Fundamental solutions for discrete dynamical systems involving the fractional Laplacian

We prove representation results for solutions of a time‐fractional differential equation involving the discrete fractional Laplace operator in terms of generalized Wright functions. Such equations arise in the modeling of many physical systems, for example, chain processes in chemistry and radioactivity. Our focus is in the problem , where 0<β ≤ 2, 0<α ≤ 1, , (?Δ_d)^α is the discrete fractional Laplacian, and is the Caputo fractional derivative of order β. We discuss important special cases as consequences of the representations obtained.     

A radial symmetry and Liouville theorem for systems involving fractional Laplacian

We investigate the nonnegative solutions of the system involving the fractional Laplacian:

$$\left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {( - \Delta )^\alpha u_i (x) = f_i (u),} & {x \in \mathbb{R}^n , i = 1,2, \ldots ,m,} \\ \end{array} } \\ {u(x) = (u_1 (x),u_2 (x), \ldots ,u_m (x)),} \\ \end{array} } \right.$$

where 0 < α < 1, n > 2, f _i (u), 1 ≤ i ≤ m, are real-valued nonnegative functions of homogeneous degree p _i ≥ 0 and nondecreasing with respect to the independent variables u ₁, u ₂,..., u _m . By the method of moving planes, we show that under the above conditions, all the positive solutions are radially symmetric and monotone decreasing about some point x ₀ if p _i = (n + 2α)/(n ? 2α) for each 1 ≤ i ≤ m; and the only nonnegative solution of this system is u ≡ 0 if 1 < p _i < (n + 2α)/(n ? 2α) for all 1 ≤ i ≤ m.

     

Integro-differential systems of mixed type involving higher order fractional Laplacian

In this paper, we deal with a new class of non-local operators that we term integro-differential systems of mixed type. We study the behaviour of solutions of this system when the diffusion term involves higher order fractional powers of the Laplacian. Moreover, we prove that the solution of the system decays faster than a power with an exponent given by the smallest index of the fractional power of the Laplacian.     

Existence and multiplicity of solutions for nonlocal systems involving fractional Laplacian with non-differentiable terms

In this paper, we devote to the study of the existence and multiplicity of solutions of nonlocal systems involving fractional Laplacian with non-differentiable terms using some extended critical point theorems for locally Lipschitz function on product spaces.     

Radial symmetry for systems of fractional Laplacian

In this paper, we consider systems of fractional Laplacian equations in Rnwith nonlinear terms satisfying some quite general structural conditions. These systems were categorized critical and subcritical cases. We show that there is no positive solution in the subcritical cases, and we classify all positive solutions ui in the critical cases by using a direct method of moving planes introduced in Chen-Li-Li [11] and some new maximum principles in Li-Wu-Xu [27].     

Semilinear elliptic equations for the fractional Laplacian involving critical exponential growth

We establish the existence and multiplicity of weak solutions for a class of nonlocal equations involving the fractional Laplacian operator, nonlinearities with critical exponential growth, and potentials that may change sign. The proofs of our existence results rely on minimization methods and the mountain pass theorem. Copyright © 2016 John Wiley & Sons, Ltd.     

Inverse source problem for a diffusion equation involving the fractional spectral Laplacian

In this paper, we consider an inverse problem related to a fractional diffusion equation. The model problem is governed by a nonlinear partial differential equation involving the fractional spectral Laplacian. This study is focused on the reconstruction of an unknown source term from a partial internal measured data. The considered ill‐posed inverse problem is formulated as a minimization one. The existence, uniqueness, and stability of the solution are discussed. Some theoretical results are established. The numerical reconstruction of the unknown source term is investigated using an iterative process. The proposed method involves a denoising procedure at each iteration step and provides a sequence of source term approximations converging in norm to the actual solution of the minimization problem. Some numerical results are presented to show the efficiency and the accuracy of the proposed approach.     

Existence results for asymmetric fractional ‐Laplacian problem

We investigated a class of quasi‐linear nonlocal problems with a right‐hand side nonlinearity which exhibits an asymmetric growth at and . Namely, it is linear at and superlinear at . However, it needs not satisfy the Ambrosetti–Rabinowitz condition on the positive semiaxis. Some existence results for nontrivial solution are established by using variational methods combined with the Moser–Trudinger inequality.     

Entire large solutions of semilinear equations involving the fractional Laplacian

We investigate the existence and nonexistence of positive entire large solutions of a class of semilinear equations involving the fractional Laplacian. Sharp lower and upper bounds of the minimal solution, if it exists, are also given.     

Nodal bound state of nonlinear problems involving the fractional Laplacian

This paper is concerned with the following nonlinear fractional Schrödinger equation where ε>0 is a small parameter, V(x) is a positive function, 0<s<1, and . Under some suitable conditions, we prove that for any positive integer k, one can construct a nonradial sign‐changing (nodal) solutions with exactly k maximum points and k minimum points near the local minimum point of V(x).     

Interior regularity results for zeroth order operators approaching the fractional Laplacian

In this article we are interested in interior regularity results for the solution \({\mu _ \in } \in C(\bar \Omega )\) of the Dirichlet problem

$$\{ _{\mu = 0in{\Omega ^c},}^{{I_ \in }(\mu ) = {f_ \in }in\Omega }$$

where Ω is a bounded, open set and \({f_ \in } \in C(\bar \Omega )\) for all ? ∈ (0, 1). For some σ ∈ (0, 2) fixed, the operator \(\mathcal{I}_{\in}\) is explicitly given by

$${I_ \in }(\mu ,x) = \int_{{R^N}} {\frac{{[\mu (x + z) - \mu (x)]dz}}{{{ \in ^{N + \sigma }} + |z{|^{N + \sigma }}}}} ,$$

which is an approximation of the well-known fractional Laplacian of order σ, as ? tends to zero. The purpose of this article is to understand how the interior regularity of u? evolves as ? approaches zero. We establish that u_? has a modulus of continuity which depends on the modulus of f_?, which becomes the expected Hölder profile for fractional problems, as ? → 0. This analysis includes the case when f? deteriorates its modulus of continuity as ? → 0.

     

Comparison and regularity results for the fractional Laplacian via symmetrization methods

In this paper we establish a comparison result through symmetrization for solutions to some boundary value problems involving the fractional Laplacian. This allows to get sharp estimates for the solutions, obtained by comparing them with solutions of suitable radial problems. Furthermore, we use such result to prove a priori estimates for solutions in terms of the data, providing several regularity results which extend the well-known ones for the classical Laplacian.     

Bifurcation and multiplicity results for critical fractional p‐Laplacian problems

We prove a bifurcation and multiplicity result for a critical fractional p‐Laplacian problem that is the analog of the Brézis‐Nirenberg problem for the nonlocal quasilinear case. This extends a result in the literature for the semilinear case to all , in particular, it gives a new existence result. When , the nonlinear operator , has no linear eigenspaces, so our extension is nontrivial and requires a new abstract critical point theorem that is not based on linear subspaces. We prove a new abstract result based on a pseudo‐index related to the ‐cohomological index that is applicable here.     

Nonzero positive solutions of fractional Laplacian systems with functional terms

We study the existence of nonzero positive solutions of a class of systems of differential equations driven by fractional powers of the Laplacian. Our approach is based on the notion of fixed point index, and allows us to deal with nonlocal functional weights and functional boundary conditions. We present two examples to shed light on the type of functionals and growth conditions that can be considered with our approach.