Solutions For Kirchhoff Type Equations Research Articles

Multiplicity of solutions for Kirchhoff type equations with critical growth in ℝ N

In this paper, we consider the following Kirchhoff type problem with critical growth: { − ( a + b ∫ R N | ∇u | 2 d x ) Δu = λk ( x ) | u | q − 2 u + μ | u | 2 ∗ − 2 u , x ∈ R N , u ∈ D 1 , 2 ( R N ) , where N ≥ 3 , a ≥ 0 , b>0, 1<q<2, λ , μ > 0 , 2 ∗ = 2 N N − 2 is the critical Sobolev exponent and k ( x ) ∈ L 2 ∗ / ( 2 ∗ − q ) ( R N ) is a positive function. Using variational methods and the concentration-compactness principle, we demonstrate the existence and multiplicity of solutions.

Concentration behavior and local uniqueness of normalized solutions for Kirchhoff type equation

Concentration behavior and local uniqueness of normalized solutions for Kirchhoff type equation

Normalized solutions for Kirchhoff type equations with combined nonlinearities: The Sobolev critical case

In this paper, we study the Kirchhoff equation with Sobolev critical exponent \begin{document}$ -\left(a+b\int_{ {\mathbb{R}}^3}|\nabla u|^2\right)\Delta u = \lambda u+\mu|u|^{q-2}u+|u|^{4}u\ \ {\rm in}\ {\mathbb{R}}^3 $\end{document} under the normalized constraint$ \int_{ {\mathbb{R}}^3}u^2 = c^2, $where \begin{document}$ a, \, b, \, c&gt;0 $\end{document} are constants, \begin{document}$ \lambda, \, \mu\in{\mathbb{R}} $\end{document} and \begin{document}$ 2&lt;q&lt;6 $\end{document}. The number \begin{document}$ 2+8/3 $\end{document} behaves as the \begin{document}$ L^2 $\end{document}-critical exponent for the above problem. When \begin{document}$ \mu&gt;0 $\end{document}, we distinguish the problem into four cases: \begin{document}$ 2&lt;q&lt;2+4/3 $\end{document}, \begin{document}$ q = 2+4/3 $\end{document}, \begin{document}$ 2+4/3&lt;q&lt;2+8/3 $\end{document} and \begin{document}$ 2+8/3\leq q&lt;6 $\end{document}, and prove the existence and multiplicity of normalized solutions under suitable assumptions on \begin{document}$ \mu $\end{document} and \begin{document}$ c $\end{document}. The solution obtained is either a minimum (local or global) or a mountain pass solution. When \begin{document}$ \mu\leq 0 $\end{document}, we establish the nonexistence of nonnegative normalized solutions. Finally, we investigate the asymptotic behavior of normalized solutions obtained above as \begin{document}$ \mu\to0^+ $\end{document} and as \begin{document}$ b\to0^+ $\end{document} respectively.

Multiplicity of nontrivial solutions for Kirchhoff type equations with zero mass and a critical term

In this paper, a class of Kirchhoff type equations in ℝN (N ⩾ 3) with zero mass and a critical term is studied. Under some appropriate conditions, the existence of multiple solutions is obtained by using variational methods and a variant of Symmetric Mountain Pass theorem. The Second Concentration Compactness lemma is used to overcome the lack of compactness in critical problem. Compared to the usual Kirchhoff-type problems, we only require the nonlinearity to satisfy the classical superquadratic condition (Ambrosetti-Rabinowitz condition).

Infinitely many solutions for Kirchhoff type equations involving degenerate operator

Infinitely many solutions for Kirchhoff type equations involving degenerate operator

Existence and concentration of ground state solutions for Kirchhoff type equations with general nonlinearities

In this paper, we study the following singularly perturbed Kirchhoff type equation in subcritical case where ε &gt; 0 is a parameter, and . Under some suitable conditions, we obtain that there exists a ground state solution of the equation concentrating at a global minimum of V in the semiclassical limit by applying some new techniques and subtle analyses. Moreover, we use Moser iteration to conquer the difficulty caused by the lack of Sobolev inequality.

Mathematical behavior of solutions of the Kirchhoff type equation with logarithmic nonlinearity

We consider the existence and decay estimates of solutions for Kirchhoff type equation with damping logarithmic source term. We proved global existence of solutions under suitable conditons by potential well method and the decay estimates result of the solutions for subcritical energy level.

Bound state solutions for Kirchhoff type equations in dimension two

In this paper, we consider the following Kirchhoff type equation:−M(∫R2|∇u|2dx)Δu+u=a(x)f(u)inR2, where M is a general function with infR+⁡M>0 and f is a superlinear subcritical term. We obtain bound state solutions by developing some techniques in variational methods.

Existence of solutions for a class of quasilinear Schrödinger equation with a Kirchhoff-type

In this paper, we discuss the existence of multiple solutions of the generalized quasilinear Schrödinger equation of Kirchhoff-type where b>0 is a parameter, . In preceding approaches, the order of the nonlinear term about t at infinity is usually supposed to be higher than . But in this paper, this condition is not necessary. Under weaker condition , where is the primitive function of , , we get the existence of infinitely many solutions of the equation. Our work can be regarded as the extension of those such as [J. Zhang et al. Existence of multiple solutions of Kirchhoff type equation with sign-changing potential. Appl Math Comput. 2014;242:491–499] and related works.

Existence of solutions to Kirchhoff type equations involving the nonlocal p 1& … &p m fractional Laplacian with critical Sobolev-Hardy exponent

The aim of this paper is to study the existence of solutions for Kirchhoff type equations involving the nonlocal fractional Laplacian with critical Sobolev-Hardy exponent where are nonnegative constants and is called the critical Sobolev-Hardy exponent, . Here with is the fractional r-Laplace operator. Ω is an open bounded subset of with smooth boundary and . are continuous functions and f is a Carathéodory function which does not satisfy the Ambrosetti-Rabinowitz condition. By using the Mountain Pass Theorem, we obtain the existence of solutions for the above problem. Furthermore, using Fountain Theorem, we get the existence of infinitely many solutions for the above problem when . We also study the existence of two nontrivial solutions for the Kirchhoff type equation involving the fractional p-Laplacian via Morse theory. Finally, we consider the case and study a degenerate Kirchhoff equation involving Trudinger-Moser nonlinearity. In our best knowledge, it is the first time our problems are studied in this area.

Ground state solutions for Kirchhoff-type equations with general nonlinearity in low dimension

This paper is dedicated to studying the following Kirchhoff-type problem: {−m(∥∇u∥L2(RN)2)Δu+V(x)u=f(u),x∈RN;u∈H1(RN),\\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}$$ \\textstyle\\begin{cases} -m ( \\Vert \\nabla u \\Vert ^{2}_{L^{2}(\\mathbb{R} ^{N})} )\\Delta u+V(x)u=f(u), & x\\in \\mathbb{R} ^{N}; \\\\ u\\in H^{1}(\\mathbb{R} ^{N}), \\end{cases} $$\\end{document} where N=1,2, m:[0,infty )rightarrow (0,infty ) is a continuous function, V:mathbb{R} ^{N}rightarrow mathbb{R} is differentiable, and fin mathcal{C}(mathbb{R} ,mathbb{R} ). We obtain the existence of a ground state solution of Nehari–Pohozaev type and the least energy solution under some assumptions on V, m, and f. Especially, the existence of nonlocal term m(|nabla u|^{2}_{L^{2}(mathbb{R} ^{N})}) and the lack of Hardy’s inequality and Sobolev’s inequality in low dimension make the problem more complicated. To overcome the above-mentioned difficulties, some new energy inequalities and subtle analyses are introduced.

Multiple solutions of Kirchhoff type equations involving Neumann conditions and critical growth

<abstract> In this paper, we consider a Neumann problem of Kirchhoff type equation <p class="disp_formula">$ \begin{equation*} \begin{cases} -\left(a+b\int_{\Omega}|\nabla u|^2dx\right)\Delta u+u = Q(x)|u|^4u+\lambda P(x)|u|^{q-2}u, &amp;\rm \mathrm{in}\ \ \Omega, \\ \frac{\partial u}{\partial v} = 0, &amp;\rm \mathrm{on}\ \ \partial\Omega, \end{cases} \end{equation*} $ where $ \Omega $ $ \subset $ $ \mathbb{R}^3 $ is a bounded domain with a smooth boundary, $ a, b &gt; 0 $, $ 1 &lt; q &lt; 2 $, $ \lambda &gt; 0 $ is a real parameter, $ Q(x) $ and $ P(x) $ satisfy some suitable assumptions. By using the variational method and the concentration compactness principle, we obtain the existence and multiplicity of nontrivial solutions. </abstract>

Critical groups and multiple solutions for Kirchhoff type equations with critical exponents

In this paper, by Morse theory we will study the Kirchhoff type equation with an additional critical nonlinear term, and the main results are to compute the critical groups including the cases where zero is a mountain pass solution and the nonlinearity is resonant at zero. As an application, the multiplicity of nontrivial solutions for this equation with the parameter across the first eigenvalue is investigated under appropriate assumptions. To our best knowledge, estimates of our critical groups are new even for the Kirchhoff type equations with subcritical nonlinearities.

Existence and Multiplicity of Positive Solutions for Kirchhoff-Type Equations with the Critical Sobolev Exponent

In this paper, we consider the following Kirchhoff-type problems involving critical exponent −a+b∫Ω∇u2dxΔu+Vxu=μu2∗−1+λgx,u, x∈Ωu&gt;0, x∈Ωu=0, x∈∂Ω. The existence and multiplicity of positive solutions for Kirchhoff-type equations with a nonlinearity in the critical growth are studied under some suitable assumptions on Vx and gx,u. By using the mountain pass theorem and Brézis–Lieb lemma, the existence and multiplicity of positive solutions are obtained.

Existence results of positive solutions for Kirchhoff type equations via bifurcation methods

In this paper we address the following Kirchhoff type problem $$\begin{aligned} \left\{ \begin{array}{ll} -\varDelta (g(|\nabla u|_2^2) u + u^r) = a u + b u^p&{} \quad \text{ in }~\varOmega , u>0&{}\quad \text{ in }~\varOmega , u= 0&{} \quad \text{ on }~\partial \varOmega , \end{array} \right. \end{aligned}$$ in a bounded and smooth domain $$\varOmega $$ in $$\mathbb {R}^{N}$$ . By using change of variables and bifurcation methods, we show, under suitable conditions on the parameters a, b, p, r and the non-linearity g, the existence of positive solutions.

Infinitely many sign-changing solutions for Kirchhoff type equations

ABSTRACTIn this paper, we study the Kirchhoff type equation where a, b>0 are constants and is a given positive potential. The nonlinearity covers the pure power type with 2<p<4, a case in which few existence results of sign-changing solutions are known. By using the method of invariant sets of descending flow, we obtain a sign-changing solution for the given problem. Furthermore, if f is odd with respect to u, we prove that the problem admits infinitely many sign-changing solutions.

Existence and multiplicity of solutions for Kirchhoff type equations involving fractional p-Laplacian without compact condition

The purpose of this paper is mainly to investigate the following fractional Kirchhoff equation in $${{\mathbb {R}}}^N$$$$\begin{aligned} \left( a+b\iint _{{{\mathbb {R}}}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\mathrm{d}x\mathrm{d}y\right) ^{p-1} (-\Delta )^s_p u+\lambda V(x)|u|^{p-2}u=f(x,u), \end{aligned}$$where $$0 0$$ are constants, $$\lambda $$ is a parameter, V is sign-changing potential function satisfying some assumptions which may not guarantee the compactness of the corresponding Sobolev embedding. Under some suitable conditions, we prove the existence and multiplicity of nontrivial solutions by applying some new tricks for the above equation.

Multiplicity and asymptotic behavior of solutions for Kirchhoff type equations involving the Hardy\u2013Sobolev exponent and singular nonlinearity

In this paper, we study a class of critical elliptic problems of Kirchhoff type: \t\t\t[a+b(∫R3|∇u|2−μu2|x|2dx)2−α2](−Δu−μu|x|2)=|u|2∗(α)−2u|x|α+λf(x)|u|q−2u|x|β,\\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}$$ \\biggl[a+b \\biggl( \\int_{\\mathbb{R}^{3}}\\vert \\nabla u\\vert ^{2}-\\mu \\frac{u^{2}}{\\vert x\\vert ^{2}}\\,dx \\biggr)^{\\frac{2-\\alpha }{2}} \\biggr]\\biggl(-\\Delta u- \\mu \\frac{u}{\\vert x\\vert ^{2}}\\biggr) = \\frac{\\vert u\\vert ^{2^{*}(\\alpha )-2}u }{\\vert x\\vert ^{\\alpha }}+\\lambda \\frac{f(x)\\vert u\\vert ^{q-2}u }{\\vert x\\vert ^{\\beta }}, $$\\end{document} where a,b>0, mu in [0,1/4), alpha , beta in [0,2), and qin (1,2) are constants and 2^{*}(alpha )=6-2alpha is the Hardy–Sobolev exponent in mathbb{R}^{3}. For a suitable function f(x), we establish the existence of multiple solutions by using the Nehari manifold and fibering maps. Moreover, we regard b>0 as a parameter to obtain the convergence property of solutions for the given problem as bsearrow 0^{+} by the mountain pass theorem and Ekeland’s variational principle.

A priori bounds for positive solutions of Kirchhoff type equations

Let Ω be a bounded smooth domain in RN. Assume that 0<α<2∗−12, a>0, and b>0. We consider the following Dirichlet problem of Kirchhoff type equation (0.1)−(a+b‖∇u‖22α)Δu=|u|p−1u+h(x,u,∇u)inΩ,u=0on∂Ωwith p∈(0,2∗)∖{1}. Where 2∗=+∞ for N=2, and 2∗=N+2N−2 for N≥3. Under suitable conditions of h(x,u,∇u) (see (A), (H1) and (H2) in Section 3), we get a priori estimates for positive solutions to problem (0.1). By making use of these estimates and the continuous method, we further get some existence results for positive solutions to problem (0.1) when 0<p<1, or 2α+1<p<2∗. Effects of the term a+b‖∇u‖22α on the solution set of problem (0.1) can be seen in an example given in Section 2.

Existence and concentrating behavior of solutions for Kirchhoff type equations with steep potential well

In this paper, we consider the following Kirchhoff type equations(0.1){−(a+b∫R3|∇u|2)Δu+λV(x)u=q(x)f(u) in R3,u∈H1(R3), where a, b, λ>0, V∈C(R3,R) is a potential well, q(x) is a positive bounded function, f(s) is either asymptotically linear or asymptotically 3-linear in s at infinity. Under some other suitable conditions on V, q and f, the existence, nonexistence and concentrating behavior of solutions to problem (0.1) are obtained by using variational methods. We mainly extend the results in J. Sun and T. Wu (2014) [26], which dealt with Kirchhoff type equations with positive potential well, to Kirchhoff type equations with sign-changing potential well.