Several equivalent definitions for this energy and the fractional Laplacian on a manifold are given, and they are shown to be identical up to explicit multiplicative constants. Moreover, the precise behavior of the kernel associated with the singular integral definition of the fractional Laplacian is...
Small Order Asymptotics of the Dirichlet Eigenvalue Problem for the Fractional Laplacian ArticleOpen access01 March 2022 1Introduction Letbe a complete, noncompact Riemannian manifold andbe its Laplace–Beltrami operator. It is well understood that the long-time behavior of solutions to the heat equat...
Nehari manifoldsign-changing weight functions35J6035J5035R11In this paper, we consider a fractional p-Laplacian system with both concave–convex nonlinearities and sign-changing weight functions in bounded domains: ( Δ ) p s u = λ f ( x ) | u | q 2 u + 2 αα + β h ( x ) ...
In this paper, we consider a fractional p-Laplacian system with both concave-convex nonlinearities and sign-changing weight functions in bounded domains. With the help of the Nehari\\ manifold, we prove that the system has at least two nontrivial solutions when the pair of the parameters (\\...
Under certain assumption on a(x) and lim|x|→∞a(x)=0, the authors proved the existence of ground state solutions. In [14], the authors studied the following nonlinear Choquard equation involving fractional Laplacian (−△)α∕2u+u=(|x|−μ∗F(u))f(u)inRn,u(x)∈Hα∕2(Rn...
The fractional (p,s)-Laplacian operator (−Δ)ps is the differential of the convex functional ≔u↦1p‖u‖p≔1p∫R2n|u(x)−u(y)|p|x−y|n+psdxdydefined on the Banach space (with respect to the norm ‖u‖ defined above) ≔W0s,p(Ω)≔{u∈Lp(Ω):‖u‖<+∞}....
A further large class of discretization methods is based on discretizing the (local, second order) Laplacian and on the subsequent computation of fractional powers of the resulting stiffness matrix. In this fifth route, various approaches to computing approximations to these powers have been proposed ...
In [23], the authors present the existence and multiplicity of solutions of the Kirchhoff ψ-Hilfer fractional p-Laplacian equation using critical point theory. Researchers worked on many models of fractional differential equations using variational problems that include fractional operators, for example,...
Assume that BΛ X B11 is given by a Lipschitz pn ´ 2q´manifold with Lipschitz constant L. Let w ě 0 in D, vanishing on Λ. Assume in addition that: Obstacle Problems Involving the Fractional Laplacian | 111 1. |La w|ď c |y|a in D; 2. nondegeneracy: w pXq ě Cdβ...
As mentioned above, a related problem is theSignorini problem3.1(orboundarythin obstacle problem),in which the manifoldMis part ofBΩand one has to minimize3.1After Fichera, see ([31]), 1963. Obstacle Problems Involving the Fractional Laplacian|83the Dirichlet integral over the closed convex ...