Kirchhoff Problems Research Articles

Infinitely many solutions for a Kirchhoff problem with a subcritical exponent

This article deals with the following Kirchhoff problem:(0.1)−(a+b∫RN|∇u|2)Δu+u=K(x)|u|p−1uinRN, where a, b>0, 1<p<2⁎−1, 2⁎=2NN−2, and K(x) is a potential function. We prove that problem (0.1) has infinitely many solutions by using Lyapunov-Schmidt reduction. Our results extend and improve the results for the Schrödinger equation obtained by Badiale (2002) [3].

Discontinuous perturbations of nonhomogeneous strongly-singular Kirchhoff problems

In this paper, we are concerned with a Kirchhoff problem in the presence of a strongly-singular term perturbed by a discontinuous nonlinearity of the Heaviside type in the setting of Orlicz–Sobolev space. The presence of both strongly-singular and non-continuous terms brings up difficulties in associating a differentiable functional to the problem with finite energy in the whole space W_0^{1,Phi }(Omega ). To overcome this obstacle, we establish an optimal condition for the existence of W_0^{1,Phi }(Omega )-solutions to a strongly-singular problem, which allows us to constrain the energy functional to a subset of W_0^{1,Phi }(Omega ) in order to apply techniques of convex analysis and generalized gradient in the sense of Clarke.

Local uniqueness of multi-bump solutions for singularly perturbed Kirchhoff problems

We consider the following Kirchhoff type problem: −ɛ2a+ɛb∫R3|∇u|2Δu+V(x)u=|u|p−2u,x∈R3,where a,b>0, p∈(4,6), V∈C1(R,[0,∞)) and ɛ>0 is a parameter. Under some local assumptions on V, we prove the uniqueness of multi-bump solutions concentrating around zero points of V by a local Pohozaev identity.

On critical Kirchhoff problems driven by the fractional Laplacian

We study a nonlocal parametric problem driven by the fractional Laplacian operator combined with a Kirchhoff-type coefficient and involving a critical nonlinearity term in the Sobolev embedding sense. Our approach is of variational and topological nature. The obtained results can be viewed as a nontrivial extension to the nonlocal setting of some recent contributions already present in the literature.

A priori bounds and existence of positive solutions to a p-Kirchhoff equations

In this paper, we consider a [Formula: see text]-Kirchhoff problem with Dirichlet boundary problem in a bounded domain. Under suitable conditions, we get a priori estimates for positive solutions to an auxiliary problem by the well-known blow-up argument. As an application, a existence result for positive solutions is proved by the topological degree theory.

Positive solutions for a degenerate Kirchhoff problem

AbstractBy assuming that the Kirchhoff term has $K$ degeneracy points and other appropriated conditions, we have proved the existence of at least $K$ positive solutions other than those obtained in Santos Júnior and Siciliano [Positive solutions for a Kirchhoff problem with vanishing nonlocal term, J. Differ. Equ. 265 (2018), 2034–2043], which also have ordered $H_{0}^{1}(\Omega )$-norms. A concentration phenomena of these solutions is verified as a parameter related to the area of a region under the graph of the reaction term goes to zero.

A Multiplicity Property for a Class of Kirchhoff Problems with Magnetic Potential

This paper is concerned with the mathematical analysis of solutions to following class of magnetic Kirchhoff problems $$\begin{aligned} \left\{ \begin{array}{ll} -K\Big (\displaystyle \int _\Omega |\nabla _A u|^2dx\Big )\Delta _Au = g(x,|u|^2)u &{} \quad \mathrm{in} \ \Omega \\ {\,u|_{\partial \Omega }=0}\\ \end{array} \right\} , \end{aligned}$$ where $$\Omega $$ is a smooth bounded domain in $${{\mathbb {R}}}^N$$ ( $$N \ge 2$$ ), A is a magnetic potential, and $$\Delta _A :=(\nabla -iA)^2$$ is the magnetic Laplace operator. Additionally, $$K:[0,+\infty )\mapsto (0,+\infty )$$ and $$g:{{\bar{\Omega }}}\times {{\mathbb {R}}}\mapsto {{\mathbb {R}}}$$ are appropriate continuous functions. Under some natural hypotheses, we obtain the existence of infinitely many solutions in a related magnetic Sobolev space.

Three positive solutions for Kirchhoff problems with steep potential well and concave–convex nonlinearities

In this paper, we study a class of Kirchhoff equation with steep potential well and concave–convex nonlinearities as follows: −a+b∫RN|∇u|2dx△u+λV(x)u=f(x)|u|p−2u+g(x)|u|q−2uinRN,where a,b>0, N≥3, 1<q<2<p<min{4,2∗}, 2∗=2N∕(N−2) and V∈C(RN,R) is a steep potential well. Such problem has been rarely studied for the case 2<p<min{4,2∗} since the appearance of the term ∫RN|∇u|2dx△u. By combining the Ekeland variational principle and the filtration of Nehari manifold, we prove the multiplicity of positive solutions for the above problem when b is sufficiently small and λ is large enough. Recent results from the literature are extensively improved and extended.

Uniqueness and concentration for a fractional Kirchhoff problem with strong singularity

In this paper, we consider the following fractional Kirchhoff problem with strong singularity: {(1+b∫R3∫R3|u(x)−u(y)|2|x−y|3+2sdxdy)(−Δ)su+V(x)u=f(x)u−γ,x∈R3,u>0,x∈R3,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} (1+ b\\int _{\\mathbb{R}^{3}}\\int _{\\mathbb{R}^{3}} \\frac{ \\vert u(x)-u(y) \\vert ^{2}}{ \\vert x-y \\vert ^{3+2s}}\\,\\mathrm{d}x \\,\\mathrm{d}y )(-\\Delta )^{s} u+V(x)u = f(x)u^{-\\gamma }, & x \\in \\mathbb{R}^{3}, \\\\ u>0,& x\\in \\mathbb{R}^{3}, \\end{cases} $$\\end{document} where (-Delta )^{s} is the fractional Laplacian with 0< s<1, b>0 is a constant, and gamma >1. Since gamma >1, the energy functional is not well defined on the work space, which is quite different with the situation of 0<gamma <1 and can lead to some new difficulties. Under certain assumptions on V and f, we show the existence and uniqueness of a positive solution u_{b} by using variational methods and the Nehari manifold method. We also give a convergence property of u_{b} as brightarrow 0, where b is regarded as a positive parameter.

A Hopf type lemma and the symmetry of solutions for a class of Kirchhoff equations

<p style='text-indent:20px;'>In this paper, we proved a fractional Kirchhoff version of Hopf lemma for anti-symmetry functions and applied it to prove the symmetry and monotonicity of solutions for fractional Kirchhoff equations in the whole space by method of moving planes. We also obtain radially symmetry and monotonicity of solutions for fractional Kirchhoff equations in the unit ball. As far as we know, this is the first time to apply direct method of moving planes to fractional Kirchhoff problems.

On the existence of solutions of a nonlocal biharmonic problem

This paper is concerned with the existence of an eigenvalue for a p(x)-biharmonic Kirchhoff problem with Navier boundary condition. Under some suitable conditions, we establish that any λ > 0 is an eigenvalue . The proofs combine variational methods with energy estimates. The main results of this paper improve and generalize the previous one introduced by Kefi and Radulescu (Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 29 (2018), 439-463).

Asymptotic behavior of the unique solution for a fractional Kirchhoff problem with singularity

In this paper, we consider the following fractional Kirchhoff problem with singularity $ \left \{\begin{array}{lcl} \Big(1+ b\int_{\mathbb{R}^3}\int_{\mathbb{R}^3} \frac{|u(x)-u(y)|^2}{|x-y|^{3+2s}}\mathrm{d}x \mathrm{d}y \Big)(-\Delta)^s u+V(x)u = f(x)u^{-\gamma}, &&\quad u>0,&&\quad x\in\mathbb{R}^3, \end{array}\right. $ where $ (-\Delta)^s $ is the fractional Laplacian with $ 0

Combined effects of Choquard and singular nonlinearities in fractional Kirchhoff problems

AbstractThe aim of this paper is to study the existence and multiplicity of solutions for a class of fractional Kirchho problems involving Choquard type nonlinearity and singular nonlinearity. Under suitable assumptions, two nonnegative and nontrivial solutions are obtained by using the Nehari manifold approach combined with the Hardy-Littlehood-Sobolev inequality.

Existence of solutions for a p-Laplacian Kirchhoff type problem with nonlinear term of superlinear and subcritical growth

"This paper is concerned by the study of the existence of nonnegative and nonpositive solutions for a nonlocal quasilinear Kirchhoff problem by using the Mountain Pass lemma technique."

Multiple Solutions for a Class of New p(x)-Kirchhoff Problem without the Ambrosetti-Rabinowitz Conditions

In this paper, we consider a nonlocal p(x)-Kirchhoff problem with a p+-superlinear subcritical Caratheodory reaction term, which does not satisfy the Ambrosetti–Rabinowitz condition. Under some certain assumptions, we prove the existence of nontrivial solutions and many solutions. Our results are an improvement and generalization of the corresponding results obtained by Hamdani et al. (2020).

Bound state solutions of Choquard equations with a nonlocal operator

In this paper, we study the following Choquard equation with Kirchhoff operator where a ≥ 0, b &gt; 0, α ∈ (0, N), is the critical exponent respect to Hardy–Littlewood–Sobolev inequality, and is a given nonnegative function. By using the classical linking theorem and global compactness theorem, we prove that equation (0.1) has at least one bound state solution if is small. More intriguingly, our result covers a novel feature of Kirchhoff problems, which is the fact that the parameter a can be zero.

High energy semiclassical states for Kirchhoff problems with critical frequency

We study the following singularly perturbed Kirchhoff equation −(ϵ2a+ϵb∫R3|∇u|2dx)Δu+V(x)u=u5,u∈D1,2(R3),where ϵ>0 is a parameter, a,b>0, V∈L32(R3) is nonnegative and of critical frequency. By means of variational methods, we investigate the relation between the number of high energy semiclassical states and the topology of the zero set of V for small ϵ.

Multiple positive solutions to the fractional Kirchhoff problem with critical indefinite nonlinearities

This article concerns the existence and multiplicity of positive solutions to the fractional Kirchhoff equation with critical indefinite nonlinearities by applying the Nehari manifold approach and fibering maps.

Solutions for a Kirchhoff type problem with critical exponent in [formula omitted

In this paper, we consider the existence of solutions of the Kirchhoff problem(K)−(a+b∫RN|∇u|2dx)Δu+u=kf(u)+|u|2⁎−2u,inRN, where a,b>0,k>0 and N≥3. We transform it into an equivalent system with respect to (u,λ), which is easier to solve,(S){−Δu+u=kf(u)+|u|2⁎−2u,λ−a−bλN−22∫RN|∇u|2dx=0,inRN×R+. With the equivalence of (K) to (S), we obtain the existence of solutions for (K) by solving (S). Existence of solutions of (S) is verified mainly by mountain pass theorem without (PS) condition. Our method works without the restriction of the (AR) condition in spaces of three or more dimensions.

Least-energy nodal solutions of critical Kirchhoff problems with logarithmic nonlinearity

In this paper, we are concerned with the existence of least energy sign-changing solutions for the following fractional Kirchhoff problem with logarithmic and critical nonlinearity: a+b[u]s,pp(-Δ)psu=λ|u|q-2uln|u|2+|u|ps∗-2uinΩ,u=0inRN\\Ω,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{ll} \\left( a+b[u]_{s,p}^p\\right) (-\\Delta )^s_pu = \\lambda |u|^{q-2}u\\ln |u|^2 + |u|^{ p_s^{*}-2 }u &{}\\quad \\text {in } \\Omega , \\\\ u=0 &{}\\quad \\text {in } {\\mathbb {R}}^N{\\setminus } \\Omega , \\end{array}\\right. \\end{aligned}$$\\end{document}where N >sp with s in (0, 1), p>1, and [u]s,pp=∬R2N|u(x)-u(y)|p|x-y|N+psdxdy,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned}{}[u]_{s,p}^p =\\iint _{{\\mathbb {R}}^{2N}}\\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy, \\end{aligned}$$\\end{document}p_s^*=Np/(N-ps) is the fractional critical Sobolev exponent, Omega subset {mathbb {R}}^N(Nge 3) is a bounded domain with Lipschitz boundary and lambda is a positive parameter. By using constraint variational methods, topological degree theory and quantitative deformation arguments, we prove that the above problem has one least energy sign-changing solution u_b. Moreover, for any lambda > 0, we show that the energy of u_b is strictly larger than two times the ground state energy. Finally, we consider b as a parameter and study the convergence property of the least energy sign-changing solution as b rightarrow 0.