Semilinear equations for non-local operators: beyond the fractional Laplacian
This paper deal with the following non-local semi-linear problem \[ \begin{cases} -Lu(x) = f(x,u(x)),&\mbox{in \(D\)},\\ u= \lambda, & \mbox{in \(D^c\)},\\ W_Du = \mu,&\mbox{on \(\partial D\)}, \end{cases}\tag{1} \] where \(D\subset \mathbb R^N\), \(N\geq 2\) is a bounded open set, \(f: D\times \mathbb R \to \mathbb R\) is continuous with respect to the second variable, and there is a function \(\rho:D\to [0, \infty)\) and a continuous function \(\Lambda: [0, \infty) \to [0, \infty)\) such that \[ |f(x,t)| \leq \rho (x)\Lambda(|t|). \] \(\lambda\) and \(\mu\) are signed measures in \(D^c=\mathbb R^N\setminus D\) and \(\partial D\) respectively. The operator \(L\) is a nonlocal operator defined as \[ Lu(x) = P.V. \int_{\mathbb R^N} (u(y)-u(x))j(|y-x|) dy, \] where the singular kernel \(j\) is completely determined by the function \(\phi: (0, \infty) \to (0, \infty)\) is a complete Bernstein function without drift satisfying certain weak scaling conditions. By using the sub and super solution the authors obtained the existence of weak dual solution to (1).
scientific article