Fractional Laplacian Problems Research Articles

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. 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.

Nontrivial solutions for the fractional Laplacian problems without asymptotic limits near both infinity and zero

<p style='text-indent:20px;'>In this paper we obtain the existence of nontrivial solutions for the fractional Laplacian equations with the nonlinearity may fail to have asymptotic limits at zero and at infinity. We make use of a combination of homotopy invariance of critical groups and the topological version of linking methods.

Existence of positive solutions for fractional Laplacian equations: theory and numerical experiments

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 positive weak solution for classes of sublinear nonlinearities including logistic type. A method of sub- and supersolution, without monotone iteration, is established to prove our existence results. We also provide numerical bifurcation diagrams and the profile of positive solutions, corresponding to the theoretical results using the finite element method in one dimension.&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2020/81/abstr.html

Existence of solution for a singular fractional Laplacian problem with variable exponents and indefinite weights

In this paper, we consider a class of fractional Laplacian problems of the form: where is a bounded domain and is the fractional -Laplacian. We assume that λ is a nonnegative parameter and is a continuous function. By using variational methods and monotonicity arguments combined with the theory of the generalized Lebesgue Sobolev spaces, we prove the existence of solutions to problem .

Nonlinear fractional Laplacian problems with nonlocal ‘gradient terms’

AbstractLet$\Omega \subset \mathbb{R}^{N} $, N ≽ 2, be a smooth bounded domain. For s ∈ (1/2, 1), we consider a problem of the form $$\left\{\begin{array}{@{}ll} (-\Delta)^s u = \mu(x)\, \mathbb{D}_s^{2}(u) + \lambda f(x), &amp; {\rm in}\,\Omega, \\ u= 0, &amp; {\rm in}\,\mathbb{R}^{N} \setminus \Omega,\end{array}\right.$$ where λ &gt; 0 is a real parameter, f belongs to a suitable Lebesgue space, $\mu \in L^{\infty}$ and $\mathbb {D}_s^2$ is a nonlocal ‘gradient square’ term given by $$\mathbb{D}_s^2 (u) = \frac{a_{N,s}}{2} \int_{\mathbb{R}^{N}} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}\,{\rm d}y.$$ Depending on the real parameter λ &gt; 0, we derive existence and non-existence results. The proof of our existence result relies on sharp Calderón–Zygmund type regularity results for the fractional Poisson equation with low integrability data. We also obtain existence results for related problems involving different nonlocal diffusion terms.

A Simple Solver for the Fractional Laplacian in Multiple Dimensions

We present a simple discretization scheme for the hypersingular integral representa- tion of the fractional Laplace operator and solver for the corresponding fractional Laplacian problem. Through s...

Infinitely many solutions for Kirchhoff-type variable-order fractional Laplacian problems involving variable exponents

ABSTRACT In this paper, we show the existence of infinitely many solutions for Kirchhoff-type variable-order fractional Laplacian problems involving variable exponents. More precisely, we consider where is a continuous function, Ω is a bounded domain in with for all is the variable-order fractional Laplace operator, and are two continuous functions, are two parameters and In addition to using the new version of Clark's theorem due to Liu and Wang to prove the existence of infinitely many solutions for the above problem, we also apply the symmetric mountain pass theorem, fountain theorem and dual fountain theorem to obtain the same conclusion. The main feature, as well as the main difficulty, of our problem is the fact that the Kirchhoff term M could be zero at zero.

Some bifurcation results for fractional Laplacian problems

We consider the following nonlocal problem (−Δ)su=λu+f(λ,x,u)inΩ;u=0inRN∖Ω,where (−Δ)s is the fractional Laplacian operator with fixed 0<s<1, Ω⊂RN(N>2s) is a bounded domain with C1,1 boundary, CN,s>0 is a constant and λ∈R is a bifurcation parameter. Here f:R×Ω×R→R is a Carathéodory function that is sublinear at infinity. We use bifurcation theory to establish the existence of continua of the solution set bifurcating from infinity at the principal eigenvalue of (−Δ)s and discuss the nodal properties of solutions on these continua. We establish the multiplicity of solutions near the resonance and the existence of solution in the resonant case. As corollaries, we obtain anti-maximum principle and solvability for the resonant case satisfying the so called Landesman–Lazer type condition.

On a nonlocal problem with critical Sobolev growth

We obtain positive solutions for a nonlinear fractional Laplacian problem with critical Sobolev growth via localization principle and variational methods.

On a class of fractional Laplacian problems with variable exponents and indefinite weights

Let $$\Omega \subset {\mathbb {R}}^N, N\ge 2$$, be a bounded smooth domain. In this paper, we consider a class of fractional Laplacian problems of the form $$\begin{aligned} \left\{ \begin{array}{ll} (\Delta )^s_{p_1(x,.)}u(x)+(\Delta )^s_{p_2(x,.)}u(x) + |u|^{q(x)-2}u = \lambda V_1(x)|u(x)|^{r_1(x)-2}u(x) \\ \qquad - \mu V_2(x)|u(x)|^{r_2(x)-2}u(x) \hbox { in }\Omega , \\ u(x) = 0 \; \hbox { in } \partial \Omega , \end{array} \right. \end{aligned}$$where $$(\Delta )^s_{p_i(.,.)} (0<s<1), i=1,2$$, are the fractional $$p_i(.,.)$$-Laplacians, $$p_i\in C({\overline{\Omega }} \times {\overline{\Omega }}), q, r_i\in C({\overline{\Omega }}), i=1,2$$ while $$\lambda , \mu $$ are two positive parameters, $$V_1, V_2$$ are weight functions in generalized Lebesgue spaces $$L^{\alpha _1(.)}(\Omega )$$ and $$L^{\alpha _2(.)}(\Omega )$$ respectively such that $$V_1$$ may change sign in $$\Omega $$ and $$V_2(x)\ge 0$$ for all $$x\in \Omega $$. Using variational techniques and Ekeland’s variational principle, we establish some existence results for the problem in an appropriate space of functions.

The fractional Allen–Cahn equation with the sextic potential

We extend the classical Allen–Cahn (AC) equation to the fractional Allen–Cahn equation (FAC) with triple-well potential. By replacing the spatial Laplacian and double-well potential with fractional Laplacian and triple-well potential, we observe different dynamics. This study leads us to understand different properties of the FAC equation. We seek the existence, boundedness, and unique solvability of numerical solutions for the FAC equation with triple-well potential. In addition, the inclusion principle for the Allen–Cahn equation is considered. These different properties make us enhance the applicability of the phase-field method to the mathematical modeling in materials science. In computation, the spectral decomposition for the fractional operator allows us to develop a numerical method for the fractional Laplacian problem. For the periodic and discrete Laplacian matrix and vector multiplication, circulant submatices are formed in more than two-dimensional case. Even if the fast Fourier transform (FFT) can be utilized in this modeling, we construct the inverse of the doubly-block-circulant matrix for the solution of the fractional Allen–Cahn equation. In doing so, it helps to straightforwardly understand the numerical treatment, and exploit the properties of the discrete Fourier transforms.

Profile of solutions for nonlocal equations with critical and supercritical nonlinearities

We study the fractional Laplacian problem [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] is a parameter. Here, [Formula: see text] is a bounded star-shaped domain with smooth boundary and [Formula: see text]. We establish existence of a variational positive solution [Formula: see text] and characterize the asymptotic behavior of [Formula: see text] as [Formula: see text]. When [Formula: see text], we describe how the solution [Formula: see text] blows up at an interior point of [Formula: see text]. Furthermore, we prove the local uniqueness of solution of the above problem when [Formula: see text] is a convex symmetric domain of [Formula: see text] with [Formula: see text] and [Formula: see text].

On nonlocal fractional Laplacian problems with oscillating potentials

In this paper, we deal with the following fractional nonlocal $p$-Laplacian problem: \begin{equation} \begin{cases}(-\Delta )^{s}_{p}u= \lambda \beta (x) u^q + f(u) &\mbox {in } \Omega ,\\ u\geq 0,\ u\not \equiv 0 &\mbox {in } \Omega ,\\ u=0 &\mbox {in } \mathbb{R} ^{N}\setminus \Omega , \end{cases} \end{equation} where $\Omega \subset \mathbb{R} ^{N}$ is a bounded domain with a smooth boun\-dary of $\mathbb{R} ^N$, $s\in (0,1)$, $p\in (1, \infty )$, $N> s p$, $\lambda $ is a real parameter, $\beta \in L^\infty (\Omega )$ is allowed to be indefinite in sign, $q>0$ and $f:[0,+\infty )\to \mathbb{R} $ is a continuous function oscillating near the origin or at infinity. By using variational and topological methods, we obtain the existence of infinitely many solutions for the problem under consideration. The main results obtained here represent some new interesting phenomena in the nonlocal setting.

Multiplicity and bifurcation of positive solutions for nonhomogeneous semilinear fractional Laplacian problems

In this paper we consider the following nonhomogeneous semilinear fractional Laplacian problem $$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^s u+u=\lambda (f(x,u)+h(x)) \,\, \text {in}\,\, \mathbb {R}^N,\\ u\in H^s(\mathbb {R}^N), u>0\,\, \text {in}\,\, \mathbb {R}^N, \end{array}\right. } \end{aligned}$$ where $$\lambda >0$$ and $$\lim _{|x|\rightarrow \infty }f(x,u)=\overline{f}(u)$$ uniformly on any compact subset of $$[0,\infty )$$ . We prove that under suitable conditions on f and h, there exists $$0<\lambda ^*<+\infty $$ such that the problem has at least two positive solutions if $$\lambda \in (0,\lambda ^*)$$ , a unique positive solution if $$\lambda =\lambda ^*$$ , and no solution if $$\lambda >\lambda ^*$$ . We also obtain the bifurcation of positive solutions for the problem at $$(\lambda ^*,u^*)$$ and further analyse the set of positive solutions.

Multiple solutions for the fractional Laplacian problems with different asymptotic limits near infinity

In this paper we obtain the existence of two one-sign nontrivial solutions for the fractional Laplacian equations with the nonlinearity having different asymptotic limits at infinity via the mountain pass theorem and the cut-off techniques.

On fractional Laplacian problems with indefinite nonlinearity

In this paper, we study the existence of positive solutions for the following Laplacian problem with indefinite nonlinearitywhere is a bounded domain and a(x) is a sign-changing function. For suitable conditions on a(x) and p, we prove: there exists such that problem has a positive solution for and has no positive solution for , where is the first eigenvalue of the fractional Laplacian operator.

Multiplicity existence for sublinear fractional Laplacian problems

The purpose of this paper is mainly to investigate the existence of weak solutions of fractional Laplacian equations with sublinear growth and oscillatory behavior. By using variational methods and abstract critical point theory, we want to establish the existence of multiplicity positive, multiplicity negative, and in particular, of multiplicity sign changing solutions. The multiplicity results depend on the real parameter . The key point is the choice of the framework to study the existence of weak solutions. In the suitable framework, by verifying that the conditions in abstract critical point theory are satisfied, we are able to fulfill our strategy.

Infinitely many solutions for fractional Laplacian problems with local growth conditions

In this paper, we deal with the fractional Laplacian equations $$(\mathrm{P})\quad \left \{ \textstyle\begin{array}{@{}l@{\quad}l} (-\Delta)^{s} u = f(x,u), & x\in \Omega, u(x)= 0, & x\in \mathbb{R}^{N}\backslash\Omega, \end{array}\displaystyle \right . $$ where $0< s<1<p<+\infty$ , $N\in\mathbb{N}, N>2s$ , $\Omega\subset \mathbb{R}^{N}$ is a bounded domain with smooth boundary. Under local growth conditions of $f(x,t)$ , infinitely many solutions for problem (P) are obtained via variational methods.

Asymptotically compatible schemes for the approximation of fractional Laplacian and related nonlocal diffusion problems on bounded domains

Approximations of solutions of fractional Laplacian equations on bounded domains are considered. Such equations allow global interactions between points separated by arbitrarily large distances. Two approximations are introduced. First, interactions are localized so that only points less than some specified distance, referred to as the interaction radius, are allowed to interact. The resulting truncated problem is a special case of a more general nonlocal diffusion problem. The second approximation is the spatial discretization of the related nonlocal diffusion problem. A recently developed abstract framework for asymptotically compatible schemes is applied to prove convergence results for solutions of the truncated and discretized problem to the solutions of the fractional Laplacian problems. Intermediate results also provide new convergence results for the nonlocal diffusion problem. Special attention is paid to limiting behaviors as the interaction radius increases and the spatial grid size decreases, regardless of how these parameters may or may not be dependent. In particular, we show that conforming Galerkin finite element approximations of the nonlocal diffusion equation are always asymptotically compatible schemes for the corresponding fractional Laplacian model as the interaction radius increases and the grid size decreases. The results are developed with minimal regularity assumptions on the solution and are applicable to general domains and general geometric meshes with no restriction on the space dimension and with data that are only required to be square integrable. Furthermore, our results also solve an open conjecture given in the literature about the convergence of numerical solutions on a fixed mesh as the interaction radius increases.

On the Existence of a Non-trivial Non-negative Global Radial Weak Solution to a Fractional Laplacian Problem with a Singular Potential

On the Existence of a Non-trivial Non-negative Global Radial Weak Solution to a Fractional Laplacian Problem with a Singular Potential