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

Existence and localization of solutions for nonlocal fractional equations