equation} where $\Omega$ is a bounded domain in $\mathbb{R}^n, p :\mathbb{R}^n \rightarrow \mathbb{R}$ 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 (1.1) 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.
- Published
- 2024