We investigate quantitative properties of nonnegative solutions u(x) ≥ 0 to the semilinear diffusion equation Lu = f(u), posed in a bounded domain Ω ⊂ R with appropriate homogeneous Dirichlet or outer boundary conditions. The operator L may belong to a quite general class of linear operators that include the standard Laplacian, the two most common definitions of the fractional Laplacian (−∆) (0...

We study semilinear problems in general bounded open sets for non-local operators with exterior and boundary conditions. The are more than the fractional Laplacian. also give results case of C1,1 sets.

Abstract In this paper, we study a class of fractional p ( x )-Laplacian Dirichlet problems in bounded domain with Lipschitz boundary. Using variational methods, prove different situations the existence and multiplicity solutions.

Abstract In this paper, we establish the existence of a nonnegative nontrivial weak solution for fractional critical ( p , q )-Laplacian problem with discontinuous nonlinearity. The approach is based on suitable variational methods.

In this paper, we provide a framework of designing the local discontinuous Galerkin scheme for integral fractional Laplacian (−Δ)s with s∈(0,1) in two dimensions. We theoretically prove and numerically verify numerical stability convergence rate no worse than O(hk+12) k≥1.

Abstract In this work, we consider the solvability of a fractional-order p -Laplacian boundary value problem on half-line where fractional differential operator is nonlinear and has kernel dimension equal to two. Due nonlinearity operator, Ge Ren extension Mawhin’s coincidence degree theory applied obtain existence results for at resonance. Two examples are used validate established results.

Under certain nonlinear growth conditions of the nonlinearity, we investigate the existence of solutions for a nonlinear Hadamard type fractional differential equation with strip condition and p-Laplacian operator. At the end, two examples are given to illustrate our main results. ©2016 All rights reserved.

By virtue of the upper and lower solutions method, as well as the Schauder fixed point theorem, the existence of positive solutions to a class of q-fractional difference boundary value problems with φ-Laplacian operator is investigated. The conclusions here extend existing results. c ©2016 All rights reserved.

We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem. We write an equivalent characterization as a thin obstacle problem. In this way we are able to apply local type arguments to obtain sharp regularity estimates for the solution and study the regularity of the free boundary.