Fractional Differential Equations

Fractional differential equations (FDEs) are used in the study of fluid flow, diffusive transport akin to diffusion, rheology, probability, electrical networks, etc. [1, 8, 14, 15, 17–23]. There is a wide range of works deal with fractional equations with fractional Laplacian, for example, readers are encouraged to study [5, 7, 10,16, 25,28]. Recently, the existence of solutions for Schrödinger–Hardy systems, p-fractional Hardy–Schrödinger–Kirchhoff systems as well as a class of systems involving fractional (p, q)-Laplacian operators (see [11–13]) are studied. Here, we consider the following fractional Schrödinger coupled system { (−∆s)u+ V1(x)u+ f(x, u, v) = λu for all x ∈ R , (−∆s)v + V2(x)v + g(x, u, v) = λv for all x ∈ R , (1.1)

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A unique weak solution for a kind of coupled system of fractional Schrodinger equations

The aim of this paper is to show that a supercritical Neumann problem has at least one positive radial solution via recent variational principle and defining an energy functional with a hidden symmetry associated with the problem. Also, we study the problem in the setting of the Heisenberg group $${\mathbb {H}}^n$$
 , which is a special case of locally compact groups.

Positive weak solutions of a generalized supercritical Neumann problem

In this paper, we study the following nonlinear Schrodinger–Bopp–Podolsky system $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+V(x)u+q\phi u=f(u)&{}\\ -\Delta \phi +a^2\Delta ^2\phi =4\pi u^2 \end{array} \right. \ \,\mathrm{in }\,{\mathbb {R}}^3, \end{aligned}$$
 where $$a>0$$
 , $$q>0$$
 and $$V\in {\mathcal {C}}({\mathbb {R}}^3,{\mathbb {R}})$$
 . By means of the variational methods, we prove the existence of infinitely many nontrivial solutions, the existence of a ground state solution for $$f(u)=|u|^{p-2}u+h(u)$$
 with $$p\in [4,6)$$
 and the existence of at least one positive solution for $$f(u)=P(x)u^5+\mu |u|^{p-2}u$$
 with $$p\in (2,6)$$
 under some certain assumptions.

The Schrödinger–Bopp–Podolsky Equation Under the Effect of Nonlinearities

To study existence of at least three weak solutions to a system of over-determined Fredholm fractional integro-differential equations

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On the weak solutions of an overdetermined system of nonlinear fractional partial integro-differential equations

A general approach is presented for proving existence of multiple solutions of the third-order nonlinear differential equation 

$$Au^{\prime\prime\prime}(x) + u^{\prime\prime}(x)u^\prime(x) + u^\prime(x)f(u(x))=0,\quad x \in [0,1] ,$$

subject to given proper boundary conditions. The proof is constructive in nature, and could be used for numerical generation of the solution or closed-form analytical solution by introducing some special functions. The only restriction is about f(u), where it is supposed to be differentiable function with continuous derivative. It is proved the problem may admit no solution, may admit unique solution or may admit multiple solutions.

On The Existence of Multiple Solutions of a Class of Third-Order Nonlinear Two-Point Boundary Value Problems

The existence of at least three weak solutions for a kind of nonlinear time-dependent equation is studied. In fact, we consider the case that the source function has singularity at origin. To this aim, the variational methods and the well-known critical points theorem are main tools.

Multiple weak solutions for a kind of time-dependent equation involving singularity

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Fatemeh Abdolrazaghi

Papers

On the weak solutions of an overdetermined system of nonlinear fractional partial integro-differential equations

A unique weak solution for a kind of coupled system of fractional Schrodinger equations

On The Existence of Multiple Solutions of a Class of Third-Order Nonlinear Two-Point Boundary Value Problems

Multiple weak solutions for a kind of time-dependent equation involving singularity

Multipleweak solutions for a kind of time-dependent equation involving singularity