Fractional Kirchhoff Equation Research Articles

Bound states for fractional Kirchhoff equations with critical growth

This paper deals with the following fractional Kirchhoff problems: ( ε 2 s a + ε 4 s − N b ∫ R N | ( − Δ ) s 2 u | 2 d x ) ( − Δ ) s u + V ( x ) u = | u | 2 s ∗ − 2 u , in R N , where a, b>0, s ∈ ( 0 , 1 ) , 2s<N<4s, ε is a positive parameter, 2 s ∗ = 2 N N − 2 s is the critical Sobolev exponent, V ( x ) ∈ L N 2 s ( R N ) and V ( x ) vanishes at some points in R N . In virtue of a global compactness result and Ljusternik–Schnirelman theory, we succeed in proving the multiplicity of bound states.

Multiplicity and Concentration Behavior of Solutions to a Class of Fractional Kirchhoff Equation Involving Exponential Nonlinearity

Multiplicity and Concentration Behavior of Solutions to a Class of Fractional Kirchhoff Equation Involving Exponential Nonlinearity

High Perturbations of a Fractional Kirchhoff Equation with Critical Nonlinearities

This paper concerns a fractional Kirchhoff equation with critical nonlinearities and a negative nonlocal term. In the case of high perturbations (large values of α, i.e., the parameter of a subcritical nonlinearity), existence results are obtained by the concentration compactness principle together with the mountain pass theorem and cut-off technique. The multiplicity of solutions are further considered with the help of the symmetric mountain pass theorem. Moreover, the nonexistence and asymptotic behavior of positive solutions are also investigated.

Existence of Ground State Solutions for a Class of Non-Autonomous Fractional Kirchhoff Equations

We take a look at the fractional Kirchhoff problem in this paper. Using a variational approach, we show that there exists a ground state solution for this problem. Furthermore, using the approach developed by Szulkin and Weth, we also find that positive ground state solutions exist for the fractional Kirchhoff equation with p=4.

The Existence Result for a Fractional Kirchhoff Equation Involving Doubly Critical Exponents and Combined Nonlinearities

The Existence Result for a Fractional Kirchhoff Equation Involving Doubly Critical Exponents and Combined Nonlinearities

Multiplicity of nontrivial solutions for a class of fractional Kirchhoff equations

&lt;abstract&gt;&lt;p&gt;In this article, we study a class of fractional Kirchhoff with a superlinear nonlinearity:&lt;/p&gt; &lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ \begin{equation} \begin{cases} M(\int_{\mathbb{R}^{N}}|(-\triangle)^{\frac{\alpha}{2}}u|^{2}dx)(-\triangle)^{\alpha}u+\lambda V(x)u = f(x, u)\; \; \mbox{in}\; \; \mathbb{R}^{N}, \\ u\in H^{\alpha}(\mathbb{R}^{N}), \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; N\geq1, \; \; \; \; \; \; \; \; (1.1)\notag \end{cases} \end{equation} $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt; &lt;p&gt;where $ \lambda &amp;gt; 0 $ is a parameter, $ a $ and $ b $ are positive numbers satisfying $ M(t) = am(t)+b $, $ m:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+} $ is continuous. $ V: \mathbb{R}^{N}\times\mathbb{R}\rightarrow \mathbb{R} $ is continuous. $ f $ satisfies $ \lim\limits_{|t|\rightarrow \infty}f(x, t)/|t|^{k-1} = Q(x) $ uniformly in $ x\in\mathbb{R}^{N} $ for each $ 2 &amp;lt; k &amp;lt; 2_{\alpha}^{\ast}, (2_{\alpha}^{\ast} = \frac{2N}{N-2\alpha}) $. We investigated the effects of functions $ m $ and $ Q $ on the solution. By applying the variational method, we obtain the existence of multiple solutions. Furthermore, it is worth mentioning that the ground state solution has also been obtained.&lt;/p&gt;&lt;/abstract&gt;

A Nonexistence Result of the Normalized Solutions to a Fractional Kirchhoff Equation with Doubly Critical Exponents

A Nonexistence Result of the Normalized Solutions to a Fractional Kirchhoff Equation with Doubly Critical Exponents

On existence and concentration of positive solutions for a fractional Kirchhoff equation with critical exponential growth

In this paper we consider the following fractional Kirchhoff type problem 0 & {\\rm in}\\ \\mathbb{R}, \\end{array}\\right. \\end{align*} $$]]> { ( ϵ a + ϵ b 2 π ∫ R 2 | u ( x ) − u ( y ) | 2 | x − y | 2 d x d y ) ( − Δ ) 1 2 u + V ( x ) u = f ( u ) i n R , u ∈ H 1 / 2 ( R ) , u > 0 i n R , where ϵ is a positive parameter and a,b>0 are constants; V is a positive continuous potential and f is a nonlinear term with critical exponential growth in the Trudinger–Moser sense. We obtain the existence and concentration of positive solutions by variational methods.

Multiple solutions for the logarithmic fractional Kirchhoff equation with critical or supercritical nonlinearity

In this paper, we study the following logarithmic fractional Kirchhoff equation: ( a + b ∫ R 3 | ( − Δ ) s / 2 u | 2 d x ) ( − Δ ) s u + V ( x ) u = | u | p − 2 u log ⁡ u 2 + λ | u | q − 2 u , in R 3 , where a, b>0, ( − Δ ) s is the fractional Laplace operator with q ≥ 2 s ∗ = 6 3 − 2 s and V satisfies suitable conditions. Since the presence of supercritical exponents leads to a lack of compactness, we introduce a truncation function to reduce the exponents, allowing compactness to be recovered. We show that the above equation has a ground state solution and a sign-changing solution via the constraint variation method, the quantitative deformation lemma and Moser's iterative method.

Normalized solutions for nonlinear Kirchhoff type equations with low-order fractional Laplacian and critical exponent

In this paper, we study the following fractional Kirchhoff equation with critical growth a+b∫R3|(−Δ)s2u|2dx(−Δ)su+λu=μ|u|q−2u+|u|2s∗−2uinR3,under the constraint ∫R3|u|2dx=c2, where s∈(0,1), a,b,c>0, 2<q<2s∗, μ>0, and λ∈R appears as a Lagrange multiplier. The equation has been extensively studied in the case s∈(34,1). In contrast, no existence result of normalized solutions is available for the case s∈(0,34]. Because the complicated competition between the Kirchhoff nonlocal term and the Sobolev critical term, such problem cannot be studied by applying standard variational methods, even by restricting its underlying energy functional on the Pohožaev manifold. In this paper, by using appropriate transform, we first get the equivalent system of the above problem. With the equivalence result, we obtain the nonexistence, existence and multiplicity of normalized solutions in the case s∈(0,34].

Properties of minimizers for the fractional Kirchhoff energy functional

In this paper, we are concerned with a fractional Kirchhoff equation with a general coercive potential. First, we consider some existence and nonexistence of L2-constraint minimizers for related constrained minimization problems. Most importantly, by constructing appropriate trial functions for some delicate energy estimates and studying decay properties of solution sequences, we then establish the concentration behaviors of L2-constraint minimizers for related constrained minimization problems.

Normalized ground states for fractional Kirchhoff equations with Sobolev critical exponent and mixed nonlinearities

In this paper, we study the existence of normalized ground states for nonlinear fractional Kirchhoff equations with Sobolev critical exponent and mixed nonlinearities in R3. To overcome the special difficulties created by the nonlocal term and fractional Sobolev critical term, we develop a perturbed Pohožaev method based on the Brézis–Lieb lemma and monotonicity trick. Using the Pohožaev manifold decomposition and fibering map, we prove the existence of a positive normalized ground state. Moreover, the asymptotic behavior of the obtained normalized solutions is also explored. These conclusions extend some known ones in previous papers.

Infinitely many positive multi-bump solutions for fractional Kirchhoff equations

The purpose of this paper is to study the existence of infinitely many non-radial positive solutions to the following Kirchhoff equation involving the fractional Laplacian in RN(K)(a+b∫RN×RN|u(x)−u(y)|2|x−y|N+2sdxdy)(−Δ)su+V(|x|)u=up,u>0,inRN where a,b>0, 0<s<1, N>2s, 1<p<2s⁎−1, 2s⁎=2NN−2s is the critical Sobolev exponent and V(|x|) is a positive function. Here we are interested in the case N≥3. For the case N=3, by reducing (K) to a finite dimensional problem by the Lyapunov-Schmidt reduction method, we show that (K) has infinitely solutions whose energy tends to infinity. In contrast to most existing multiplicity results in the literature, we do not need the restriction p>3. For the case N≥4, we use a scaling technique to prove that the energy on the set of solutions to (K) is bounded. It turns out that N≤3 is necessary for the existence of solutions with large energy.

Maximum principles and Liouville theorems for fractional Kirchhoff equations

Maximum principles and Liouville theorems for fractional Kirchhoff equations

Sign-changing solutions for fractional Kirchhoff equations with cubic growth in bounded domains

Sign-changing solutions for fractional Kirchhoff equations with cubic growth in bounded domains

Multiplicity and concentration results for fractional Kirchhoff equation with magnetic field

ABSTRACT In the paper, we considered the fractional Kirchhoff equation with magnetic field ( a ε 2 s + b ε 4 s − 3 [ u ] A / ε 2 ) ( − Δ ) A / ε s u + V ( x ) u = f ( | u | 2 ) u in R 3 , where ε is a small parameter, a, b are positive constants, s ∈ ( 3 4 , 1 ) , A : R 3 → R 3 is a Hölder continuous magnetic potential with exponent α ∈ ( 0 , 1 ] , [ u ] A / ε 2 = ∬ R 6 | u ( x ) − ε ı ( x − y ) ⋅ A ε ( x + y 2 ) u ( y ) | 2 | x − y | 3 + 2 s d x d y . With proper assumptions on V and f, we develop a new approach to prove the Palais–Smale condition for the corresponding energy functional and establish the multiplicity and concentration of solutions by using perturbation techniques and Ljusternik–Schnirelmann theory.

Concentration of solutions for fractional Kirchhoff equations with discontinuous reaction

In this paper, we consider the following fractional Kirchhoff equation with discontinuous nonlinearity ε2αa+ε4α-3b∫R3|(-Δ)α2u|2dx(-Δ)αu+V(x)u=H(u-β)f(u)inR3,u∈Hα(R3),u>0inR3,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{ll} \\left( \\varepsilon ^{2\\alpha }a+\\varepsilon ^{4\\alpha -3}b\\int _{{\\mathbb {R}}^3}|(-\\Delta )^{\\frac{\\alpha }{2}} u|^2{{\\mathrm{d}}}x\\right) (-\\Delta )^\\alpha {u}+V(x)u = H(u-\\beta )f(u) &{} \\quad \ ext{ in }\\,\\,{\\mathbb {R}}^3, \\\\ u\\in H^\\alpha ({\\mathbb {R}}^3),\\quad u>0 &{} \\quad \ ext{ in }\\,\\, {\\mathbb {R}}^3, \\end{array} \\right. \\end{aligned}$$\\end{document}where varepsilon ,beta >0 are small parameters, alpha in (frac{3}{4},1) and a, b are positive constants, (-Delta )^{alpha } is the fractional Laplacian operator, H is the Heaviside function, V is a positive continuous potential, and f is a superlinear continuous function with subcritical growth. By using minimax theorems together with the non-smooth theory, we obtain existence and concentration properties of positive solutions to this non-local system.

A priori bounds and existence of positive solutions for fractional Kirchhoff equations

A priori bounds and existence of positive solutions for fractional Kirchhoff equations

Ground state solution for a critical fractional Kirchhoff equation with L2-constraint

In this paper, we study the existence of ground state solution to the following fractional Kirchhoff equation with Sobolev critical exponent in RN:(1+b∫RN|(−Δ)s2u|2dx)(−Δ)su+λu=μ|u|p−2u+|u|2⁎−2u, under the L2-norm constraint:∫RN|u|2dx=a2, where 0<s<1 and N>2s. b,μ are positive numbers and a>0 is a prescribed constant. The existence results are obtained when 2<p<2+4sN.

On the singularly perturbation fractional Kirchhoff equations: Critical case

Abstract This article deals with the following fractional Kirchhoff problem with critical exponent a + b ∫ R N ∣ ( − Δ ) s 2 u ∣ 2 d x ( − Δ ) s u = ( 1 + ε K ( x ) ) u 2 s ∗ − 1 , in R N , \left(a+b\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}| {\left(-\Delta )}^{\tfrac{s}{2}}u\hspace{-0.25em}{| }^{2}{\rm{d}}x\right){\left(-\Delta )}^{s}u=\left(1+\varepsilon K\left(x)){u}^{{2}_{s}^{\ast }-1},\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}, where a , b &gt; 0 a,b\gt 0 are given constants, ε \varepsilon is a small parameter, 2 s ∗ = 2 N N − 2 s {2}_{s}^{\ast }=\frac{2N}{N-2s} with 0 &lt; s &lt; 1 0\lt s\lt 1 and N ≥ 4 s N\ge 4s . We first prove the nondegeneracy of positive solutions when ε = 0 \varepsilon =0 . In particular, we prove that uniqueness breaks down for dimensions N &gt; 4 s N\gt 4s , i.e., we show that there exist two nondegenerate positive solutions which seem to be completely different from the result of the fractional Schrödinger equation or the low-dimensional fractional Kirchhoff equation. Using the finite-dimensional reduction method and perturbed arguments, we also obtain the existence of positive solutions to the singular perturbation problems for ε \varepsilon small.