Paper detail

Existence and localization of solutions for nonlocal fractional equations

This work is devoted to the study of the existence of at least one weak solution to nonlocal equations involving a general integro-differential operator of fractional type. As a special case, we derive an existence theorem for the fractional Laplacian, finding a nontrivial weak solution of the equation \begin{eqnarray*} \begin{cases} (-Δ)^s u=h(x)f(u) & {\mbox{ in }} Ω\\ u=0 & {\mbox{ in }} \mathbb{R}^n\setminus Ω, \end{cases} \end{eqnarray*} where $h\in L^{\infty}_+(Ω)\setminus\{0\}$ and $f:\mathbb{R}\rightarrow\mathbb{R}$ is a suitable continuous function. These problems have a variational structure and we find a nontrivial weak solution for them by exploiting a recent local minimum result for smooth functionals defined on a reflexive Banach space. To make the nonlinear methods work, some careful analysis of the fractional spaces involved is necessary.