We study the existence and multiplicity results for the following nonlocal $p(x)$-Kirchhoff problem: \begin{equation}
\label{10} \begin{cases} -\left(a-b\int_Ω\frac{1}{p(x)}| \nabla u| ^{p(x)}dx\right)div(|\nabla u| ^{p(x)-2}\nabla u)=λ|u| ^{p(x)-2}u+g(x,u) \mbox{ in } Ω, \\ u=0,\mbox{ on } \partialΩ, \end{cases} \end{equation} where $a\geq b > 0$ are constants, $Ω\subset \mathbb{R}^N$ is a bounded smooth domain, $p\in C(\overlineΩ)$ with $N>p(x)>1$, $λ$ is a real parameter and $g$ is a continuous function. The analysis developed in this paper proposes an approach based on the idea of considering a new nonlocal term which presents interesting difficulties.
In this paper, we study two classes of Kirchhoff type problems set on a double phase framework. That is, the functional space where finding solutions coincides with the Musielak-Orlicz-Sobolev space $W^{1,\mathcal H}_0(Ω)$, with modular function $\mathcal H$ related to the so called double phase operator. Via a variational approach, we provide existence and multiplicity results.
Firstly, we use Nehari manifold and Mountain Pass Lemma to prove an existence result of positive solutions for a class of nonlocal elliptic system with Kirchhoff type. Then a multiplicity result is established by cohomological index of Fadell and Rabinowitz. We also consider the critical case and prove existence of positive least energy solution when the parameter $β$ is sufficiently large.
The fact that potentially skilled, but biologically later-maturing athletes are less likely to be selected into talent development programmes can represent a failure of Talent Identification in sports. In this article, we prove the existence of solutions for Kirchhoff type equations with Dirichlet boundaryvalue condition. We use the Mountain Pass Theorem in critical point theory, without the (PS) condition.
In this paper, we study the existence and multiplicity of weak solutions to a general type of Kirchhoff equations in magnetic fractional Orlicz-Sobolev spaces. Specifically, we appeal to Critical Point Theory to prove the existence of non-trivial solutions under the so-called Ambrosetti-Rabinowitz condition. We also state the existence of ground-state solutions. Moreover, multiplicity results which yield the existence of an unbounded sequence of solutions are also provided. Finally, we show existence under a weak-type Ambrosetti-Rabinowitz condition formulated in the framework of Orlicz spaces…
In this paper we deal with the following fractional Kirchhoff equation \begin{equation*} \left(p+q(1-s) \iint_{\mathbb{R}^{2N}} \frac{|u(x)- u(y)|^{2}}{|x-y|^{N+2s}} \, dx\,dy \right)(-Δ)^{s}u = g(u) \mbox{ in } \mathbb{R}^{N}, \end{equation*} where $s\in (0,1)$, $N\geq 2$, $p>0$, $q$ is a small positive parameter and $g: \mathbb{R}\rightarrow \mathbb{R}$ is an odd function satisfying Berestycki-Lions type assumptions. By using minimax arguments, we establish a multiplicity result for the above equation, provided that $q$ is sufficiently small.
In this paper, we study the multiplicity and concentration of the positive solutions to the following critical Kirchhoff type problem: \begin{equation*} -\left(\varepsilon^2 a+\varepsilon b\int_{\R^3}|\nabla u|^2\mathrm{d} x\right)Δu + V(x) u = f(u)+u^5\ \ {\rm in } \ \ \R^3, \end{equation*} where $\varepsilon$ is a small positive parameter, $a$, $b$ are positive constants, $V \in C(\mathbb{R}^3)$ is a positive potential, $f \in C^1(\R^+, \R)$ is a subcritical nonlinear term, $u^5$ is a pure critical nonlinearity. When $\varepsilon>0$ small, we establish the relationship between the number of …
In this paper, we show the existence and multiplicity of nontrivial, non-negative solutions of the fractional $p$-Kirchhoff problem \begin{equation*} \begin{array}{rllll} M\left(\displaystyle\int_{\mathbb{R}^{2n}}\frac{|u(x)-u(y)|^p}{\left|x-y\right|^{n+ps}}dx\,dy\right)(-Δ)^{s}_p u &=λf(x)|u|^{q-2}u+ g(x)\left|u\right|^{r-2}u\, \text{in} Ω,\\ u&=0 \;\mbox{in} \mathbb{R}^{n}\setminus Ω, \end{array} \end{equation*} where $(-Δ)^{s}_p$ is the fractional $p$-Laplace operator, $Ω$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary, $f \in L^{\frac{r}{r-q}}(Ω)$ and $g\in L^\infty(Ω)$ are sig…
In this paper, we consider the following Kirchhoff type equation $$ -\left(a+ b\int_{\R^3}|\nabla u|^2\right)\triangle {u}+V(x)u=f(u),\,\,x\in\R^3, $$ where $a,b>0$ and $f\in C(\R,\R)$, and the potential $V\in C^1(\R^3,\R)$ is positive, bounded and satisfies suitable decay assumptions. By using a new perturbation approach together with a new version of global compactness lemma of Kirchhoff type, we prove the existence and multiplicity of bound state solutions for the above problem with a general nonlinearity. We especially point out that neither the corresponding Ambrosetti-Rabinowitz conditio…
This paper is concerned with the following fractional $p$-Kirchhoff equation \begin{eqnarray*} \varepsilon ^{sp}M\left( {\varepsilon ^{sp - N}}\iint_{\mathbb{R}^{2N}}\frac{{{{\left| {u(x) - u(y)} \right|}^p}}}{{{{\left| {x - y} \right|}^{N + sp}}}}dxdy\right)(-Δ)_p^su + V(x){u^{p - 1}} = {u^{p_s^* - 1}}+f(u),\ \ u>0, \ \mbox{in}\ {\mathbb{R}^N}, %u \in {W^{s,p}}(\mathbb{R}^N), \end{eqnarray*} where $\varepsilon>0$ is a parameter, $M(t)=a+bt^{θ-1}$ with $a>0$, $b>0$, $θ>1$, $(-Δ)_p^s$ denotes the fractional $p$-Laplacian operator with $0<s<1$ and $1<p<\infty$, $N>sp$, $θp<p_s^*$ with $p_s^*=\fr…
This paper deals with the existence and multiplicity of solutions for a class of Kirchhoff type elliptic system involving the Trudinger-Moser exponential growth nonlinearities. We first study the existence of solutions for the following system \begin{eqnarray*} \left\{ \arraycolsep=1.5pt \begin{array}{ll} -\big(a_1+b_1\|u\|^{2(θ_1-1)}\big)Δu= λH_u(x,u,v)\ \ \ &\ \mbox{in}\ \ \ Ω,\\[2mm] -\big(a_2+b_2\|v\|^{2(θ_2-1)}\big)Δv= λH_v(x,u,v)\ \ \ &\ \mbox{in}\ \ \ Ω,\\[2mm] u=0, v=0\ \ \ \ &\ \mbox{on}\ \ \ \partialΩ, \end{array} \right. \end{eqnarray*} where $Ω$ is a bounded domain in $\mathbb{R}^2…
In this article, we investigate the existence and multiplicity of solutions of Kirchhoff equation \begin{equation*} \left\{ \begin{aligned} -(1+b \int_{\mathbb{R}^3}|\nabla u|^2)Δu= k(x)\frac{|u|^2 u}{|x|} +λh(x)u,~~x\in\mathbb{R}^3\\ u(x)\rightarrow 0 ~~~~~~~~~~~~~~~~~~~~~~~as ~~~~|x|\rightarrow\infty \end{aligned} \right. \end{equation*} where the potential $k(x)$ allows sign changing. Making use of Nehari manifold method and Concentration-compactness principle, we obtain the existence and multiplicity of solutions for this equation.
In this paper we are concerned with questions of multiplicity and concentration behavior of positive solutions of the elliptic problem $$\left\{\begin{array}{rcl} \mathcal{L}_{\varepsilon}u = f(u) \ \ \mbox{in} \ \ \mathbb{R}^3,\\ u>0 \ \ \mbox{in} \ \ \mathbb{R}^3,\\ u \in H^1 (\mathbb{R}^3), \end{array} \right.$$ where $\varepsilon$ is a small positive parameter, $f:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous function, $\mathcal{L}_{\varepsilon}$ is a nonlocal operator defined by $$ \mathcal{L}_{\varepsilon} u = M \left(\frac{1}{\varepsilon} \int_{\mathbb{R}^3} |\nabla u|^2 + \frac{1}{…
In this paper, we prove the existence of infinitely many solutions for a class of quasilinear elliptic $m(x)$-polyharmonic Kirchhoff equations where the nonlinear function has a quasicritical growth at infinity and without assuming the Ambrosetti and Rabinowitz type condition. The new aspect consists in employing the notion of a Schauder basis to verify the geometry of the symmetric mountain pass theorem. Furthermore, for the case $m(x)\equiv Const$, we introduce a positive quantity $λ_M$ similar to the first eigenvalue of the $m$-polyharmonic operator to find a mountain pass solution, and als…
In the present note we prove a multiplicity result for a Kirchhoff type problem involving a critical term, giving a partial positive answer to a problem raised by Ricceri.
We investigate the existence, multiplicity and concentration of nontrivial solutions for the following fractional magnetic Kirchhoff equation with critical growth: \begin{equation*} \left(a\varepsilon^{2s}+b\varepsilon^{4s-3} [u]_{A/\varepsilon}^{2}\right)(-Δ)_{A/\varepsilon}^{s}u+V(x)u=f(|u|^{2})u+|u|^{\2-2}u \quad \mbox{ in } \mathbb{R}^{3}, \end{equation*} where $\varepsilon$ is a small positive parameter, $a, b>0$ are fixed constants, $s\in (\frac{3}{4}, 1)$, $2^{*}_{s}=\frac{6}{3-2s}$ is the fractional critical exponent, $(-Δ)^{s}_{A}$ is the fractional magnetic Laplacian, $A:\mathbb{R}^{…
We investigate the existence and multiplicity of weak solutions for a nonlinear Kirchhoff type quasilinear elliptic system on the whole space $\mathbb{R}^N$. We assume that the nonlinear term satisfies the locally super-$(m_1,m_2)$ condition, that is, $\lim_{|(u,v)|\rightarrow+\infty}\frac{F(x,u,v)}{|u|^{m_1}+|v|^{m_2}}=+\infty \mbox{ for a.e. } x \in G$ where $G$ is a domain in $\mathbb{R}^N$, which is weaker than the well-known Ambrosseti-Rabinowitz condition and the naturally global restriction, $\lim_{|(u,v)|\rightarrow+\infty}\frac{F(x,u,v)}{|u|^{m_1}+|v|^{m_2}}=+\infty \mbox{ for a.e. } …
We study a nonlocal elliptic equation of $p$-Kirchhoff type involving the critical Sobolev exponent. First we give sufficient conditions for the (PS) condition to hold. Then we prove some existence and multiplicity results using tools from Morse theory, in particular, the notion of a cohomological local splitting and eigenvalues based on the Fadell-Rabinowitz cohomological index.
We study a class of critical Kirchhoff problems with a general nonlocal term. The main difficulty here is the absence of a closed-form formula for the compactness threshold. First we obtain a variational characterization of this threshold level. Then we prove a series of existence and multiplicity results based on this variational characterization.
We are interested in the existence of solutions for the following fractional $p(x,\cdot)$-Kirchhoff type problem $$ \left\{\begin{array}{ll} M \, \left(\displaystyle\int_{Ω\times Ω} \ \displaystyle{\frac{|u(x)-u(y)|^{p(x,y)}}{p(x,y) \ |x-y|^{N+p(x,y)s}}} \ dx \, dy\right)(-Δ)^{s}_{p(x,\cdot)}u = f(x,u), \quad x\in Ω, \\ \\ u= 0, \quad x\in \partialΩ, \end{array}\right.$$ where $Ω\subset\mathbb{R}^{N}$, $N\geq 2$ is a bounded smooth domain, $s\in(0,1),$ $p: \overlineΩ\times \overlineΩ \rightarrow (1, \infty)$, $(-Δ)^{s}_{p(x,\cdot)}$ denotes the $p(x,\cdot)$-fractional Laplace operator, $M: [0,…
This paper is concerned with the multiplicity of nontrivial solutions in an Orlicz-Sobolev space for a nonlocal problem with critical growth, involving N-functions and theory of locally Lispchitz continuous functionals.
We establish a concentration-compactness principle for the Sobolev space $W^{2,p(\cdot)}(Ω)\cap W_0^{1,p(\cdot)}(Ω)$ that is a tool for overcoming the lack of compactness of the critical Sobolev imbedding. Using this result we obtain several existence and multiplicity results for a class of Kirchhoff type problems involving $p(\cdot)$-biharmonic operator and critical growth.
This article deals with the study of the following Kirchhoff equation with exponential nonlinearity of Choquard type (see $(KC)$ below). We use the variational method in the light of Moser-Trudinger inequality to show the existence of weak solutions to $(KC)$. Moreover, analyzing the fibering maps and minimizing the energy functional over suitable subsets of the Nehari manifold, we prove existence and multiplicity of weak solutions to convex-concave problem $(\mathcal{P}_{\la,M})$ below.
In this paper, we study multiplicity results for double phase problems of Kirchhoff type with right-hand sides that include a parametric singular term and a nonlinear term of subcritical growth. Under very general assumptions on the data, we prove the existence of at least two weak solutions that have different energy sign. Our treatment is based on the fibering method in form of the Nehari manifold. We point out that we cover both the non-degenerate as well as the degenerate Kirchhoff case in our setting.
In this paper, we study the existence of infinitely many weak solutions to a fractional Kirchhoff-Schrödinger-Poisson system involving the weak singularity, i.e. when $0<γ<1$. Further, we obtain the existence of a solution with the strong singularity, i.e. when $γ>1$. We employ variational techniques to prove the existence and multiplicity results. Moreover, a $L^{\infty}$ estimate is obtained by using the Moser iteration method.
In this article, we establish the existence of solutions to the fractional $p-$Kirchhoff type equations with a generalized Choquard nonlinearities without assuming the Ambrosetti-Rabinowitz condition.
This paper deals with some classes of Kirchhoff type problems on a double phase setting and with nonlinear boundary conditions. Under general assumptions, we provide multiplicity results for such problems in the case when the perturbations exhibit a suitable behavior in the origin and at infinity, or when they do not necessarily satisfy the Ambrosetti-Rabinowitz condition. To this aim, we combine variational methods, truncation arguments and topological tools.
In this paper we study the Kirchhoff problem \begin{equation*} \left \{ \begin{array}{ll} -m(\| u \|^{2})Δu = f(u) & \mbox{in $Ω$,} u=0 & \mbox{on $\partialΩ$,} \end{array}\right. \end{equation*} in a bounded domain, allowing the function $m$ to vanish in many different points. Under an appropriated {\sl area condition}, by using a priori estimates, truncation techniques and variational methods, we prove a multiplicity result of positive solutions which are ordered in the $H_{0}^{1}(Ω)$-norm.
The present paper deals with a parametrized Kirchhoff type problem involving a critical nonlinearity in high dimension. Existence, non existence and multiplicity of solutions are obtained under the effect of a subcritical perturbation by combining variational properties with a careful analysis of the fiber maps of the energy functional associated to the problem. The particular case of a pure power perturbation is also addressed. Through the study of the Nehari manifolds we extend the general case to a wider range of the parameters.
The paper deals with the study of the existence result for a Kirchhoff elliptic system with additive right hand side and variable parameters by using the sub-super solutions method. Our study is the second result of our previous once in (Math. Methods Appl. Sci. 41 (2018), 5203-5210).
In this note we present some recent results for Kirchhoff equations in generalized Gevrey spaces. We show that these spaces are the natural framework where classical results can be unified and extended. In particular we focus on existence and uniqueness results for initial data whose regularity depends on the continuity modulus of the nonlinear term, both in the strictly hyperbolic case, and in the degenerate hyperbolic case.
In this paper, we consider a viscoelastic kirchhoff equation with a delay term in the internal feedback. By using the Faedo-Galarkin approximation method we prove the well-posedness of the global solutions. Introducing suitable energy, we prove the general uniform decay results.
The aim of this paper is to study the multiplicity of solutions for the following Kirchhoff type elliptic systems \begin{eqnarray*} \left\{ \arraycolsep=1.5pt \begin{array}{ll} -m\left(\sum^k_{j=1}\|u_j\|^2\right)Δu_i=\frac{f_i(x,u_1,\ldots,u_k)}{|x|^β}+\varepsilon h_i(x),\ \ & \mbox{in}\ \ Ω, \ \ i=1,\ldots,k ,\\[2mm] u_1=u_2=\cdots=u_k=0,\ \ & \mbox{on}\ \ \partialΩ, \end{array} \right. \end{eqnarray*} where $Ω$ is a bounded domain in $\mathbb{R}^2$ containing the origin with smooth boundary, $β\in [0,2)$, $m$ is a Kirchhoff type function, $\|u_j\|^2=\int_Ω|\nabla u_j|^2dx$, $f_i$ behaves li…
In this paper, we study the existence of least energy nodal solutions for some class of Kirchhoff type problems. Since Kirchhoff equation is a nonlocal one, the variational setting to look for sign-changing solutions is different from the local cases. By using constrained minimization on the sign-changing Nehari manifold, we prove the Kirchhoff problem has a least energy nodal solution with its energy exceeding twice the least energy. As a co-product of our approaches, we obtain the existence of least energy sign-changing solution for Choquard equations and show that the sign-changing solution…
We study an initial-boundary-value problem for a quasilinear thermoelastic plate of Kirchhoff \& Love-type with parabolic heat conduction due to Fourier, mechanically simply supported and held at the reference temperature on the boundary. For this problem, we show the short-time existence and uniqueness of classical solutions under appropriate regularity and compatibility assumptions on the data. Further, we use barrier techniques to prove the global existence and exponential stability of solutions under a smallness condition on the initial data. It is the first result of this kind established…
In this paper we study the geometry of certain functionals associated to quasilinear elliptic boundary value problems with a degenerate nonlocal term of Kirchhoff type. Due to the degeneration of the nonlocal term it is not possible to directly use classical results such as uniform a-priori estimates and "Sobolev versus Hölder local minimizers" type of results. We prove that results similar to these hold true or not, depending on how degenerate the problem is. We apply our findings in order to show existence and multiplicity of solutions for the associated quasilinear equations, considering se…
We carry out an investigation of the existence of infinitely many solutions to a fractional $p$-Kirchhoff type problem with a singularity and a superlinear nonlinearity with a homogeneous Dirichlet boundary condition. Further the solution(s) will be proved to be bounded and a weak comparison principle has also been proved. A {\it `$C^1$ versus $W_0^{s,p}$'} analysis has also been discussed.
In this paper, we consider the multiplicity of solutions for a class of Kirchhoff type problems with sub-linear and critical terms on an unbounded domain. With the aid of Ekeland's variational principle and the concentration compactness principle we prove that the Kirchhoff problem has at least two solutions.
We consider the Cauchy problem for the Kirchhoff equation on $\mathbb{T}^d$ with initial data of small amplitude $\varepsilon$ in Sobolev class. We prove a lower bound $\varepsilon^{-4}$ for the existence time, which improves the bound $\varepsilon^{-2}$ given by the standard local theory. The proof relies on a normal form transformation, preceded by a nonlinear transformation that diagonalizes the operator at the highest order, which is needed because of the quasilinear nature of the equation.
In this paper we establish the orbital stability of standing wave solutions associated to the one-dimensional Schrödinger-Kirchhoff equation. The presence of a mixed term gives us more dispersion, and consequently, a different scenario for the stability of solitary waves in contrast with the corresponding nonlinear Schrödinger equation. For periodic waves, we exhibit two explicit solutions and prove the orbital stability in the energy space.
We study the existence and multiplicity of solutions for a class of fractional Schrödinger-Kirchhoff type equations with the Trudinger-Moser nonlinearity. More precisely, we consider \begin{gather*} \begin{cases} M\big(\|u\|^{N/s}\big)\left[(-Δ)^s_{N/s}u+V(x)|u|^{\frac{N}{s}-1}u\right]= f(x,u) +λh(x)|u|^{p-2}u\, &{\rm in}\ \ \mathbb{R}^N,\\ \|u\|=\left(\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{N/s}}{|x-y|^{2N}}dxdy+\int_{\mathbb{R}^N}V(x)|u|^{N/s}dx\right)^{s/N}, \end{cases}\end{gather*} where $M:[0,\infty]\rightarrow [0,\infty)$ is a continuous function, $s\in (0,1)$, $N\geq2$, $λ>0$ is a par…
We consider the hyperbolic-parabolic singular perturbation problem for a degenerate quasilinear Kirchhoff equation with weak dissipation. This means that the coefficient of the dissipative term tends to zero when t tends to +infinity. We prove that the hyperbolic problem has a unique global solution for suitable values of the parameters. We also prove that the solution decays to zero, as t tends to +infinity, with the same rate of the solution of the limit problem of parabolic type.
In this article we study the existence of weak solution, existence of ground state solution using Nehari manifold and existence of infinitely many solutions using Fountain theorem and Dual fountain theorem for a class of doubly nonlocal Kirchhoff-Choquard type equations involving the variable-order fractional $p(\cdot)-$ Laplacian operator. Here the nonlinearity does not satisfy the well known Ambrosetti-Rabinowitz type condition.
In this paper we address the following Kirchhoff type problem \begin{equation*} \left\{ \begin{array}{ll} -Δ(g(|\nabla u|_2^2) u + u^r) = a u + b u^p& \mbox{in}~Ω, u>0& \mbox{in}~Ω, u= 0& \mbox{on}~\partialΩ, \end{array} \right. \end{equation*} in a bounded and smooth domain $Ω$ in ${\rm I}\hskip -0.85mm{\rm R}$. By using change of variables and bifurcation methods, we show, under suitable conditions on the parameters $a,b,p,r$ and the nonlinearity $g$, the existence of positive solutions.
For a class of Kirchhoff functional, we first give a complete classification with respect to the exponent $p$ for its $L^2$-normalized critical points, and show that the minimizer of the functional, if exists, is unique up to translations. Secondly, we search for the mountain pass type critical point for the functional on the $L^2$-normalized manifold, and also prove that this type critical point is unique up to translations. Our proof relies only on some simple energy estimates and avoids using the concentration-compactness principles. These conclusions extend some known results in previous p…
It is well-known that the life span of solutions to Kirchhoff equations tends to infinity when initial data tend to zero. These results are usually referred to as almost global existence, at least in a neighborhood of the null solution. Here we extend this result by showing that the life span of solutions is lower semicontinuous, and in particular it tends to infinity whenever initial data tend to some limiting datum that originates a global solution. We also provide an estimate from below for the life span of solutions when initial data are close to some of the classes of data for which globa…
We study the existence of ground states to a nonlinear fractional Kirchhoff equation with an external potential $V$. Under suitable assumptions on $V$, using the monotonicity trick and the profile decomposition, we prove the existence of ground states. In particular, the nonlinearity does not satisfy the Ambrosetti-Rabinowitz type condition or monotonicity assumptions.
