Existence of solutions for critical fractional Kirchhoff problems

Tempered fractional Sobolev spaces

This paper designs a new finite difference compact reconstruction unequal-sized weighted essentially nonoscillatory scheme (CRUS-WENO) for solving fractional differential equations containing the fractional Laplacian operator. This new CRUS-WENO scheme uses stencils of different sizes to achieve fifth-order accuracy in smooth regions and maintain nonoscillatory properties near discontinuities. The fractional Laplacian operator of order β(0<β<1) is split into the integral part and the first derivative term. Using the Gauss–Jacobi quadrature method to solve the integral part of the fractional Laplacian operators, a new finite difference CRUS-WENO scheme is presented to discretize the first derivative term of the fractional equation. This new CRUS-WENO scheme has the advantages of a narrower large stencil and high spectral resolution. In addition, the linear weights of the new CRUS-WENO scheme can be any positive numbers whose sum is one, which greatly reduces the calculation cost. Some numerical examples are given to show the effectiveness and feasibility of this new CRUS-WENO scheme in solving fractional equations containing the fractional Laplacian operator.

The paper deals with the following Kirchhoff‐type problem where M models a Kirchhoff coefficient, is a variable s(·)‐order p(·)‐fractional Laplace operator, with and . Here, is a bounded smooth domain with N > p(x, y)s(x, y) for any , μ is a positive parameter, g is a continuous and subcritical function, while variable exponent r(x) could be close to the critical exponent , given with and for . We prove the existence and asymptotic behavior of at least one non‐trivial solution. For this, we exploit a suitable tricky step analysis of the critical mountain pass level, combined with a Brézis and Lieb‐type lemma for fractional Sobolev spaces with variable order and variable exponent.

This paper deals with a class of nonlocal Schrodinger equations with critical exponent $$\begin{aligned} \left\{ \begin{array}{llll} (-\Delta )^{s} u+V(x)u-K(x)\phi |u|^{2^*_s-3}u=f(x,u),&{}\quad \mathrm{in}\ \mathbb {R}^3,\\ (-\Delta )^s \phi =K(x)|u|^{2^*_s-1},&{}\quad \mathrm{in}\ \mathbb {R}^3. \end{array}\right. \end{aligned}$$ By employing the mountain pass theorem, concentration-compactness principle and approximation method, the existence of nontrivial solution is obtained under appropriate assumptions on V, K and f.

In this paper, we prove the existence of positive ground state solutions of the Schrödinger–Poisson system involving a negative nonlocal term and critical exponent on a bounded domain. The main tools are the mountain pass theorem and the concentration compactness principle.

Aiming at the heat conduction equation, the heat conduction model is a very typical example of the partial differential equation. This paper focuses on the development and preliminary knowledge of the partial differential equation, the establishment of the heat conduction model and the solution of the heat conduction model. The partial differential equation with fractional Laplace operator is a kind of typical fractional partial differential equation, which has important applications in the fields of science and engineering. Fractional Laplace operator is a kind of nonlocal quasi-differential operator, which is the infinitesimal generator of L é vy steady-state process. It is essentially different from the classical Laplace operator, which leads to the disappearance of some classical properties, which generally brings difficulties to the study of this kind of problems. It is very difficult to solve the explicit solution of partial differential equation with fractional Laplace operator. In order to improve the accuracy and speed of solving partial differential equation, a high precision method for solving partial differential equation based on deep learning network is proposed. A multilayer radial basis function neural network is constructed and its structural layers are determined. By introducing subnetworks into each layer of the multilayer radial basis function neural network, a composite multilayer radial basis function neural network is constructed to improve the real function approximation performance and operation accuracy of the multilayer radial basis function neural network. The high-precision compound multi-layer radial basis function neural network is used to solve the partial differential equation. By giving a specific example of solving the partial differential equation, the solution effect of the method is tested. The results show that the method has very high solution accuracy under the four-layer network structure, which can improve about 1.5 orders of magnitude compared with the first layer. Under different number of training samples, it has higher solution accuracy and speed, and the comprehensive performance is superior. Therefore, it is a practical but challenging work to study the existence of solutions of partial differential equations with fractional Laplace operators. This paper mainly describes the research progress and trends of the existence of solutions of several classes of partial differential equations with fractional Laplace operators, including some of the work done by the author in this field in recent years.

In this paper, we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator where and the fractional derivatives , , are in the Caputo sense. Applying integral transform methods, we describe a complete family of eigenfunctions and fundamental solutions of the operator in classes of functions admitting a summable fractional derivative. The solutions are expressed using the Mittag–Leffler function. From the family of fundamental solutions obtained, we deduce a family of fundamental solutions of the corresponding fractional Dirac operator, which factorizes the fractional Laplace operator introduced in this paper.

The fractional Laplacian operator is a very important fractional operator that is often used to describe several anomalous diffusion phenomena. In this paper, we develop some numerical schemes, including a finite difference scheme and finite volume scheme for the fractional Laplacian operator, and apply the resulting numerical schemes to solve some fractional diffusion equations. First, the fractional Laplacian operator can be characterized as the weak singular integral by an integral operator with zero boundary condition. Second, because the solutions of fractional diffusion equations are usually singular near the boundary, we use a fractional interpolation function in the region near the boundary and use a classical interpolation function in other intervals. Then, we apply a finite difference scheme to the discrete fractional Laplacian operator and fractional diffusion equation with the above fractional interpolation function and classical interpolation function. Moreover, it is found that the differential matrix of the above scheme is a symmetric matrix and strictly row-wise diagonally dominant in special fractional interpolation functions. Third, we show a finite volume scheme for a discrete fractional diffusion equation by fractional interpolation function and classical interpolation function and analyze the properties of the differential matrix. Finally, the numerical experiments are given, and we verify the correctness of the theoretical results and the efficiency of the schemes.

This paper presents three new families of fractional Sobolev spaces and their accompanying theory in one dimension. The new construction and theory are based on a newly developed notion of weak fractional derivatives, which are natural generalizations of the well-established integer order Sobolev spaces and theory. In particular, two new families of one-sided fractional Sobolev spaces are introduced and analyzed, and they reveal more insights about another family of so-called symmetric fractional Sobolev spaces. Many key theorems/properties, such as density/approximation theorem, extension theorems, one-sided trace theorem, and various embedding theorems and Sobolev inequalities in those Sobolev spaces are established. Moreover, a few relationships with existing fractional Sobolev spaces are also uncovered. The results of this paper lay down a solid theoretical foundation for systematically developing a fractional calculus of variations theory and a fractional PDE theory as well as their numerical solutions in subsequent works.

In this paper, we consider a class of fractional Kirchhoff equations with L^{2} critical exponents. By using the scaling technique and concentration-compactness principle we obtain the existence and nonexistence of ground state for fractional Kirchhoff equation with L^{2} critical exponent.

ABSTRACTIn this paper, we are concerned with questions of the existence of solution for a class of nonlocal and nonhomogeneous Neumann boundary value problems involving the -Laplacian in which the nonlinear terms assume both critical growth. The main tools used are the Lions' Concentration-Compactness Principle [Lions PL. The concentration-compactness principle in the calculus of variations. The limit case I, part 1. Rev Mat Iberoam. 1985;1(1):145–201.] for variable exponent spaces and the Mountain Pass Theorem.

The aim of this thesis is to prove the existence of a weak solution for semilinear fractional elliptic equations given by \begin{eqnarray*}\left\{ \begin{array}{ll} (-\Delta)^s u=|u|^{p-2} u+ f(x),& \quad x\in \Omega,\\ u=0, & \quad x \in \R^n \backslash \Omega, %0, & \quad x \in \partial \Oemga, \end{array} \right. \end{eqnarray*} where $(-\Delta)^s$ denotes the fractional Laplace operator with $s \in (0,1],$ $n > 2s,$ $\Omega$ is an open bounded domain in $\R^n$ with $C^2$-boundary and $f \in L^2(\Omega).$ We are interested in extending this result for $s \in (0, 1]$ to $p$ greater than the critical Sobolev exponent where the compact embedding fails to hold. We shall make use of a new variational principle presented in \cite{Mo1} that allows one to deal with problems well beyond the compact structure.

We are concerned with the multiplicity of solutions to the system driven by a fractional operator with homogeneous Dirichlet boundary conditions. Namely, using fibering maps and the Nehari manifold, we obtain multiple solutions to the following fractional elliptic system:where Ω is a smooth bounded set in ℝn, n > 2s, with s ∈ (0, 1); (–Δ)s is the fractional Laplace operator;, λ, μ > 0 are two parameters; the exponent n/(n – 2s) ⩽ q < 2; α > 1, β > 1 satisfy is the fractional critical Sobolev exponent.

The purpose of this paper is to investigate the existence and multiplicity of weak solutions for a class of nonlocal problems involving the fractional magnetic operator and nonlinearities which may have critical or subcritical growth in the sense of Trudinger–Moser inequality. By using variational methods based on minimax argument and assuming suitable conditions on the nonlinearity, we address essential geometric cases of the associated energy functional, such as the mountain pass geometry and the linking geometry. Some of the main results established in this work are new even when the magnetic potential is equal to zero, which corresponds to the usual fractional Laplacian operator.

In this article, we study regularity of weak solutions to a class of nonlinear parabolic equations in divergence form. The main purpose is to present a regularity estimate with more general conditions on coefficients, N-functions and non-homogeneous terms in the fractional Sobolev spaces. By deriving a higher integrability estimate of weak solutions, we obtain the desired regularity estimate. In addition, the results of this article expand the regularity theory of parabolic equations in fractional Sobolev spaces and Besov spaces.