THE RADIAL SYMMETRY OF POSITIVE SOLUTIONS FOR SEMILINEAR PROBLEMS INVOLVING WEIGHTED FRACTIONAL LAPLACIANS

This paper deals with the radial symmetry of positive solutions to the nonlocal problem(-Δ)_(γ)~su=b(x)f(u)in B_(1){0},u=h in R~N B_(1),where b:B_1→R is locally Holder continuous,radially symmetric and decreasing in the|x|direction,F:R→R is a Lipschitz function,h:B_1→R is radially symmetric,decreasing with respect to|x|in R^(N)/B_(1),B_(1) is the unit ball centered at the origin,and(-Δ)_γ~s is the weighted fractional Laplacian with s∈(0,1),γ∈[0,2s)defined by(-△)^(s)_(γ)u(x)=CN,slimδ→0+∫R^(N)/B_(δ)(x)u(x)-u(y)/|x-y|N+2s|y|^(r)dy.We consider the radial symmetry of isolated singular positive solutions to the nonlocal problem in whole space(-Δ)_(γ)^(s)u(x)=b(x)f(u)in R^(N)\{0},under suitable additional assumptions on b and f.Our symmetry results are derived by the method of moving planes,where the main difficulty comes from the weighted fractional Laplacian.Our results could be applied to get a sharp asymptotic for semilinear problems with the fractional Hardy operators(-Δ)^(s)u+μ/(|x|^(2s))u=b(x)f(u)in B_(1)\{0},u=h in R^(N)\B_(1),under suitable additional assumptions on b,f and h.

ON A SUPER POLYHARMONIC PROPERTY OF A HIGHER-ORDER FRACTIONAL LAPLACIAN

Let 0<α<2,p≥1,m∈ℕ_(+).Consider the positive solution u of the PDE(-△)^(α/2+m)u(x)=u^(p)(x) in R^(n).(0.1) In[1](Transactions of the American Mathematical Society,2021),Cao,Dai and Qin showed that,under the condition u∈Lα,(0.1)possesses a super polyharmonic property (-△)^(k+α/2)u≥0 for k=0,1,⋯,m−1.In this paper,we show another kind of super polyharmonic property(−Δ)^(k)u>0 for k=1,⋯,m−1,under the conditions and(−Δ)^(m)u≥0.Both kinds of super polyharmonic properties can lead to an equivalence between(0.1)and the integral equation u(x)=∫_(R^(n))u^(p)(y)/|x-y|^(n-2m-α)dy.One can classify solutions to(0.1)following the work of[2]and[3]by Chen,Li,Ou.

ON SINGULAR EQUATIONS INVOLVING FRACTIONAL LAPLACIAN

We study the existence and the regularity of solutions for a class of nonlocal equations involving the fractional Laplacian operator with singular nonlinearity and Radon measure data.

THE EXISTENCE OF A NONTRIVIAL WEAK SOLUTION TO A DOUBLE CRITICAL PROBLEM INVOLVING A FRACTIONAL LAPLACIAN IN R^N WITH A HARDY TERM

In this paper,we consider the existence of nontrivial weak solutions to a double critical problem involving a fractional Laplacian with a Hardy term:(−Δ)s u−γu|x|2s=|u|2∗s(β)−2 u|x|β+[Iμ∗Fα(⋅,u)](x)fα(x,u),u∈H˙s(R n),(0.1)(1)where s∈(0,1),0≤α,β<2s<n,μ∈(0,n),γ<γH,Iμ(x)=|x|−μ,Fα(x,u)=|u(x)|2#μ(α)|x|δμ(α),fα(x,u)=|u(x)|2#μ(α)−2 u(x)|x|δμ(α),2#μ(α)=(1−μ2n)⋅2∗s(α),δμ(α)=(1−μ2n)α,2∗s(α)=2(n−α)n−2s andγH=4 sΓ2(n+2s4)Γ2(n−2s4).We show that problem(0.1)admits at least a weak solution under some conditions.To prove the main result,we develop some useful tools based on a weighted Morrey space.To be precise,we discover the embeddings H˙s(R n)↪L 2∗s(α)(R n,|y|−α)↪L p,n−2s2 p+pr(R n,|y|−pr),(0.2)(2)where s∈(0,1),0<α<2s<n,p∈[1,2∗s(α))and r=α2∗s(α).We also establish an improved Sobolev inequality,(∫R n|u(y)|2∗s(α)|y|αdy)12∗s(α)≤C||u||θH˙s(R n)||u||1−θL p,n−2s2 p+pr(R n,|y|−pr),∀u∈H˙s(R n),(0.3)(3)where s∈(0,1),0<α<2s<n,p∈[1,2∗s(α)),r=α2∗s(α),0<max{22∗s(α),2∗s−12∗s(α)}<θ<1,2∗s=2nn−2s and C=C(n,s,α)>0 is a constant.Inequality(0.3)is a more general form of Theorem 1 in Palatucci,Pisante[1].By using the mountain pass lemma along with(0.2)and(0.3),we obtain a nontrivial weak solution to problem(0.1)in a direct way.It is worth pointing out that(0.2)and 0.3)could be applied to simplify the proof of the existence results in[2]and[3].

RADIAL SYMMETRY FOR SYSTEMS OF FRACTIONAL LAPLACIAN

In this paper, we consider systems of fractional Laplacian equations in ]I^n with 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].

MULTIPLICITY RESULTS FOR A NONLINEAR ELLIPTIC PROBLEM INVOLVING THE FRACTIONAL LAPLACIAN

In this paper, we consider a class of superlinear elliptic problems involving trac- tional Laplacian （-△）s/2u = λf（u） in a bounded smooth domain with zero Diriehlet bound- ary condition. We use the method on harmonic extension to study the dependence of the number of sign-changing solutions on the parameter λ.

NONEXISTENCE OF POSITIVE SOLUTIONS FOR A SEMI-LINEAR EQUATION INVOLVING THE FRACTIONAL LAPLACIAN IN R^N

In this paper, we consider the semilinear equation involving the fractional Laplacian in the Euclidian space R^n：（-△）^α/2u（x） ： f（xn）u^p（x）, x ∈R^n（0.1）in the subcritical case with 1〈 p〈n＋a/n-a.Instead of carrying out direct investigations on pseudo-differential equation （0.1）, we first seek its equivalent form in an integral equation as below：u（x）=∫R^nG∞（x, y） f（yn） u^p（y）dy,where G∞（x, y） is the Green＇s function associated with the fractional Laplacian in R^n. Employing the method of moving planes in integral forms, we are able to derive the nonexistence of positive solutions for （0.2） in the subcritical case. Thanks to the equivalence, same con- clusion is true for （0.1）.

BLOW-UP IN A FRACTIONAL LAPLACIAN MUTUALISTIC MODEL WITH NEUMANN BOUNDARY CONDITIONS

In this paper,a fractional Laplacian mutualistic system under Neumann boundary conditions is studied.Using the method of upper and lower solutions,it is proven that the solutions of the fractional Laplacian strong mutualistic model with Neumann boundary conditions will blow up when the intrinsic growth rates of species are large.

NONEXISTENCE AND SYMMETRY OF SOLUTIONS TO SOME FRACTIONAL LAPLACIAN EQUATIONS IN THE UPPER HALF SPACE

In this article, we consider the fractional Laplacian equation {（-△）α/2u=k（x）f（u）,x∈Rn＋, u=0, x Rn＋, where 0 〈α 〈 2,En＋：= {x = （x1,x2,… ,xn）｜xn〉 0}. When K is strictly decreasing with respect to ｜x＇｜, the symmetry of positive solutions is proved, where x＇ = （x1, x2,…, xn-1） ∈Rn- 1. When K is strictly increasing with respect to xn or only depend on xn, the nonexistence of positive solutions is obtained.

EXISTENCE AND NONEXISTENCE OF GLOBAL SOLUTIONS FOR A SEMI-LINEAR HEAT EQUATION WITH FRACTIONAL LAPLACIAN

In this paper, we are concerned with the existence and non-existence of global solutions of a semi-linear heat equation with fractional Laplacian. We obtain some extem sion of results of Weissler who considered the case α = 1, and h ≡ 1.

A solution to parabolic system with the fractional Laplacian

The existence of a solution to the parabolic system with the fractional Laplacian （-△） α/2, α 〉 0 is proven, this solution decays at different rates along different time sequences going to infinity. As an application, the existence of a solution to the generalized Navier-Stokes equations is proven, which decays at different rates along different time sequences going to infinity. The generalized Navier-Stokes equations are the equations resulting from replacing -△ in the Navier-Stokes equations by （-△）^m, m〉 0. At last, a similar result for 3-D incompressible anisotropic Navier-Stokes system is obtained.

A Finite-Difference Approximation for the Oneand Two-Dimensional Tempered Fractional Laplacian

This paper provides a finite-difference discretization for the one-and two-dimensional tempered fractional Laplacian and solves the tempered fractional Poisson equation with homogeneous Dirichlet boundary conditions.The main ideas are to,respectively,use linear and quadratic interpolations to approximate the singularity and non-singularity of the one-dimensional tempered fractional Laplacian and bilinear and biquadratic interpolations to the two-dimensional tempered fractional Laplacian.Then,we give the truncation errors and prove the convergence.Numerical experiments verify the convergence rates of the order O(h^2−2s).

Multiple Solutions for a Class of Variable-Order Fractional Laplacian Equations with Concave-Convex Nonlinearity

This paper is concerned with the following variable-order fractional Laplacian equations , where N ≥ 1 and N > 2s(x,y) for (x,y) ∈ Ω × Ω, Ω is a bounded domain in RN, s(⋅) ∈ C (RN × RN, (0,1)), (-Δ)s(⋅) is the variable-order fractional Laplacian operator, λ, μ > 0 are two parameters, V: Ω → [0, ∞) is a continuous function, f ∈ C(Ω × R) and q ∈ C(Ω). Under some suitable conditions on f, we obtain two solutions for this problem by employing the mountain pass theorem and Ekeland’s variational principle. Our result generalizes the related ones in the literature.

Spectral Galerkin Approximation of Fractional Optimal Control Problems with Fractional Laplacian

In this paper spectral Galerkin approximation of optimal control problem governed by fractional elliptic equation is investigated.To deal with the nonlocality of fractional Laplacian operator the Caffarelli-Silvestre extension is utilized.The first order optimality condition of the extended optimal control problem is derived.A spectral Galerkin discrete scheme for the extended problem based on weighted Laguerre polynomials is developed.A priori error estimates for the spectral Galerkin discrete scheme is proved.Numerical experiments are presented to show the effectiveness of our methods and to verify the theoretical findings.

The Heat Equation for the Dirichlet Fractional Laplacian with Negative Potentials:Existence and Blow-up of Nonnegative Solutions

We establish conditions ensuring either existence or blow-up of nonnegative solutions for the heat equation generated by the Dirichlet fractional Laplacian perturbed by negative potentials on bounded sets. The elaborated theory is supplied by some examples.

Sharp heat kernel estimates for spectral fractional Laplacian perturbed by gradients

Using Duhamel’s formula,we prove sharp two-sided estimates for the spectral fractional Laplacian’s heat kernel with time-dependent gradient perturbation in bounded C^1,1 domains.In addition,we obtain a gradient estimate as well as the Holder continuity of the heat kernel’s gradient.

Extension Problems Related to the Higher Order Fractional Laplacian

Caffarelli and Silvestre [Comm. Part. Diff. Eqs., 32, 1245-1260 （2007）] characterized the fractional Laplacian （-△）s as an operator maps Dirichlet boundary condition to Neumann condition via the harmonic extension problem to the upper half space for 0 〈 s 〈 1. In this paper, we extend this result to all s 〉 0. We also give a new proof to the dissipative a priori estimate of quasi-geostrophic equations in the framework of Lp norm using the Caffarelli-Silvestre＇s extension technique.

A radial symmetry and Liouville theorem for systems involving fractional Laplacian

We investigate the nonnegative solutions of the system involving the fractional Laplacian：{（-△）^αui（x）=fi（u）,x∈R^n,i=1,2,…,m, u（x）=（u1（x）,u2（x）,……,um（x））,where 0 〈 α 〈 1, n 〉 2, fi（u）, 1 4 ≤ 4 ≤m, are real-valued nonnegative functions of homogeneous degree Pi ≥0 and nondecreasing with respect to the independent variables ul, u2,..., urn. 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 x0 if Pi = （n ＋ 2α）/（n- 2α） for each 1 ≤ i ≤ m; and the only nonnegative solution of this system is u ≡ 0 if 1〈pi〈（n＋2α）/（n-2α） for all 1≤i≤m.

POSITIVE SOLUTIONS TO HYBRID SCHR?DINGER EQUATION WITH NORMAL AND FRACTIONAL LAPLACIANS

In this paper, we study the hybrid Schrodinger equation involving normal and fractional Laplace operator, and obtain the existence of the solutions to this class of the hybrid partial differential equation. Our main argument is variational methods.

On a Rayleigh-Faber-Krahn Inequality for the Regional Fractional Laplacian

We study a Rayleigh-Faber-Krahn inequality for regional fractional Laplacian operators.In particular,we show that there exists a compactly supported nonnegative Sobolev function u_(0)that attains the infimum(which will be a positive real number)of the set{{∫∫(u>0)×(u>0)|u(x)-u(y)|^(2)/|x-y|^(n+2σ)dxdy:u∈^(σ)(R^(n)),∫R^(n)u^(2)=1,|{u>0}|≤1}.Unlike the corresponding problem for the usual fractional Laplacian,where the domain of the integration is R^(n)×R^(n),symmetrization techniques may not apply here.Our approach is instead based on the direct method and new a priori diameter estimates.We also present several remaining open questions concerning the regularity and shape of the minimizers,and the form of the Euler-Lagrange equations.