Nonlocal Integro-differential Operator Research Articles

Existence results for Cahn–Hilliard-type systems driven by nonlocal integrodifferential operators with singular kernels

We introduce a fractional variant of the Cahn–Hilliard equation settled in a bounded domain and with a possibly singular potential. We first focus on the case of homogeneous Dirichlet boundary conditions, and show how to prove the existence and uniqueness of a weak solution. The proof relies on the variational method known as minimizing movements scheme, which fits naturally with the gradient-flow structure of the equation. The interest of the proposed method lies in its extreme generality and flexibility. In particular, relying on the variational structure of the equation, we prove the existence of a solution for a general class of integrodifferential operators, not necessarily linear or symmetric, which include fractional versions of the q-Laplacian.In the second part of the paper, we adapt the argument in order to prove the existence of solutions in the case of regional fractional operators. As a byproduct, this yields an existence result in the interesting cases of homogeneous fractional Neumann boundary conditions or periodic boundary conditions.

Concentration-compactness type principle for systems with critical terms in [formula omitted

In this paper, we obtain some important variants of the Lions and Chabrowski Concentration-compactness principle, for nonlinear systems in the context of fractional Sobolev spaces with variable exponents. As an application of the results, we show the existence and asymptotic behaviour of nontrivial solutions for elliptic systems involving a new class of general nonlocal integrodifferential operators with variable exponents and critical growth conditions in RN.

Resonant problems for non-local elliptic operators with unbounded nonlinearites

<abstract><p>In this paper we study the existence of nontrivial solutions of a class of asymptotically resonant problems driven by a non-local integro-differential operator with homogeneous Dirichlet boundary conditions by applying Morse theory and critical groups for a $ C^{2} $ functional at both isolated critical points and infinity.</p></abstract>

Multiplicity of solutions for a singular Kirchhoff-type problem

This paper deals with some Kirchhoff-type problems driven by a non-local integrodifferential operator of singular elliptic type with combined nonlinearities which generalizes the fractional Laplacian operator. Our main result is to give and prove the existence of weak solutions for such problems with homogeneous Dirichlet boundary conditions. The proof is based on a variational method, precisely, we use the Nehari manifold method and the analysis of the fibering maps.

Harmonic extension technique for non-symmetric operators with completely monotone kernels

We identify a class of non-local integro-differential operators K in mathbb {R} with Dirichlet-to-Neumann maps in the half-plane mathbb {R}times (0, infty ) for appropriate elliptic operators L. More precisely, we prove a bijective correspondence between Lévy operators K with non-local kernels of the form nu (y - x), where nu (x) and nu (-x) are completely monotone functions on (0, infty ), and elliptic operators L= a(y) partial _{xx} + 2 b(y) partial _{x y} + partial _{yy}. This extends a number of previous results in the area, where symmetric operators have been studied: the classical identification of the Dirichlet-to-Neumann operator for the Laplace operator in mathbb {R}times (0, infty ) with -sqrt{-partial _{xx}}, the square root of one-dimensional Laplace operator; the Caffarelli–Silvestre identification of the Dirichlet-to-Neumann operator for nabla cdot (y^{1 - alpha } nabla ) with (-partial _{xx})^{alpha /2} for alpha in (0, 2); and the identification of Dirichlet-to-Neumann maps for operators a(y) partial _{xx} + partial _{yy} with complete Bernstein functions of -partial _{xx} due to Mucha and the author. Our results rely on recent extension of Krein’s spectral theory of strings by Eckhardt and Kostenko.

Variational framework and Lewy-Stampacchia type estimates for nonlocal operators on Heisenberg group

The aim of this article is to derive some Lewy-Stampacchia estimates and the existence of solutions for equations driven by a nonlocal integro-differential operator on the Heisenberg group.

Multiplicity results for elliptic problems involving nonlocal integrodifferential operators without Ambrosetti-Rabinowitz condition

<p style='text-indent:20px;'>In this paper, we study the existence and multiplicity of weak solutions for a general class of elliptic equations <inline-formula><tex-math id="M1">\begin{document} $( \mathscr{P}_\lambda)$\end{document}</tex-math></inline-formula> in a smooth bounded domain, driven by a nonlocal integrodifferential operator <inline-formula><tex-math id="M2">\begin{document}$ \mathscr{L}_{\mathcal{A}K} $\end{document}</tex-math></inline-formula> with Dirichlet boundary conditions involving variable exponents without Ambrosetti and Rabinowitz type growth conditions. Using different versions of the Mountain Pass Theorem, as well as, the Fountain Theorem and Dual Fountain Theorem with Cerami condition, we obtain the existence of weak solutions for the problem <inline-formula><tex-math id="M3">\begin{document} $( \mathscr{P}_\lambda)$\end{document}</tex-math></inline-formula> and we show that the problem treated has at least one nontrivial solution for any parameter <inline-formula><tex-math id="M4">\begin{document}$ \lambda >0 $\end{document}</tex-math></inline-formula> small enough as well as that the solution blows up, in the fractional Sobolev norm, as <inline-formula><tex-math id="M5">\begin{document}$ \lambda \to 0 $\end{document}</tex-math></inline-formula>. Moreover, for the sublinear case, by imposing some additional hypotheses on the nonlinearity <inline-formula><tex-math id="M6">\begin{document}$ f(x,\cdot) $\end{document}</tex-math></inline-formula>, and by using a new version of the symmetric Mountain Pass Theorem due to Kajikiya [<xref ref-type="bibr" rid="b18">18</xref>], we obtain the existence of infinitely many weak solutions which tend to zero, in the fractional Sobolev norm, for any parameter <inline-formula><tex-math id="M7">\begin{document}$ \lambda >0 $\end{document}</tex-math></inline-formula>. As far as we know, the results of this paper are new in the literature.</p>

Schrödinger–Kirchhof-type problems involving the fractional p-Laplacian with exponential growth

In this paper, we study the existence of a solution to Schrödinger–Kirchhoff-type problems involving a nonlocal integro-differential operator with the Trudinger–Moser nonlinearity. As a particular case, we consider the following fractional problem: where is a continuous function with some appropriate assumptions, is the fractional p-Laplacian, with sp = N, K, V are positive continuous functions satisfying some additional conditions, f is a continuous function on with exponential growth. By using the mountain pass theorem, we obtain the existence of solutions to the above problem in suitable Sobolev space. A novel feature of our paper is that the above problem may be degenerate, that is, the Kirchhoff function .

On fractional p-Laplacian type equations with general nonlinearities

In this paper, we study the existence and multiplicity of solutions for a class of quasi-linear elliptic problems driven by a nonlocal integro-differential operator with homogeneous Dirichlet boundary conditions. As a particular case, we study the following problem: \begin{equation*} \left\{ \begin{array}{l} (-\Delta)_p^s u= f(x,u) \quad \hfill \textrm{in} \ \Omega,\\ \quad u=0 \ \hfill \textrm{in} \ R^N \setminus \Omega, \end{array} \right.\\ \end{equation*} where $(-\Delta)_p^s$ is the fractional p-Laplacian operator, $\Omega$ is an open bounded subset of $R^N$ with Lipschitz boundary and $f:\Omega \times R \to R$ is a generic Carath\'eodory function satisfying either a $p-$sublinear or a $p-$superlinear growth condition.

Multiplicity solutions to non-local problems with general potentials and combined nonlinearities

In this paper, we study the existence and multiplicity of nontrivial solutions for equations driven by a nonlocal integrodifferential operator LK with homogeneous Dirichlet boundary conditions. Taking general potentials and combined nonlinearities into consideration, at least two weak solutions are obtained by abstract critical point results.

Kinetic maximal [formula omitted]-regularity for the fractional Kolmogorov equation with variable density

We consider the Kolmogorov equation, where the right-hand side is given by a non-local integro-differential operator comparable to the fractional Laplacian in velocity with possibly time, space and velocity dependent density. We prove that this equation admits kinetic maximal Lμp-regularity under suitable assumptions on the density and on p and μ. We apply this result to prove short-time existence of strong Lμp-solutions to quasilinear non-local kinetic partial differential equations.

Qualitative Analysis for a Degenerate Kirchhoff-Type Diffusion Equation Involving the Fractional p-Laplacian

This paper is devoted to study an anomalous diffusion model of Kirchhoff type driven by a nonlocal integro-differential operator. The properties of solutions, such as vacuum isolating phenomena, global existence, extinction, exponentially decay, exponentially growth, and finite time blow-up were studied by potential well method and energy estimate method. The results of this paper extend and complete the recent studies on this model.

Existence of Ground State Sign-Changing Solutions of Fractional Kirchhoff-Type Equation with Critical Growth

In this paper, we study the following fractional Kirchhoff-type equation $$\begin{aligned}{\left\{ \begin{array}{ll} -(a+ b\int _{{\mathbb {R}}^{N}}\int _{{\mathbb {R}}^{N}}|u(x)-u(y)|^{2}K(x-y)dxdy){\mathcal {L}}_{K}u=|u|^{2_{\alpha }^{*}-2}u+\mu f(u), ~\ x\in \Omega ,\\ u=0, ~\ x\in {\mathbb {R}}^{N}\backslash \Omega , \end{array}\right. } \end{aligned}$$ where $$\Omega \subset {\mathbb {R}}^{N}$$ is a bounded domain with a smooth boundary, $$\alpha \in (0,1)$$ , $$2\alpha<N<4\alpha $$ , $$2_{\alpha }^{*}$$ is the fractional critical Sobolev exponent and $$\mu , a, b>0$$ ; $${\mathcal {L}}_{K}$$ is non-local integrodifferential operator. Under suitable conditions on f, for $$\mu $$ large enough, by using constraint variational method and the quantitative deformation lemma, we obtain a ground state sign-changing (or nodal) solution to this problem, and its energy is strictly larger than twice that of the ground state solutions.

Ground state sign-changing solutions for fractional Laplacian equations with critical nonlinearity

<abstract> In this paper, we investigate the existence of the least energy sign-changing solutions for nonlinear elliptic equations driven by nonlocal integro-differential operators with critical nonlinearity. By using constrained minimization method and topological degree theory, we obtain a least energy sign-changing solution for them under much weaker conditions. As a particular case, we drive an existence theorem of sign-changing solutions for the fractional Laplacian equations with critical growth. </abstract>

On Critical Schrödinger–Kirchhoff-Type Problems Involving the Fractional p-Laplacian with Potential Vanishing at Infinity

The aim of this paper is to study the existence of solutions for critical Schrodinger–Kirchhoff-type problems involving a nonlocal integro-differential operator with potential vanishing at infinity. As a particular case, we consider the following fractional problem: $$\begin{aligned}&M\left( \iint _{{\mathbb {R}}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\mathrm{dxdy}+\int _{{\mathbb {R}}^N}V(x)|u(x)|^{p}\mathrm{{d}}x\right) ((-\Delta )_p^{s}u(x)+V(x)|u|^{p-2}u)\\&\quad =K(x)(\lambda f(x,u)+|u|^{p_s^{*}-2}u), \end{aligned}$$ where $$M:[0, \infty )\rightarrow [0, \infty )$$ is a continuous function, $$(-\Delta )_p^{s}$$ is the fractional p-Laplacian, $$0<s<1<p<\infty $$ with $$sp<N,$$ $$p_s^{*}=Np/(N-ps),$$ K, V are nonnegative continuous functions satisfying some conditions, and f is a continuous function on $${\mathbb {R}}^N\times {\mathbb {R}}$$ satisfying the Ambrosetti–Rabinowitz-type condition, $$\lambda >0$$ is a real parameter. Using the mountain pass theorem, we obtain the existence of the above problem in suitable space W. For this, we first study the properties of the embedding from W into $$L_{K}^{\alpha }({\mathbb {R}}^N), \alpha \in [p, p_s^{*}].$$ Then, we obtain the differentiability of energy functional with some suitable conditions on f. To the best of our knowledge, this is the first existence results for degenerate Kirchhoff-type problems involving the fractional p-Laplacian with potential vanishing at infinity. Finally, we fill some gaps of papers of do O et al. (Commun Contemp Math 18: 150063, 2016) and Li et al. (Mediterr J Math 14: 80, 2017).

Sign-changing and constant-sign solutions for elliptic problems involving nonlocal integro-differential operators

This work is devoted to study the existence of sign-changing and constant-sign solutions for nonlinear elliptic problems driven by nonlocal integro-differential operators with subcritical nonlinearity which is not locally Lipschitz continuous and there is no (AR) condition and no any monotony properties. By using some variants of the Mountain Pass Lemma, we show the existence of a positive solution, a negative solution and a sign-changing solution.

Existence and Multiplicity of Solutions for the Equation with Nonlocal Integrodifferential Operator

We are concerned with the following elliptic equation with a general nonlocal integrodifferential operator $${\mathcal {L}}_K$$ $$\begin{aligned} \begin{aligned} \left\{ \begin{array}{ll} -{\mathcal {L}}_Ku=\lambda u+f(x,u), &{}\quad \text {in}\quad \Omega ,\\ u=0, &{} \quad \text {in}\quad {\mathbb {R}}^n{\setminus }\Omega , \end{array}\right. \end{aligned} \end{aligned}$$ where $$\Omega $$ be an open-bounded set of $${\mathbb {R}}^n$$ with continuous boundary, $$\lambda \in {\mathbb {R}}$$ is a real parameter, and f is a nonlinear term with subcritical growth. We show the existence of a ground state and infinitely many pairs of solutions. The proof is based on the method of Nehari manifold for the equation with $$\lambda <\lambda _1$$ , where $$\lambda _1$$ is the first eigenvalue of the nonlocal operator $$-{\mathcal {L}}_K$$ with homogeneous Dirichlet boundary condition, and the method of generalized Nehari manifold for the equation with $$\lambda \ge \lambda _1$$ . As a concrete example, we derive the existence and multiplicity of solutions for the equation driven by fractional Laplacian $$\begin{aligned} \begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^\alpha u=\lambda u+f(x,u),&{}\quad \text {in}\quad \Omega ,\\ u=0, &{}\quad \text {in}\quad {\mathbb {R}}^n{\setminus }\Omega , \end{array}\right. \end{aligned} \end{aligned}$$ where $$0<\alpha <1$$ . The results presented here may be viewed as the extension of some classical results for the Laplacian to nonlocal fractional setting.

A class of elliptic equations involving nonlocal integrodifferential operators with sign-changing weight functions

In this paper, we investigate the existence of nontrivial weak solutions to a class of elliptic equations involving a general nonlocal integrodifferential operator LAK, two real parameters, and two weight functions, which can be sign-changing. Considering different situations concerning the growth of the nonlinearities involved in the problem (P), we prove the existence of two nontrivial distinct solutions and the existence of a continuous family of eigenvalues. The proofs of the main results are based on ground state solutions using the Nehari method, Ekeland’s variational principle, and the direct method of the calculus of variations. The difficulties arise from the fact that the operator LAK is nonhomogeneous and the nonlinear term is undefined.

Local existence, global existence and blow-up of solutions to a nonlocal Kirchhoff diffusion problem**This work is supported by NSFC (Grant No. 11201380).

In this paper, we study the following diffusion model of Kirchhoff-type driven by a nonlocal integro-differential operator where [u]s is the Gagliardo seminorm of u, is a bounded domain with Lipschitz boundary, , is a nonlocal integro-differential operator defined in (), which generalizes the fractional Laplace operator , is the initial function, and is a continuous function and there exist two constants m0 > 0 and such that As is well-known, the nonlocal Kirchhoff problem was first introduced and motivated in Fiscella and Valdinoci (2014 Nonlinear Anal. 94 156–70) and the above problem was studied by Xiang et al (2018 Nonlinearity 31 3228–50), the main results of Xiang et al (2018 Nonlinearity 31 3228–50) are as follows: The local existence of nontrivial, nonnegative weak solution for , where . The blow-up conditions for nontrivial, nonnegative weak solution when J(u0) < 0, where J(u0) denotes the initial energy.The main purpose of this paper is to extend the above results and we get: The global existence of nontrivial, nonnegative weak solution for any . The global existence and blow-up conditions for nontrivial, nonnegative weak solution when for the case , where d is a positive constant given in ().

A priori Lipschitz estimates for solutions of local and nonlocal Hamilton–Jacobi equations with Ornstein–Uhlenbeck operator

We establish a priori Lipschitz estimates for unbounded solutions of second-order Hamilton–Jacobi equations in \mathbb R^N in presence of an Ornstein–Uhlenbeck drift. We generalize the results obtained by Fujita, Ishii and Loreti (2006) in several directions. The first one is to consider more general operators. We first replace the Laplacian by a general diffusion matrix and then consider a nonlocal integro-differential operator of fractional Laplacian type. The second kind of extension is to deal with more general Hamiltonians which are merely sublinear. These results are obtained for both degenerate and nondegenerate equations.