Fractional Laplacian Problems Research Articles

Positivity of nonnegative solutions to a system of fractional Laplacian problems in a ball

Positivity of nonnegative solutions to a system of fractional Laplacian problems in a ball

Error analysis of a collocation method on graded meshes for a fractional Laplacian problem

Error analysis of a collocation method on graded meshes for a fractional Laplacian problem

Existence of positive weak solutions for stationary fractional Laplacian problem by using sub-super solutions

Abstract In this work, we establish a theorem concerning the extension of positive weak solutions for a stationary fractional Laplacian problem featuring weight functions that change sign. Additionally, we introduce novel conditions to ensure the existence of a positive solution for the given problem. These conditions are derived utilizing the approach of sub-super solutions, thereby extending and complementing existing results in the literature.

Sign-changing solutions for Kirchhoff-type variable-order fractional Laplacian problems

In this paper, we are concerned with the Kirchhoff-type variable-order fractional Laplacian problems involving critical exponents and logarithmic nonlinearity. By using the constraint variational method, we show the existence of one least energy sign-changing solution. Moreover, we show that this energy is strictly larger than twice the ground energy.

Multiplicity of solutions for nonlocal fractional equations with nonsmooth potentials

This paper is concerned a specific category of nonlocal fractional Laplacian problems that involve nonsmooth potentials. By utilizing an abstract critical point theorem for nonsmooth functionals and combining it with the analytical framework on fractional Sobolev spaces developed by Servadei and Valdinoci, we are able to establish the existence of at least three weak solutions for nonlocal fractional problems. Moreover, this work also generalizes and improves upon certain results presented in the existing literature.

Multiplicity of solutions for nonlocal fractional equations with nonsmooth potentials

This paper is concerned a specific category of nonlocal fractional Laplacian problems that involve nonsmooth potentials. By utilizing an abstract critical point theorem for nonsmooth functionals and combining it with the analytical framework on fractional Sobolev spaces developed by Servadei and Valdinoci, we are able to establish the existence of at least three weak solutions for nonlocal fractional problems. Moreover, this work also generalizes and improves upon certain results presented in the existing literature.

On the fractional P–Q laplace operator with weights

We exploit the existence and non-existence of positive solutions to the eigenvalue problem driven by the nonhomogeneous fractional p &q Laplacian operator with indefinite weights ( − Δ p ) α u + ( − Δ q ) β u = λ [ a | u | p − 2 u + b | u | q − 2 u ] in Ω , where Ω ⊆ R N is a smooth bounded domain that has been extended by zero. We further show the existence of a continuous family of eigenvalues in the case Ω = R N and b ≡ 0 a.e. Our approach relies strongly on variational Analysis, in which the Mountain pass theorem plays the key role. Due to the lack of spatial compactness and the embedding W α , p ( R N ) ↪ W β , q ( R N ) in R N , we employ the concentration-compactness principle of P.L. Lions [The concentration-compactness principle in the calculus of variations. The limit case. II, Rev Mat Iberoamericana. 1985;1(2):45–121]. to overcome the difficulty. Our paper can be considered as a counterpart to the important works [Alves et al. Existence, multiplicity and concentration for a class of fractional p &q Laplacian problems in R N , Commun Pure Appl Anal, 2019;18(4):2009–2045], [Benci et al. An eigenvalue problem for a quasilinear elliptic field equation. J Differ Equ, 2002;184(2):299–320], [Bobkov et al. On positive solutions for ( p , q ) -Laplace equations with two parameters, Calc Var Partial Differ Equ, 2015;54(3):3277–3301], [Colasuonno and Squassina. Eigenvalues for double phase variational integrals, Ann Mat Pura Appl (4), 2016;195(6):1917–1956], [Papageorgiou et al. Positive solutions for nonlinear Neumann problems with singular terms and convection, J Math Pures Appl (9), 2020;136:1–21], [Papageorgiou et al. Ground state and nodal solutions for a class of double phase problems, Z Angew Math Phys, 2020;71:1–15], and may have further applications to deal with other problems.

Fractional potential: A new perspective on the fractional Laplacian problem on bounded domains

In this work, the fractional Laplacian operator is studied based on its spectral definition for a bounded, one-dimensional domain. We first note that both the analytical and numerical solutions of the fractional diffusion equation subject to Neumann boundary conditions do not exist due to the need to raise the zero eigenvalue to a negative fractional index. To overcome this issue, we use the concept of regularisation and modify the Neumann boundary conditions to Robin type by introducing regularisation parameters that ensure the smallest eigenvalue of both the Sturm–Liouville system and the matrix representation of the Laplacian operator is nonzero. We then observe that these regularisation parameters can have a significant impact on the solution behaviour of the fractional diffusion model. This finding motivates the use of what we call the ‘fractional potential’ to define a new fractional model that is far less sensitive to the regularisation parameter and makes imposing fractional Neumann boundary conditions also feasible. Next, we compare the results of this new fractional model with a fractional Riesz problem subject to Neumann boundary conditions on a bounded symmetric domain. An interesting finding is that the steady-state solution for the fractional Laplacian problem leads to a constant function, while the fractional Riesz operator results in a function exhibiting boundary singularities. Finally, we propose a three-layered fractional model based on the Laplacian operator to naturally impose the traditional boundary conditions by using the idea of ‘geometric reconstruction’. We derive semi-analytical and numerical solutions for this model. An important outcome is that by choosing an extremely small scale for the boundary transition layer on both sides of the domain, this three-layered model can be used to approximate the fractional Laplacian problem on a bounded domain. The proposed model captures the fractional diffusion behaviour in the inner layer well while describing the classical diffusion behaviour near the boundary. We show that this model is ideal for modelling anomalous diffusion in a binary medium exhibiting distinct heterogeneity.

Positive solutions for nonlinear fractional Laplacian problems

We consider a class of nonlinear fractional Laplacian problems satisfying the homogeneous Dirichlet condition on the exterior of a bounded domain. We prove the existence of a positive weak solution for classes of nonlinearities which are either sublinear or asymptotically linear at infinity. We use the method of sub-and-supersolutions to establish the results. We also provide numerical bifurcation diagrams, corresponding to the theoretical results, using the finite element method in one dimension. See also https://ejde.math.txstate.edu/special/02/h1/abstr.html

Kirchhoff-Type Fractional Laplacian Problems with Critical and Singular Nonlinearities

Kirchhoff-Type Fractional Laplacian Problems with Critical and Singular Nonlinearities

Asymptotic Radial Solution of Parabolic Tempered Fractional Laplacian Problem

We study parabolic equation with the tempered fractional Laplacian and logarithmic nonlinearity by the direct method of moving planes. We first prove several important theorems, such as asymptotic maximum principle, asymptotic narrow region principle and asymptotic strong maximum principle for antisymmetric functions, which are critical factors in the process of moving planes. Then, we further derive some properties of asymptotic radial solution to parabolic equation with the tempered fractional Laplacian and logarithmic nonlinearity in a unit ball. These consequences can be applied to investigate more nonlinear nonlocal parabolic equations.

Computing solution landscape of nonlinear space-fractional problems via fast approximation algorithm

The nonlinear space-fractional problems often allow multiple stationary solutions, which can be much more complicated than the corresponding integer-order problems. In this paper, we systematically compute the solution landscapes of nonlinear constant/variable-order space-fractional problems on one- and two-dimensional rectangular domains. A fast approximation algorithm is developed to deal with the variable-order spectral fractional Laplacian by approximating the variable-indexing Fourier modes, and then combined with saddle dynamics to construct the solution landscape of variable-order space-fractional phase field model. Numerical experiments are performed to substantiate the accuracy and efficiency of fast approximation algorithm and elucidate essential features of the stationary solutions of space-fractional phase field model. Furthermore, we demonstrate that the solution landscapes of spectral fractional Laplacian problems can be reconfigured by varying the diffusion coefficients in the corresponding integer-order problems.

Nehari manifold for singular fractional p(x,.)-Laplacian problem

In this paper, we consider a class of fractional Laplacian problems of the form: where is a bounded domain and is the fractional -Laplacian operator. We assume that λ and μ are positive parameters and is a continuous function. By opting for the Nehari manifold method combined with the theory of generalized Lebesgue Sobolev spaces, we will prove the existence of solutions to the above problem.

Sign-changing solutions for Kirchhoff-type problems involving variable-order fractional Laplacian and critical exponents

In this paper, we are concerned with the Kirchhoff-type variable-order fractional Laplacian problem with critical variable exponent. By using constraint variational method and quantitative deformation lemma we show the existence of one least energy solution, which is strictly larger than twice of that of any ground state solution.

Existence and regularity of solutions for a class of fractional Laplacian problems

In this paper, we first establish a modified Marcinkiewicz interpolation result, and using this result, we obtain a new regularity result for a fractional Laplacian problem on a bounded C1,1-domain Ω in RN, and we also obtain a new regularity result for then fractional problem on unbounded Ω=RN by using the Stampacchia truncation method. Next, by the Leray-Schauder fixed point theorem we obtain the existence of solutions for a class of fractional Laplacian problems with weak growth conditions on the nonlinearities. Finally, as an application, we prove the existence of positive solutions for a fractional Laplacian problem which has an exponential growth nonlinearity.

Continuum of positive solutions of superlinear fractional Laplacian problems

Continuum of positive solutions of superlinear fractional Laplacian problems

A fractional Laplacian problem with mixed singular nonlinearities and nonregular data

In this note, we study on the existence and uniqueness of a positive solution to the following doubly singular fractional problem: $$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^s u = \dfrac{K(x)}{u^q} + \dfrac{f(x)}{u^{\gamma }}+\mu &{} \mathrm {in} \,\, \Omega ,\\ u>0 &{} \mathrm {in} \,\, \Omega , \\ u=0 &{} \mathrm {in} \,\, \big ({\mathbb {R}}^N \setminus \Omega \big ). \end{array}\right. } \end{aligned}$$Here \(\Omega \subset {\mathbb {R}}^N\) (\(N > 2s\)) is an open bounded domain with smooth boundary, \( s \in (0,1)\), \(q>0\), \(\gamma >0\), and K(x) is a positive Hölder continuous function in which behaves as \(\mathrm {dist}(x, \partial \Omega )^{-\beta }\) near the boundary with \(0 \le \beta <2s\). Also, \(0 \le f, \mu \in L^1(\Omega )\), or non-negative bounded Radon measures in \(\Omega \). Moreover, we assume that \(0<\frac{\beta }{s}+q < 1\), or \(\frac{\beta }{s}+q > 1\) with \(2\beta +q(2s-1)<(2s+1)\). For \(s \in (0,\frac{1}{2})\), we take advantage of the convexity of \(\Omega \). For any \(\gamma >0\), we will prove the existence of a positive weak (distributional) solution to the above problem. Besides, for the case \( 0 < \gamma \le 1\), \(\mu \in L^1(\Omega )+X^{-s}(\Omega )\), and some weighted integrable functions f, we will show the existence and uniqueness of another notion of a solution, so-called entropy solution. Also, we will discuss the uniqueness of the weak solution for the case \(\gamma >1\), and also the equivalence of entropy and weak solutions for the case \(0< \gamma \le 1\). Finally, we will have some relaxation on the assumptions of f in order to prove the existence of solutions.

Combined effects of logarithmic and critical nonlinearities in fractional Laplacian problems

The aim of this paper is to study the following fractional Laplacian problems with logarithmic and critical nonlinearities: \begin{align*} \begin{cases} (-\Delta)^{s}u=\lambda a(x)u\ln|u|+\mu|u|^{2_s^*-2}u \ \ & x\in\Omega,\\ u=0\ \ & x\in\mathbb{R}^N\setminus\Omega, \end{cases} \end{align*} where $s\in(0,1),\ N > 2s,$ $\lambda,\mu > 0$, $\Omega\subset \mathbb{R}^N$ is a bounded domain with Lipschitz boundary, $a\in L^\infty(\Omega)$ is a sign-changing function, $2_s^*$ is the critical Sobolev exponent and $(-\Delta)^{s}$ is the fractional Laplacian. When $\lambda$ and $\mu$ satisfy suitable assumptions, the existence and multiplicity of nontrivial and nonnegative continuous solutions for the above problem are obtained by applying the Nehari manifold approach.

A multigrid-reduction-in-time solver with a new two-level convergence for unsteady fractional Laplacian problems

The multigrid-reduction-in-time (MGRIT) technique has proven to be successful in achieving higher run-time speedup by exploiting parallelism in time. The goal of this article is to develop and analyze a MGRIT algorithm using FCF-relaxation with time-dependent time-grid propagators to seek the finite element approximations of unsteady fractional Laplacian problems. The multigrid with line smoother proposed in Chen et al. (2016) is chosen to be the spatial solver. Motivated by Southworth (2019), we provide a new temporal eigenvalue approximation property and then deduce a generalized two-level convergence theory which removes the previous unitary diagonalization assumption on the fine and coarse time-grid propagators required in Yue et al. (2019). Numerical computations are included to confirm the theoretical predictions and demonstrate the sharpness of the derived convergence upper bound.

A fractional Laplacian problem with critical nonlinearity

<abstract> In this paper, using the mountain pass theorem we obtain a positive solution to the fractional Laplacian problem <p class="disp_formula">$ \begin{equation*} \label{eq: main problem} \begin{cases} (-\Delta)^su = g(x)(u-k)_{+}^{q-1}+u^{\frac{n+2s}{n-2s}}&amp; \rm{in}\ \,\Omega,\\ u&gt;0&amp; \rm{in}\ \,\Omega,\\ u = 0&amp; \rm{on}\ \,\partial\Omega, \end{cases} \end{equation*} $ where $ \Omega\subset {\mathbb R}^{n} $ is a bounded smooth domain, $ 0 &lt; s &lt; 1, 2\leq q &lt; {2n}/(n-2s) $ and $ k\in(0, \infty) $ is an arbitrary number. The function $ g:{\Omega}\to {\mathbb R} $ is a nonnegative continuous function satisfying some integrability condition. </abstract>