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

2018 ◽  
Vol 149 (04) ◽  
pp. 1061-1081 ◽  
Author(s):  
Zhang Binlin ◽  
Vicenţiu D. Rădulescu ◽  
Li Wang
Keyword(s):  
Fractional Laplacian ◽  
Differential Operators ◽  
Smooth Boundary ◽  
Critical Groups ◽  
Kirchhoff Type ◽  
Existence And Multiplicity ◽  
Non Local ◽  
Image Position ◽  
Open Bounded Subset ◽  
Fractional Laplace Operator

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:$$\matrix{ {\left\{ {\matrix{ {M\left( {\int\!\!\!\int\limits_{{\open R}^{2N}} {\displaystyle{{ \vert u(x)-u(y) \vert ^2} \over { \vert x-y \vert ^{N + 2s}}}} {\rm d}x{\rm d}y} \right){(-\Delta )}^su = f(x,u)\quad } \hfill & {{\rm in }\Omega ,} \hfill \cr {u = 0\quad } \hfill & {{\rm in }{\open R}^N{\rm \setminus }\Omega {\mkern 1mu} ,} \hfill \cr } } \right.} \hfill \cr } $$where ( − Δ)sis the fractional Laplace operator,s∈ (0, 1),N> 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.

Non-Local Elliptic Boundary-Value Problems

1968 ◽  
Vol 20 ◽  
pp. 1365-1382 ◽  
Author(s):  
Bui An Ton
Keyword(s):  
Differential Operators ◽  
Smooth Boundary ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Linear Operators ◽  
Elliptic Boundary Value Problem ◽  
Elliptic Boundary Value ◽  
Open Set ◽  
Non Local ◽  
Image Position

Let G be a bounded open set of Rn with a smooth boundary ∂G. We consider the following elliptic boundary-value problem:where A and Bj are, respectively singular integro-differential operators on G and on ∂G, of orders 2m and rj with rj < 2m; Ck are boundary differential operators, and Ljk are linear operators, bounded in a sense to be specified.

scholarly journals Existence of multiple solutions of p-fractional Laplace operator with sign-changing weight function

2015 ◽  
Vol 4 (1) ◽  
pp. 37-58 ◽  
Author(s):  
Sarika Goyal ◽  
Konijeti Sreenadh
Keyword(s):  
Continuous Function ◽  
Weight Function ◽  
Multiple Solutions ◽  
Fractional Laplacian ◽  
Smooth Boundary ◽  
Existence Results ◽  
Laplacian Equation ◽  
Existence And Multiplicity ◽  
Beta 2 ◽  
Fractional Laplace Operator

AbstractIn this article, we study the following p-fractional Laplacian equation: $ (P_{\lambda }) \quad -2\int _{\mathbb {R}^n}\frac{|u(y)-u(x)|^{p-2}(u(y)-u(x))}{|x-y|^{n+p\alpha }} dy = \lambda |u(x)|^{p-2} u(x) + b(x)|u(x)|^{\beta -2}u(x) \quad \text{in } \Omega , \quad u = 0 \quad \text{in }\mathbb {R}^n \setminus \Omega ,\, u\in W^{\alpha ,p}(\mathbb {R}^n), $ where Ω is a bounded domain in ℝn with smooth boundary, n > pα, p ≥ 2, α ∈ (0,1), λ > 0 and b : Ω ⊂ ℝn → ℝ is a sign-changing continuous function. We show the existence and multiplicity of non-negative solutions of (Pλ) with respect to the parameter λ, which changes according to whether 1 < β < p or p < β < p* with p* = np(n-pα)-1 respectively. We discuss both cases separately. Non-existence results are also obtained.

EXISTENCE OF A WEAK SOLUTION FOR A CLASS OF FRACTIONAL LAPLACIAN EQUATIONS

2016 ◽  
Vol 102 (3) ◽  
pp. 392-404
Author(s):  
V. RAGHAVENDRA ◽  
RASMITA KAR
Keyword(s):  
Weak Solution ◽  
Fractional Laplacian ◽  
Nonlocal Problem ◽  
Bounded Subset ◽  
Lipschitz Boundary ◽  
Integrodifferential Operator ◽  
Image Position ◽  
Laplacian Equations ◽  
Open Bounded Subset ◽  
Fractional Type

We study the existence of a weak solution of a nonlocal problem$$\begin{eqnarray}\displaystyle & \displaystyle -{\mathcal{L}}_{K}u-\unicode[STIX]{x1D707}ug_{1}+h(u)g_{2}=f\quad \text{in }\unicode[STIX]{x1D6FA}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle u=0\quad \text{in }\mathbb{R}^{n}\setminus \unicode[STIX]{x1D6FA}, & \displaystyle \nonumber\end{eqnarray}$$where${\mathcal{L}}_{k}$is a general nonlocal integrodifferential operator of fractional type,$\unicode[STIX]{x1D707}$is a real parameter and$\unicode[STIX]{x1D6FA}$is an open bounded subset of$\mathbb{R}^{n}$($n>2s$, where$s\in (0,1)$is fixed) with Lipschitz boundary$\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$. Here$f,g_{1},g_{2}:\unicode[STIX]{x1D6FA}\rightarrow \mathbb{R}$and$h:\mathbb{R}\rightarrow \mathbb{R}$are functions satisfying suitable hypotheses.

New Results from Old Investigation: A Note on Fractional M-Dimensional Differential Operators. The Fractional Laplacian

2015 ◽  
Vol 18 (2) ◽  
Author(s):  
Humberto Prado ◽  
Margarita Rivero ◽  
Juan J. Trujillo ◽  
M. Pilar Velasco
Keyword(s):  
Fractional Integral ◽  
Fractional Laplacian ◽  
Singular Integrals ◽  
Differential Operators ◽  
Riesz Potential ◽  
Fractional Power ◽  
Potential Operators ◽  
Relevant Role ◽  
Non Local ◽  
Fractional Power Of Operators

AbstractThe non local fractional Laplacian plays a relevant role when modeling the dynamics of many processes through complex media. From 1933 to 1949, within the framework of potential theory, the Hungarian mathematician Marcel Riesz discovered the well known Riesz potential operators, a generalization of the Riemann-Liouville fractional integral to dimension higher than one. The scope of this note is to highlight that in the above mentioned works, Riesz also gave the necessary tools to introduce several new definitions of the generalized coupled fractional Laplacian which can be applied to much wider domains of functions than those given in the literature, which are based in both the theory of fractional power of operators or in certain hyper-singular integrals. Moreover, we will introduce the corresponding fractional hyperbolic differential operator also called fractional Lorentzian Laplacian.

Semi-classical solutions for Kirchhoff type problem with a critical frequency

2020 ◽  
pp. 1-38 ◽  
Author(s):  
Qilin Xie ◽  
Xu Zhang
Keyword(s):  
Variational Method ◽  
Critical Frequency ◽  
Classical Solutions ◽  
Positive Parameter ◽  
Type Problem ◽  
Kirchhoff Type ◽  
Classical State ◽  
Existence And Multiplicity ◽  
Kirchhoff Type Problem ◽  
Image Position

Abstract In the present paper, we consider the following Kirchhoff type problem $$ -\Big(\varepsilon^2+\varepsilon b \int_{\mathbb R^3} | \nabla v|^2\Big) \Delta v+V(x)v=|v|^{p-2}v \quad {\rm in}\ \mathbb{R}^3, $$ where b > 0, p ∈ (4, 6), the potential $V\in C(\mathbb R^3,\mathbb R)$ and ɛ is a positive parameter. The existence and multiplicity of semi-classical state solutions are obtained by variational method for this problem with several classes of critical frequency potentials, i.e., $\inf _{\mathbb R^N} V=0$ . As to Kirchhoff type problem, little has been done for the critical frequency cases in the literature, especially the potential may vanish at infinity.

scholarly journals On the spectrum of two different fractional operators

2014 ◽  
Vol 144 (4) ◽  
pp. 831-855 ◽  
Author(s):  
Raffaella Servadei ◽  
Enrico Valdinoci
Keyword(s):  
Boundary Data ◽  
Fractional Laplacian ◽  
Dirichlet Boundary ◽  
Fractional Operators ◽  
Local Operators ◽  
Non Local ◽  
Definition Of ◽  
Image Position ◽  
The Laplace Operator ◽  
Fractional Type

In this paper we deal with two non-local operators that are both well known and widely studied in the literature in connection with elliptic problems of fractional type. More precisely, for a fixed s ∈ (0,1) we consider the integral definition of the fractional Laplacian given bywhere c(n, s) is a positive normalizing constant, and another fractional operator obtained via a spectral definition, that is,where ei, λi are the eigenfunctions and the eigenvalues of the Laplace operator −Δ in Ω with homogeneous Dirichlet boundary data, while ai represents the projection of u on the direction ei.The aim of this paper is to compare these two operators, with particular reference to their spectrum, in order to emphasize their differences.

Existence and multiplicity of solutions for an indefinite Kirchhoff-type equation in bounded domains

2016 ◽  
Vol 146 (2) ◽  
pp. 435-448 ◽  
Author(s):  
Juntao Sun ◽  
Tsung-fang Wu
Keyword(s):  
Variational Principle ◽  
Mountain Pass Theorem ◽  
Type Equation ◽  
Bounded Domains ◽  
Type Problem ◽  
Smooth Bounded Domain ◽  
Kirchhoff Type ◽  
Existence And Multiplicity ◽  
Kirchhoff Type Problem ◽  
Image Position

We study the indefinite Kirchhoff-type problem where Ω is a smooth bounded domain in and . We require that f is sublinear at the origin and superlinear at infinity. Using the mountain pass theorem and Ekeland variational principle, we obtain the multiplicity of non-trivial non-negative solutions. We improve and extend some recent results in the literature.

On the fractional Lazer-McKenna conjecture with critical growth

2021 ◽  
pp. 1-33
Author(s):  
Qi Li ◽  
Shuangjie Peng
Keyword(s):  
Elliptic Equation ◽  
Critical Exponent ◽  
Bounded Domain ◽  
Laplace Operator ◽  
Reduction Method ◽  
Smooth Boundary ◽  
Infinitely Many Solutions ◽  
Positive Eigenfunction ◽  
Image Position ◽  
Fractional Laplace Operator

This paper deals with the following fractional elliptic equation with critical exponent \[ \begin{cases} \displaystyle (-\Delta )^{s}u=u_{+}^{2_{s}^{*}-1}+\lambda u-\bar{\nu}\varphi_{1}, & \text{in}\ \Omega,\\ \displaystyle u=0, & \text{in}\ {{\mathfrak R}}^{N}\backslash \Omega, \end{cases}\] where $\lambda$ , $\bar {\nu }\in {{\mathfrak R}}$ , $s\in (0,1)$ , $2^{*}_{s}=({2N}/{N-2s})\,(N>2s)$ , $(-\Delta )^{s}$ is the fractional Laplace operator, $\Omega \subset {{\mathfrak R}}^{N}$ is a bounded domain with smooth boundary and $\varphi _{1}$ is the first positive eigenfunction of the fractional Laplace under the condition $u=0$ in ${{\mathfrak R}}^{N}\setminus \Omega$ . Under suitable conditions on $\lambda$ and $\bar {\nu }$ and using a Lyapunov-Schmidt reduction method, we prove the fractional version of the Lazer-McKenna conjecture which says that the equation above has infinitely many solutions as $|\bar \nu | \to \infty$ .

scholarly journals EXISTENCE AND CONCENTRATION OF SOLUTION FOR A NON-LOCAL REGIONAL SCHRÖDINGER EQUATION WITH COMPETING POTENTIALS

2018 ◽  
Vol 61 (2) ◽  
pp. 441-460 ◽  
Author(s):  
CLAUDIANOR O. ALVES ◽  
CÉSAR E. TORRES LEDESMA
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Fractional Laplacian ◽  
Positive Parameter ◽  
Concentration Phenomena ◽  
Formula Group ◽  
Non Local ◽  
Image Position

AbstractIn this paper, we study the existence and concentration phenomena of solutions for the following non-local regional Schrödinger equation $$\begin{equation*} \left\{ \begin{array}{l} \epsilon^{2\alpha}(-\Delta)_\rho^{\alpha} u + Q(x)u = K(x)|u|^{p-1}u,\;\;\mbox{in}\;\; \mathbb{R}^n,\\ u\in H^{\alpha}(\mathbb{R}^n) \end{array} \right. \end{equation*}$$ where ϵ is a positive parameter, 0 < α < 1, $1<p<\frac{n+2\alpha}{n-2\alpha}$, n > 2α; (−Δ)ρα is a variational version of the regional fractional Laplacian, whose range of scope is a ball with radius ρ(x) > 0, ρ, Q, K are competing functions.

A bifurcation result for non-local fractional equations

2015 ◽  
Vol 13 (04) ◽  
pp. 371-394 ◽  
Author(s):  
Giovanni Molica Bisci ◽  
Raffaella Servadei
Keyword(s):  
Existence Result ◽  
Growth Conditions ◽  
Bounded Subset ◽  
Bifurcation Theorem ◽  
Fractional Equations ◽  
Non Local ◽  
Critical Point Result ◽  
Open Bounded Subset ◽  
Fractional Laplace Operator ◽  
Fractional Equation

In the present paper, we consider problems modeled by the following non-local fractional equation [Formula: see text] where s ∈ (0, 1) is fixed, (-Δ)sis the fractional Laplace operator, λ and μ are real parameters, Ω is an open bounded subset of ℝn, n > 2s, with Lipschitz boundary and f is a function satisfying suitable regularity and growth conditions. A critical point result for differentiable functionals is exploited, in order to prove that the problem admits at least one non-trivial and non-negative (non-positive) solution, provided the parameters λ and μ lie in a suitable range. The existence result obtained in the present paper may be seen as a bifurcation theorem, which extends some results, well known in the classical Laplace setting, to the non-local fractional framework.

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