We consider an elliptic partial differential equation driven by higher order fractional Laplacian $(-Δ)^{s}$, $s \in (1,2)$ with homogeneous Dirichlet boundary condition
\begin{equation*}
\left\{%
\begin{array}{ll}
(-Δ)^{s} u=f(x,u) & \text{ in }Ω,
u=0 & \text{ in } \mathbb{R}^n \setminus Ω.
\end{array}%
\right.
\end{equation*}
The above equation has a variational nature, and we investigate the existence and multiplicity results for its weak solutions under various conditions on the nonlinear term $f$: superlinear growth, concave-convex and symmetric conditions and their combinations.
The existence of two different non-trivial weak solutions is established by Mountain Pass Theorem and Ekeland's variational principle, respectively.
Furthermore, due to Fountain Theorem and its dual form, both infinitely many weak solutions with positive energy and infinitely many weak solutions with negative energy are obtained.
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In this paper, we present an approach to the fractional Dunkl Laplacian in a framework emerging from certain reflection symmetries in Euclidean spaces. Our main result is pointwise formulas, Bochner subordination, and an extension problem for the fractional Dunkl Laplacian as well
In this note, we study the asymptotic behavior of eigenvalues and eigenfunctions of the regional fractional Laplacian $(-Δ)^s$ as $ s \to 0^+$. Our analysis leads to a study of the regional logarithmic Laplacian, which arises as a formal derivative of regional fractional Laplacians at $s = 0$.
In this paper, we generalize the direct method of moving planes for the fractional Laplacian to the system case. Considering a coupled nonlinear system with fractional Laplacian, we first establish a decay at infinity principle and a narrow region principle. Using these principles, we obtain two radial symmetry results for the decaying solutions of the fractional Laplacian systems. Our method can be applied to fractional Schrödinger systems and fractional Hénon systems.
In this paper, we prove the existence and uniqueness of nonnegative renormalized solutions for the fractional p(x)-Laplacian problem with L1 data. Our results are new even in the constant exponent fractional p-Laplacian equation case.
In this paper we study the fractional Dirichlet p-sub-Laplacian in a Haar measurable set on homogeneous Lie groups. We prove fractional Sobolev and Hardy inequalities and we also present a Lyapunov-type inequality for the fractional p-sub-Laplacian. As a consequence of the Lyapunov-type inequality we show an estimate of the first eigenvalue in a quasi-ball for the Dirichlet fractional p-sub-Laplacian.
In this paper, we give some properties and remarks of the new fractional Sobolev spaces with variable exponents. We also study the eigenvalue problem involving the new fractional $p(\cdot)$-Laplacian.
In this work we analyze the behavior of the spectrum of the peridynamic fractional $p$-Laplacian, $(-Δ_p)_δ^s$, under the limit process $δ\to0^+$ or $δ\to+\infty$. We prove spectral convergence to the classical $p$-Laplacian under a suitable scaling as $δ\to0^+$ and to the fractional $p$-Laplacian as $δ\to+\infty$.
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In this paper, we consider a Lewy-Stampacchia-type inequality for the fractional Laplacian on a bounded domain in Euclidean space. Using this inequality, we can show the well-posedness of fractional-type anomalous unidirectional diffusion equations. This study is an extension of the work by Akagi-Kimura (2019) for the standard Laplacian. However, there exist several difficulties due to the nonlocal feature of the fractional Laplacian. We overcome those difficulties employing the Caffarelli-Silvestre extension of the fractional Laplacian.
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We show a convergence result of the fractional Laplacian for sequences of nonnegative functions without uniform boundedness near infinity. As an application, we construct a sequence of solutions to the fractional Nirenberg problem that blows up in the region where the prescribed functions are negative. This is a different phenomenon from the classical Nirenberg problem.
In this paper we study the obstacle problems for the Navier (spectral) fractional Laplacian $\left(-Δ_Ω\right)^{\!s}$ of order $s\in(0,1)$, in a bounded domain $Ω\subset\mathbb R^n$.
In this paper, we study qualitative properties of the fractional $p$-Laplacian. Specifically, we establish a Hopf type lemma for positive weak super-solutions of the fractional $p-$Laplacian equation with Dirichlet condition. Moreover, an optimal condition is obtained to ensure $(-\triangle)_p^s u\in C^1(\mathbb{R}^n)$ for smooth functions $u$.
We study various qualitative and quantitative (global) unique continuation properties for the fractional discrete Laplacian. We show that while the fractional Laplacian enjoys striking rigidity properties in the form of (global) unique continuation properties, the fractional discrete Laplacian does not enjoy these in general. While discretization thus counteracts the strong rigidity properties of the continuum fractional Laplacian, by discussing quantitative forms of unique continuation, we illustrate that these properties can be recovered if exponentially small (in the lattice size) correctio…
In this work we prove the non-degeneracy of the critical points of the Robin function for the Fractional Laplacian under symmetry and convexity assumptions on the domain $Ω$. This work extends to the fractional setting the results of M. Grossi concerning the classical Laplace operator.
We obtain some existence theorems for periodic solutions to several linear equations involving fractional Laplacian. We also prove that the lower bound of all periods for semilinear elliptic equations involving fractional Laplacian is not larger than some exact positive constant. Hamiltonian identity, Modica-type inequalities and an estimate of the energy for periodic solutions are also established.
We obtain fundamental imbeddings for the fractional Sobolev space with variable exponent that is a generalization of well-known fractional Sobolev spaces. As an application, we obtain a-priori bounds and multiplicity of solutions to some nonlinear elliptic problems involving the fractional $p(\cdot)$-Laplacian.
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In this paper we give sufficient conditions on the approximating domains in order to obtain the continuity of solutions for the fractional $p-$laplacian. These conditions are given in terms of the fractional capacity of the approximating domains.
In this paper, we are interested to analyze a nonlocal nonlinear parabolic equation with fractional Laplacian. We show that there are no nontrivial global weak solutions using the test function method.
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The short note here is to give a few heuristic arguments on the weird looking fractional Laplacian operator. This is certainly going to expand the vision of a reader who is looking to develope a taste for research in this direction.
We prove the attainability of the best constant in the fractional Hardy--Sobolev inequality with boundary singularity for the Spectral Dirichlet Laplacian. The main assumption is the average concavity of the boundary at the origin.
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In this paper we consider a class of critical concave convex Ambrosetti-Prodi type problems for the fractional $p$-Laplacian operator. By applying the Linking Theorem and the Mountain Pass Theorem as well, the interaction of the nonlinearities with the first eigenvalue of fractional $p$-Laplacian will be used to prove existence and multiplicity of solutions.
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