Nonlocal Integro-differential Operator Research Articles

Infinitely Many Solutions for a Superlinear Fractional p-Kirchhoff-Type Problem without the (AR) Condition

In this paper, we investigate the existence of infinitely many solutions to a fractional p-Kirchhoff-type problem satisfying superlinearity with homogeneous Dirichlet boundary conditions as follows: [a+b(∫R2Nux-uypKx-ydxdy)]Lpsu-λ|u|p-2u=gx,u, in Ω, u=0, in RN∖Ω, where Lps is a nonlocal integrodifferential operator with a singular kernel K. We only consider the non-Ambrosetti-Rabinowitz condition to prove our results by using the symmetric mountain pass theorem.

A remark on fractional p–Kirchhoff problems involving multiple zeros

ABSTRACTIn this paper, we study the multiplicity of solutions for a p–Kirchhoff type problem driven by a nonlocal integro-differential operator. As a particular case, we consider the following problem: where is the fractional p–Laplacian, with sp<N, , is a continuous function vanishing in many different points, is a continuous function, and is a Carathéodory function for each . Under some suitable assumptions, we obtain the multiplicity of solutions for the above problem by applying the mountain pass theorem. Moreover, the asymptotic behavior of solutions is also investigated. A distinguished feature of this paper is that the Kirchhoff function M has multiple zeros.

Asymptotical dynamics for non-autonomous stochastic equations driven by a non-local integro-differential operator of fractional type

Asymptotical dynamics for non-autonomous stochastic equations driven by a non-local integro-differential operator of fractional type

Existence results for Kirchhoff–type superlinear problems involving the fractional Laplacian

AbstractIn this paper, we study the existence and multiplicity of solutions for Kirchhoff-type superlinear problems involving non-local integro-differential operators. As a particular case, we consider the following Kirchhoff-type fractional Laplace equation:$$\matrix{ {\left\{ {\matrix{ {M\left( {\int\!\!\!\int\limits_{{\open R}^{2N}} {\displaystyle{{ \vert u(x)-u(y) \vert ^2} \over { \vert x-y \vert ^{N + 2s}}}} {\rm d}x{\rm d}y} \right){(-\Delta )}^su = f(x,u)\quad } \hfill &amp;amp; {{\rm in }\Omega ,} \hfill \cr {u = 0\quad } \hfill &amp;amp; {{\rm in }{\open R}^N{\rm \setminus }\Omega {\mkern 1mu} ,} \hfill \cr } } \right.} \hfill \cr } $$where ( − Δ)sis the fractional Laplace operator,s∈ (0, 1),N&amp;gt; 2s, Ω is an open bounded subset of ℝNwith smooth boundary ∂Ω,$M:{\open R}_0^ + \to {\open R}^ + $is a continuous function satisfying certain assumptions, andf(x,u) is superlinear at infinity. By computing the critical groups at zero and at infinity, we obtain the existence of non-trivial solutions for the above problem via Morse theory. To the best of our knowledge, our results are new in the study of Kirchhoff–type Laplacian problems.

Nonlocal Kirchhoff diffusion problems: local existence and blow-up of solutions

In this paper, we study a diffusion model of Kirchhoff-type driven by a nonlocal integro-differential operator. As a particular case, we consider the following diffusion problem where [u]s is the Gagliardo seminorm of u, is a bounded domain with Lipschitz boundary, is the fractional Laplacian with , is the initial function, and is continuous. Under some appropriate conditions, the local existence of nonnegative solutions is obtained by employing the Galerkin method. Then, by virtue of a differential inequality technique, we prove that the local nonnegative solutions blow-up in finite time with arbitrary negative initial energy and suitable initial values. Moreover, we give an estimate for the lower and upper bounds of the blow-up time. The main novelty is that our results cover the degenerate case, that is, the coefficient of could be zero at the origin.

Infinitely many sign-changing solutions for a nonlocal problem

In this paper, we consider the following general nonlocal problem $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\mathcal {L}_K u=f(x,u)&{}\text {in} \,\,\varOmega ,\\ u=0&{}\text {in}\,\,\mathbb {R}^N\backslash \varOmega , \end{array} \right. \end{aligned}$$ where $$\varOmega \subset \mathbb {R}^N$$ is a bounded domain with Lipschitz boundary $$\partial \varOmega $$ , $$s\in (0,1)$$ with $$2s<N$$ and $$\mathcal {L}_K$$ is a nonlocal integrodifferential operator of fractional Laplacian type. We obtain the existence of infinitely many sign-changing solutions by combining critical point theory and invariant sets of descending flow.

Combined effects for fractional Schrödinger–Kirchhoff systems with critical nonlinearities

In this paper, we investigate the existence of solutions for critical Schrödinger–Kirchhoff type systems driven by nonlocal integro–differential operators. As a particular case, we consider the following system:[see formula in PDF]where (–Δ)spis the fractionalp–Laplace operator with 0 &lt;s&lt; 1 &lt;p&lt;N/s,α,β&gt; 1 withα+β=p*s,M: ℝ+0 → ℝ+0is a continuous function,V: ℝN→ ℝ+is a continuous function, λ &gt; 0 is a real parameter. By applying the mountain pass theorem and Ekeland’s variational principle, we obtain the existence and asymptotic behaviour of solutions for the above systems under some suitable assumptions. A distinguished feature of this paper is that the above systems are degenerate, that is, the Kirchhoff function could vanish at zero. To the best of our knowledge, this is the first time to exploit the existence of solutions for fractional Schrödinger–Kirchhoff systems involving critical nonlinearities in ℝN.

Sign-changing solutions for non-local elliptic equations with asymptotically linear term

In this article, we study the existence of sign-changing solutions for a problem driven by a non-local integrodifferential operator with homogeneous Dirichlet boundary condition \begin{document}$\left\{ \begin{array}{ll}-\mathcal{L}_Ku = f(x,u) &\text{in}~Ω, \\u = 0 &\text{in}~\mathbb{R}^n\setminusΩ, \end{array} \right. \left( 1 \right)$ \end{document} where $Ω\subset\mathbb{R}^n(n≥2)$ is a bounded, smooth domain and $f(x, u)$ is asymptotically linear at infinity with respect to $u$. By introducing some new ideas and combining constraint variational method with the quantitative deformation lemma, we prove that there exists a sign-changing solution of problem (1).

Degenerate Kirchhoff-type hyperbolic problems involving the fractional Laplacian

In this paper, we are concerned with a wave problem of Kirchhoff type driven by a nonlocal integro-differential operator. As a particular case, we consider the following hyperbolic problem involving the fractional Laplacian $$\begin{aligned} {\left\{ \begin{array}{ll} u_{tt} +[u]^{2 (\theta -1)}_{s}(-\Delta )^su=|u|^{p-1}u,\ &{}\text{ in } \Omega \times {\mathbb {R}}^{+}, \\ u(\cdot ,0)=u_0,\quad u_t(\cdot ,0)=u_1,&{} \text{ in } \Omega ,\\ u=0,&{} \text{ in } ({\mathbb {R}}^N {\setminus } \Omega )\times {\mathbb {R}}^{+}_0, \end{array}\right. } \end{aligned}$$ where $$[u]_{s}$$ is the Gagliardo seminorm of u, $$s\in (0,1)$$ , $$\theta \in [1, 2_s^*/2)$$ , with $$2_s^*=2N/(N-2s)$$ , $$p\in (2\theta -1, 2_s^*-1]$$ , $$\Omega \subset {\mathbb {R}}^N$$ is a bounded domain with Lipschitz boundary $$\partial \Omega $$ , $$(-\Delta )^s$$ is the fractional Laplacian. Under some appropriate assumptions, we obtain the global existence, vacuum isolating and blowup of solutions for the above problem by combining the Galerkin method with potential wells theory. Finally, we investigate the existence of global solutions for the above problem with the critical initial conditions. The significant feature and difficulty of the above problem are that the coefficient of $$(-\Delta )^s$$ can vanish at zero.

A critical Kirchhoff type problem involving the fractional Laplacian in

In this paper, we are concerned with the existence of solutions for a critical p-Kirchhoff type problem driven by a nonlocal integro-differential operator:where is a continuous function, is a singular kernel function, is a nonlocal fractional operator, with , , f is a Carathéodory function on satisfying the Ambrosetti–Rabinowitz type condition. Under some suitable assumptions, we obtain the existence of nontrivial solutions for above problem by applying the mountain pass theorem. A distinguished feature of this paper is that M(0) may be zero, which means that the problem is degenerate. Consequently, the main theorem extends in several directions the recent results of Autuori, Fiscella and Pucci [Nonlinear Anal. 2015;125:699–714].

Asymptotic stability for nonlinear damped Kirchhoff systems involving the fractional p-Laplacian operator

This paper is devoted to the question of global and local asymptotic stability for nonlinear damped Kirchhoff systems, with homogeneous Dirichlet boundary conditions, under fairly natural assumptions on the external force f=f(t,x,u), the distributed damping Q=Q(t,x,u,ut), the perturbation term μ|u|p−2u and the dissipative term ϱ(t)M([u]sp)|ut|p−2ut, with ϱ≥0 and in Lloc1(R0+), when the initial data are in a special region. Here u=(u1,…,uN)=u(t,x) represents the vectorial displacement, with N≥1. Particular attention is devoted to the asymptotic behavior of the solutions in the linear case specified in Section 5. Finally, the results are extended to problems where the fractional p-Laplacian is replaced by a more general elliptic nonlocal integro-differential operator. The paper extends in several directions recent theorems and covers also the so-called degenerate case, that is the case in which M is zero at zero.

Existence of entire solutions for Schrödinger–Hardy systems involving two fractional operators

This paper deals with the existence of nontrivial nonnegative solutions of Schrödinger–Hardy systems driven by two possibly different fractional ℘-Laplacian operators, via various variational methods. The main features of the paper are the presence of the Hardy terms and the fact that the nonlinearities do not necessarily satisfy the Ambrosetti–Rabinowitz condition. Moreover, we consider systems including critical nonlinear terms, as treated very recently in literature, and present radial versions of the main theorems. Finally, we briefly show how to extend the previous results when the fractional Laplacian operators are replaced by more general elliptic nonlocal integro–differential operators.

Fundamental solutions of nonlocal Hörmander’s operators II

Consider the following nonlocal integro-differential operator: for $\alpha\in(0,2)$: \[\mathcal{L}^{(\alpha)}_{\sigma,b}f(x):=\mbox{p.v.}\int_{|z|<\delta}\frac{f(x+\sigma(x)z)-f(x)}{|z|^{d+\alpha}}\,\mathrm{d}z+b(x)\cdot\nabla f(x)+{\mathscr{L}}f(x),\] where $\sigma:\mathbb{R}^{d}\to\mathbb{R}^{d}\otimes\mathbb{R}^{d}$ and $b:\mathbb{R}^{d}\to\mathbb{R}^{d}$ are smooth functions and have bounded partial derivatives of all orders greater than $1$, $\delta$ is a small positive number, p.v. stands for the Cauchy principal value and ${\mathscr{L}}$ is a bounded linear operator in Sobolev spaces. Let $B_{1}(x):=\sigma(x)$ and $B_{j+1}(x):=b(x)\cdot\nabla{B}_{j}(x)-\nabla{b(x)}\cdot B_{j}(x)$ for $j\in\mathbb{N}$. Suppose $B_{j}\in C_{b}^{\infty}(\mathbb{R}^{d};\mathbb{R}^{d}\otimes\mathbb{R}^{d})$ for each $j\in\mathbb{N}$. Under the following uniform Hörmander’s type condition: for some $j_{0}\in\mathbb{N}$, \[\inf_{x\in\mathbb{R}^{d}}\inf_{|u|=1}\sum_{j=1}^{j_{0}}|uB_{j}(x)|^{2}>0,\] by using Bismut’s approach to the Malliavin calculus with jumps, we prove the existence of fundamental solutions to operator $\mathcal{L}^{(\alpha)}_{\sigma,b}$. In particular, we answer a question proposed by Nualart [Sankhyā A 73 (2011) 46–49] and Varadhan [Sankhyā A 73 (2011) 50–51].

Degenerate Kirchhoff-type diffusion problems involving the fractional [formula omitted]-Laplacian

In this paper we study the existence of a global solution for a diffusion problem of Kirchhoff type driven by a nonlocal integro-differential operator. As a particular case, we consider the following parabolic equation involving the fractional p-Laplacian: {∂tu+[u]s,p(λ−1)p(−Δ)psu=|u|q−2u,in Ω×R+,∂tu=∂u/∂t,u(x,0)=u0(x),in Ω,u(x,t)=0,in (RN∖Ω)×R0+, where [u]s,p is the Gagliardo p–seminorm of u, Ω⊂RN is a bounded domain with Lipschitz boundary ∂Ω, p<q<Np/(N−sp) with 1<p<N/s and s∈(0,1), 1≤λ<N/(N−sp), (−Δ)ps is the fractional p-Laplacian. Under some appropriate assumptions, we obtain the existence of a global solution for the problem above by the Galerkin method and potential well theory. It is worth pointing out that the main result covers the degenerate case, that is the coefficient of (−Δ)ps can vanish at zero.

Multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractional p-Laplacian

In this paper, we investigate the multiplicity of solutions for a p-Kirchhoff system driven by a nonlocal integro-differential operator with zero Dirichlet boundary data. As a special case, we consider the following fractional p-Kirchhoff system{(∑i=1k[ui]s,pp)θ−1(−Δ)psuj(x)=λj|uj|q−2uj+∑i≠jβij|ui|m|uj|m−2ujin Ω,uj=0in RN\\Ω,where , , , , is an open bounded subset of with Lipschitz boundary , N > ps with , is the fractional p-Laplacian, and for , . When and for all , two distinct solutions are obtained by using the Nehari manifold method. When and for all or and for all , the existence of infinitely many solutions is obtained by applying the symmetric mountain pass theorem. To our best knowledge, our results for the above system are new in the study of Kirchhoff problems.

Existence of solutions for fractional p-Laplacian problems via Leray-Schauder’s nonlinear alternative

In this paper, we are concerned with the existence of solutions for a class of quasilinear elliptic problems driven by a nonlocal integro-differential operator with homogeneous Dirichlet boundary data. As a particular case, we study the following problem: $$\begin{aligned}& (-\Delta)_{p}^{s} u =f (x,u )\quad \mbox{in } \Omega, \\& u=0 \quad \mbox{in } \mathbb{R}^{N}\setminus\Omega, \end{aligned}$$ where $(-\Delta)_{p}^{s}$ is the fractional p-Laplace operator, Ω is an open bounded subset of $\mathbb{R}^{N}$ with Lipschitz boundary, and $f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Caratheodory function. The existence of nonnegative solutions is obtained by using Leray-Schauder’s nonlinear alternative.

Saddle point solutions for non-local elliptic operators

The paper deals with equations driven by a non-local integrodifferential operator $\mathcal L_K$ with homogeneous Dirichlet boundary conditions. These equations have a variational structure and we find a solution for them using the Saddle Point Theorem. We prove this result for a general integrodifferential operator of fractional type and from this, as a particular case, one can derive an existence theorem for the fractional Laplacian, finding solutions of the equation $$ \begin{cases} (-\Delta)^s u=f(x,u) & {\mbox{in }} \Omega,\\ u=0 & {\mbox{in }} \mathbb{R}^n\setminus \Omega, \end{cases} $$ where the nonlinear term $f$ satisfies a linear growth condition.

Critical stationary Kirchhoff equations in $\mathbb R^N$ involving nonlocal operators

In this paper we establish existence and multiplicity of nontrivial non-negative entire (weak) solutions of a stationary Kirchhoff eigenvalue problem, involving a general nonlocal integro-differential operator. The model under consideration depends on a real parameter Λ and involves two superlinear nonlinearities, one of which could be critical or even supercritical.

Existence of solutions for a bi-nonlocal fractional [formula omitted]-Kirchhoff type problem

In this paper, we are concerned with the existence of nonnegative solutions for a p-Kirchhoff type problem driven by a non-local integro-differential operator with homogeneous Dirichlet boundary data. As a particular case, we study the following problem M(x,[u]s,pp)(−Δ)psu=f(x,u,[u]s,pp)inΩ,u=0inRN∖Ω,[u]s,pp=∬R2N|u(x)−u(y)|p|x−y|N+psdxdy, where (−Δ)ps is a fractional p-Laplace operator, Ω is an open bounded subset of RN with Lipschitz boundary, M:Ω×R0+→R+ is a continuous function and f:Ω×R×R0+→R is a continuous function satisfying the Ambrosetti–Rabinowitz type condition. The existence of nonnegative solutions is obtained by using the Mountain Pass Theorem and an iterative scheme. The main feature of this paper lies in the fact that the Kirchhoff function M depends on x∈Ω and the nonlinearity f depends on the energy of solutions.

Asymptotic symmetries for fractional operators

In this paper, we study equations driven by a non-local integrodifferential operator LK with homogeneous Dirichlet boundary conditions. More precisely, we study the problem {−LKu+V(x)u=|u|p−2u,in Ω,u=0,in RN∖Ω, where 2<p<2s∗=2NN−2s, Ω is an open bounded domain in RN for N⩾2 and V is a L∞ potential such that −LK+V is positive definite. As a particular case, we study the problem {(−Δ)su+V(x)u=|u|p−2u,in Ω,u=0,in RN∖Ω, where (−Δ)s denotes the fractional Laplacian (with 0<s<1). We give assumptions on V, Ω and K such that ground state solutions (resp. least energy nodal solutions) respect the symmetries of some first (resp. second) eigenfunctions of −LK+V, at least for p close to 2. We study the uniqueness, up to a multiplicative factor, of those types of solutions. The results extend those obtained for the local case.