In this paper, we consider the following Kirchhoff type problem $$\left\{\aligned&-\biggl(a + b\int_{\mathbb{R}^N} |\nabla u|^2 dx \biggr) Δu + V(x) u = |u|^{p-2}u &\text{ in } \mathbb{R}^N,\cr &u\in H^1(\mathbb{R}^N), \endaligned\right. \eqno{(\mathcal{P}_{a,b})} $$ where $N\geq3$, $2<p<2^*=\frac{2N}{N-2}$, $a,b>0$ are parameters and $V(x)$ is a potential function. Under some mild conditions on $V(x)$, we prove that $(\mathcal{P}_{a,b})$ has a positive solution for $b$ small enough by the variational method, a non-existence result is also established in the cases $N\geq4$. Our results in the case $N=3$ partial improve the results in \cite{G15,LY14} and our results in the cases $N\geq4$ are totally new to the best of our knowledge. By combining the scaling technique, we also give a global description on the structure of the positive solutions to the autonomous form of $(\mathcal{P}_{a,b})$, that is $V(x)\equivλ>0$. This result can be seen as a partial complement of the studies in \cite{A12,A13}.
We study the one-dimensional nonlocal elliptic equation of Kirchhoff type with logarithmic Kirchhoff function. We establish the precise asymptotic formulas for the solution $u_λ(x)$ as $λ\to \infty$. Here, $λ> 0$ is the bifurcation parameter.
In this paper, we consider the Dirichlet problem associated to an elliptic Kirchhoff-type equation depending on two parameters. Under rather general and natural assumptions, we prove that, for certain values of the parameters, the problem has at least three solutions.
In this note we complete the study made in a previous paper on a Kirchhoff type equation with a Berestycki-Lions nonlinearity. We also correct Theorem 0.6 inside.
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).
We study a generalized Kirchhoff type equation with trapping potential. The existence and blow-up behavior of solutions with normalized L2-norm for this problem are discussed.
In this paper we present a very simple proof of the existence of at least one non trivial solution for a Kirchhoff type equation on $\RN$, for $N\ge 3$. In particular, in the first part of the paper we are interested in studying the existence of a positive solution to the elliptic Kirchhoff equation under the effect of a nonlinearity satisfying the general Berestycki-Lions assumptions. In the second part we look for ground states using minimizing arguments on a suitable natural constraint.
This paper deals with the existence and the asymptotic behavior of non-negative solutions for a class of stationary Kirchhoff problems driven by a fractional integro-differential operator $\mathcal L_K$ and involving a critical nonlinearity. The main feature, as well as the main difficulty, of the analysis is the fact that the Kirchhoff function $M$ can be zero at zero, that is the problem is degenerate. The adopted techniques are variational and the main theorems extend in several directions previous results recently appeared in the literature.
Using minimax methods and Lusternik-Schnirelmann theory, we study multiple positive solutions for the Schrödinger - Kirchhoff equation $$ M\left(\dis\int_{Ω_λ}|\nabla u|^{2}dx+\dis\int_{Ω_λ}u^{2}dx\right)\left[-Δu + u \right]= f(u) $$ in $Ω_λ = λΩ$. The set $Ω\subset \mathbb{R}^3$ is a smooth bounded domain, $λ>0$ is a parameter, $M$ is a general continuous function and $f$ is a superlinear continuous function with subcritical growth. Our main result relates, for large values of $λ$, the number of solutions with the least number of closed and contractible in $\barΩ$ which cover $\barΩ$.
We study a superlinear and subcritical Kirchhoff type equation which is variational and depends upon a real parameter $λ$. The nonlocal term forces some of the fiber maps associated with the energy functional to have two critical points. This suggest multiplicity of solutions and indeed we show the existence of a local minimum and a mountain pass type solution. We characterize the first parameter $λ_0^*$ for which the local minimum has non-negative energy when $λ\ge λ_0^*$. Moreover we characterize the extremal parameter $λ^*$ for which if $λ>λ^*$, then the only solution to the Kirchhoff equat…
We consider a class of fourth-order elliptic equations of Kirchhoff type with critical growth in $\mathbb{R}^N$. By using constrained minimization in the Nehari manifold, we establish sufficient conditions for the existence of nodal (that is, sign-changing) solutions.
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 this paper, we establish a type of uniqueness and nondegeneracy results for positive solutions to the following nonlocal Kirchhoff equations \begin{eqnarray*} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\text{d} x\right)Δu+u=|u|^{p-1}u & & \text{in }\mathbb{R}^{3}, \end{eqnarray*} where $a,b$ are positive constants and $1<p<5$. Before this paper, it seems that there have no this type of results even on positive ground states solutions to Kirchhoff type equations, much less on general positive solutions. To overcome the difficulty brought by the nonlocality, some new observation on Kirchhof…
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 establish the existence of mountain pass and negative energy weak solutions for a Kirchhoff-Schrödinger type problem in $\mathbb R^4$ involving a critical nonlinearity and a suitable small perturbation. The arisen competition between the terms due to the nonlocal coefficient and critical nonlinearity turns out to be rather interesting. The main tools used in the present work are variational methods and the Lions' Concentration Compactness Principle.
In this paper we study the following class of fractional Kirchhoff problems: \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s}M(\varepsilon^{2s-N}[u]^{2}_{s})(-Δ)^{s}u + V(x) u= f(u) &\mbox{ in } \mathbb{R}^{N}, \\ u\in H^{s}(\mathbb{R}^{N}), \quad u>0 &\mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $s\in (0, 1)$, $N\geq 2$, $(-Δ)^{s}$ is the fractional Laplacian, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a positive continuous function, $M: [0, \infty)\rightarrow \mathbb{R}$ is a Kirchhoff function satisfying suitable co…
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.
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…
This article is concerned with the existence and multiplicity of positive weak solutions for the following fractional Kirchhoff-Choquard problem: \begin{equation*} \begin{array}{cc} \displaystyle M\left( \|u\|^2\right) (-Δ)^s u = \dsλf(x)|u|^{q-2}u + \left( \int\limits_Ω \frac{|u(y)|^{2^{*}_{μ,s}}}{|x-y|^ μ}\, dy\right) |u|^{2^{*}_{μ,s}-2}u \;\text{in} \; Ω, u > 0\quad \text{in} \; Ω, \,\, u = 0\quad \text{in} \; \mathbb{R}^{N}\backslashΩ, \end{array} \end{equation*} where $Ω$ is open bounded domain of $\mathbb{R}^{N}$ with $C^2$ boundary, $N > 2s$ and $s \in (0,1)$, here $M$ models Kirchhoff-…
In this paper we prove the existence of small-amplitude quasi-periodic solutions with Sobolev regularity, for the $d$-dimensional forced Kirchhoff equation with periodic boundary conditions. This is the first result of this type for a quasi-linear equations in high dimension. The proof is based on a Nash-Moser scheme in Sobolev class and a regularization procedure combined with a multiscale analysis in order to solve the linearized problem at any approximate solution.
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 are concerned with the following Kirchhoff type equation $$-\varepsilon^2 M \left(\varepsilon^{2-N} \int_{\mathbb{R}^N} | \nabla u|^2\, \mathrm{d} x\right) Δu+V(x)u = f(u),\ x \in \mathbb{R}^N,\ \ N\ge2, $$ where $M \in C(\mathbb{R}^+,\mathbb{R}^+)$, $V\in C(\mathbb{R}^N,\mathbb{R}^+)$ and $f(s)$ is of critical growth. In this paper, we construct a localized bound state solution concentrating at a local minimum of $V$ as $\varepsilon\to 0$ under certain conditions on $f(s)$, $M$ and $V$. In particular, the monotonicity of $f(s)/s$ and the Ambrosetti-Rabinowitz condition are not required.
In this paper, we prove uniqueness and nondegeneracy of positive solutions to the following Kirchhoff equations with critical growth \begin{eqnarray*} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\right)Δu=u^{5}, & u>0 & \text{in }\mathbb{R}^{3},\end{eqnarray*} where $a,b>0$ are positive constants. This result has potential applications in singular perturbation problems concerning Kirchhoff equaitons.
We consider the following $(p, q)$-Laplacian Kirchhoff type problem \begin{align*} \begin{split} &-\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{p}\, dx \right)Δ_{p}u - \left(c+d\int_{\mathbb{R}^{3}}|\nabla u|^{q}\, dx \right ) Δ_{q}u + V(x) (|u|^{p-2}u + |u|^{q-2}u)= K(x) f(u) \quad \mbox{ in } \mathbb{R}^{3}, \end{split} \end{align*} where $a, b, c, d>0$ are constants, $\frac{3}{2}< p< q<3$, $V: \mathbb{R}^{3}\rightarrow \mathbb{R}$ and $K: \mathbb{R}^{3}\rightarrow \mathbb{R}$ are positive continuous functions allowed vanishing behavior at infinity, and $f$ is a continuous function with quasicr…
We consider the following autonomous Kirchhoff-type equation \begin{equation*} -\left(a+b\int_{\mathbb{R}^N}|\nabla{u}|^2\right)Δu= f(u),~~~~u\in H^1(\mathbb{R}^N), \end{equation*} where $a\geq0,b>0$ are constants and $N\geq1$. Under general Berestycki-Lions type assumptions on the nonlinearity $f$, we establish the existence results of a ground state and multiple radial solutions for $N\geq2$, and obtain a nontrivial solution and its uniqueness, up to a translation and up to a sign, for $N=1$. The proofs are mainly based on a rescaling argument, which is specific for the autonomous case, and …
In this paper, we consider a class of quasilinear stationary Kirchhoff type potential systems in unbounded domains, which involves a general variable exponent elliptic operator. Under some suitable conditions on the nonlinearities, we establish existence of at least three weak solutions for the problem. The proof of our main result uses variational methods and the critical theorem of Bonanno and Marano.
Existence of sign-changing solutions for a logarithmic weighted Kirchhoff problem in the whole of $\mathbb{R}^{N}$ with exponential growth non-linearity
In this paper, we show the nonexistence results for the Kirchhoff elliptic, parabolic, and hyperbolic type equations on the Heisenberg groups. Also, the pseudo-parabolic and pseudo-hyperbolic equations of the Kirchhoff-type are under consideration. To prove these results we use the test function method. In addition, the analogous results are transferred to the cases of systems. Also, we give some examples of non-local nonlinearities.
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.
We are concerned with a class of Kirchhoff type equations in $\mathbb{R}^{N}$ as follows: \begin{equation*} \left\{ \begin{array}{ll} -M\left( \int_{\mathbb{R}^{N}}|\nabla u|^{2}dx\right) Δu+λV\left( x\right) u=f(x,u) & \text{in }\mathbb{R}^{N}, \\ u\in H^{1}(\mathbb{R}^{N}), & \end{array}% \right. \end{equation*}% where $N\geq 1,$ $λ>0$ is a parameter, $M(t)=am(t)+b$ with $a,b>0$ and $m\in C(\mathbb{R}^{+},\mathbb{R}^{+})$, $V\in C(\mathbb{R}^{N},\mathbb{R}^{+})$ and $f\in C(\mathbb{R}^{N}\times \mathbb{R}, \mathbb{R})$ satisfying $\lim_{|u|\rightarrow \infty }f(x,u) /|u|^{k-1}=q(x)$ uniforml…
Consider a nonlinear Kirchhoff type equation as follows \begin{equation*} \left\{ \begin{array}{ll} -\left( a\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx+b\right) Δu+u=f(x)\left\vert u\right\vert ^{p-2}u & \text{ in }\mathbb{R}^{N}, \\ u\in H^{1}(\mathbb{R}^{N}), & \end{array}% \right. \end{equation*}% where $N\geq 1,a,b>0,2<p<\min \left\{ 4,2^{\ast }\right\}$($2^{\ast }=\infty $ for $N=1,2$ and $2^{\ast }=2N/(N-2)$ for $N\geq 3)$ and the function $f\in C(\mathbb{R}^{N})\cap L^{\infty }(\mathbb{R}^{N})$. Distinguishing from the existing results in the literature, we are more interested in the geometr…
This paper is concerned with the multiplicity and concentration behavior of nontrivial solutions for the following fractional Kirchhoff equation in presence of a magnetic field: \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 \quad \mbox{ in } \mathbb{R}^{3}, \end{equation*} where $\varepsilon>0$ is a small parameter, $a, b>0$ are constants, $s\in (\frac{3}{4}, 1)$, $(-Δ)^{s}_{A}$ is the fractional magnetic Laplacian, $A:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ is a smooth magnetic potential, $V:\mathbb{R}…
In this paper, we consider the following nonlinear Kirchhoff type problem: \[ \left\{\begin{array}{lcl}-\left(a+b\displaystyle\int_{\mathbb{R}^3}|\nabla u|^2\right)Δu+V(x)u=f(u), & \textrm{in}\,\,\mathbb{R}^3,\\ u\in H^1(\mathbb{R}^3), \end{array}\right. \] where $a,b>0$ are constants, the nonlinearity $f$ is superlinear at infinity with subcritical growth and $V$ is continuous and coercive. For the case when $f$ is odd in $u$ we obtain infinitely many sign-changing solutions for the above problem by using a combination of invariant sets method and the Ljusternik-Schnirelman type minimax metho…
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 $$ \mathcal{L}_{\varepsilon} u = M \left(\frac{1}{\varepsilon} \int_{\mathbb{R}^3} |\nabla u|^2 + \frac{1}{…
This paper is concerned with an elliptic system of Kirchhoff type, driven by the variable-order fractional $p(x)$-operator. With the help of the direct variational method and Ekeland variational principle, we show the existence of a weak solution. This is our first attempt to study this kind of system, in the case of variable-order fractional variable exponents. Our main theorem extends in several directions previous results.
In this paper, we are going to study the existence of solution for the following Kirchhoff problem $$ \left\{ \begin{array}{l} M\biggl(\displaystyle\int_{\mathbb{R}^{3}}|\nabla u|^{2} dx +\displaystyle\int_{\mathbb{R}^{3}} λa(x)+1)u^{2} dx\biggl) \biggl(- Δu + (λa(x)+1)u\biggl) = f(u) \mbox{ in } \,\,\, \mathbb{R}^{3}, \\ \mbox{}\\ u \in H^{1}(\mathbb{R}^{3}). \end{array} \right. $$ Assuming that the nonnegative function $a(x)$ has a potential well with $int (a^{-1}(\{0\}))$ consisting of $k$ disjoint components $Ω_1, Ω_2, ....., Ω_k$ and the nonlinearity $f(t)$ has a subcritical growth, we ar…
We prove the existence of multiple solutions for the following sixth-order $p(x)$-Kirchhoff-type problem: $-M(\int_Ω\frac{1}{p(x)}|\nabla Δu|^{p(x)}dx)Δ^3_{p(x)} u = λf(x)|u|^{q(x)-2}u + g(x)|u|^{r(x)-2}u + h(x) \ \ \mbox{on} \ Ω$ and $ \ u=Δu=Δ^2 u=0 \ \ \mbox{on} \ \partialΩ,$ where $Ω\subset \mathbb{R}^N$ is a smooth bounded domain, $N > 3$, $Δ_{p(x)}^3u = \operatorname{div}\Big(Δ(|\nabla Δu|^{p(x)-2}\nabla Δu)\Big)$ is the $p(x)$-triharmonic operator, $p,q,r \in C(\overlineΩ)$, $1< p(x) < \frac N3$ for all $x\in \overlineΩ$, $M(s) = a - bs^γ$, $a,b,γ>0$, $λ>0$, $g: Ω\times \mathbb{R} \to \…
In this paper, we consider the following critical fractional Kirchhoff equation \begin{equation*} \Big(a+b{\int_{\mathbb{R}^{N}}}|(-Δ)^{\frac{s}{2}}u|^2dx\Big)(-Δ)^su=|u|^{2^*_s-2}u,\quad \text{in}\ \mathbb{R}^{N}, \end{equation*} where $a,b>0$, $\frac{N}{4}<s<1$, $2^*_s=\frac{2N}{N-2s}$ and $(-Δ)^s$ is the fractional Laplacian. We prove the uniqueness and nondegeneracy of positive solutions to the problem, which can be used to study the singular perturbation problems concerning fractional Kirchhoff equations.
The paper deals with existence and multiplicity of solutions of the fractional Schrödinger--Kirchhoff equation involving an external magnetic potential. As a consequence, the results can be applied to the special case \begin{equation*} (a+b[u]_{s,A}^{2θ-2})(-Δ)_A^su+V(x)u=f(x,|u|)u\,\, \quad \text{in $\mathbb{R}^N$}, \end{equation*} where $s\in (0,1)$, $N>2s$, $a\in \mathbb{R}^+_0$, $b\in \mathbb{R}^+_0$, $θ\in[1,N/(N-2s))$, $A:\mathbb{R}^N\rightarrow\mathbb{R}^N$ is a magnetic potential, $V:\mathbb{R}^N\rightarrow \mathbb{R}^+$ is an electric potential, $(-Δ)_A^s$ is the fractional magnetic o…
In this paper, we study the existence and asymptotic properties of solutions to the following fractional Kirchhoff equation \begin{equation*} \left(a+b\int_{\mathbb{R}^{3}}|(-Δ)^{\frac{s}{2}}u|^{2}dx\right)(-Δ)^{s}u=λu+μ|u|^{q-2}u+|u|^{p-2}u \quad \hbox{in $\mathbb{R}^3$,} \end{equation*} with a prescribed mass \begin{equation*} \int_{\mathbb{R}^{3}}|u|^{2}dx=c^{2}, \end{equation*} where $s\in(0, 1)$, $a, b, c>0$, $2<q<p<2_{s}^{\ast}=\frac{6}{3-2s}$, $μ>0$ and $λ\in\mathbb{R}$ as a Lagrange multiplier. Under different assumptions on $q<p$, $c>0$ and $μ>0$, we prove some existence results about…
We obtain a sequence of solutions converging to zero for the Kirchhoff equation $$-\left( 1+\int_Ω\left\vert \nabla u\right\vert^2\right) Δu+V(x)u=f(u)\text{,\qquad}u\in H_{0}^{1}(Ω)$$ via truncating technique and a variant of Clark's theorem due to Liu--Wang, where $Ω$ is a bounded smooth domain $Ω\subset\mathbb{R}^N$. Similar result for Schrödinger-Poisson system on a bounded smooth domain $Ω\subset\mathbb{R}^3$ is also presented.
The purpose of this paper is to study the indefinite Kirchhoff type problem: \begin{equation*} \left\{ \begin{array}{ll} M\left( \int_{\mathbb{R}^{N}}(|\nabla u|^{2}+u^{2})dx\right) \left[ -Δu+u\right] =f(x,u) & \text{in }\mathbb{R}^{N}, \\ 0\leq u\in H^{1}\left( \mathbb{R}^{N}\right), & \end{array} \right. \end{equation*} where $N\geq 1$, $M(t)=am\left( t\right) +b$, $m\in C(\mathbb{R}^{+})$ and $ f(x,u)=g(x,u)+h(x)u^{q-1}$. We require that $f$ is \textquotedblleft local\textquotedblright\ sublinear at the origin and \textquotedblleft local\textquotedblright\ linear at infinite. Using the mou…
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.
We consider the fractional Schrödinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity $\varepsilon^{2s}M([u]_{s,A_\varepsilon}^2)(-Δ)_{A_\varepsilon}^su + V(x)u =$ $|u|^{2_s^\ast-2}u + h(x,|u|^2)u,$ $\ \ x\in \mathbb{R}^N,$ where $ u(x) \rightarrow 0$ as $|x| \rightarrow \infty,$ and $(-Δ)_{A_\varepsilon}^s$ is the fractional magnetic operator with $0<s<1$, $2_s^\ast = 2N/(N-2s),$ $M : \mathbb{R}^{+}_{0} \rightarrow \mathbb{R}^{+}$ is a continuous nondecreasing function, $V:\mathbb{R}^N \rightarrow \mathbb{R}^+_0,$ and $A: \mathbb{R}^N \rightarrow \mathbb{R}^N$ are t…
In this paper, we study the fractional Kirchhoff equation with critical nonlinearity \begin{align*} \left(a+b\int_{\mathbb R^N}|(-Δ)^{\frac{s}{2}}u|^2dx\right)(-Δ)^su+u=f(u)\ \ \mbox{in}\ \ \mathbb R^N, \end{align*} where $N>2s$ and $(-Δ)^s$ is the fractional Laplacian with $0<s<1$. By using a perturbation approach, we prove the existence of solutions to the above problem without the Ambrosetti-Rabinowitz condition when the parameter $b$ small. What's more, we obtain the asymptotic behavior of solutions as $b\to 0$.
We study the following Kirchhoff equation $$- \left(1 + b \int_{\mathbb{R}^3} |\nabla u|^2 dx \right) Δu + V(x) u = f(x,u), \ x \in \mathbb{R}^3.$$ A special feature of this paper is that the nonlinearity $f$ and the potential $V$ are indefinite, hence sign-changing. Under some appropriate assumptions on $V$ and $f$, we prove the existence of two different solutions of the equation via the Ekeland variational principle and Mountain Pass Theorem.
In this paper we deal with the multiplicity and concentration of positive solutions for the following fractional Schrödinger-Kirchhoff type equation \begin{equation*} M\left(\frac{1}{\varepsilon^{3-2s}} \iint_{\mathbb{R}^{6}}\frac{|u(x)- u(y)|^{2}}{|x-y|^{3+2s}} dxdy + \frac{1}{\varepsilon^{3}} \int_{\mathbb{R}^{3}} V(x)u^{2} dx\right)[\varepsilon^{2s} (-Δ)^{s}u+ V(x)u]= f(u) \, \mbox{in} \mathbb{R}^{3} \end{equation*} where $\varepsilon>0$ is a small parameter, $s\in (\frac{3}{4}, 1)$, $(-Δ)^{s}$ is the fractional Laplacian, $M$ is a Kirchhoff function, $V$ is a continuous positive potential …
In this article, we deal with the existence of non-negative solutions of the class of following non local problem $$ \left\{ \begin{array}{lr} \quad - M\left(\displaystyle\int_{\mathbb R^n}\int_{\mathbb R^{n}} \frac{|u(x)-u(y)|^{\frac{n}{s}}}{|x-y|^{2n}}~dxdy\right) (-Δ)^{s}_{n/s} u=\left(\displaystyle\int_Ω\frac{G(y,u)}{|x-y|^μ}~dy\right)g(x,u) \; \text{in}\; Ω,\\ \quad \quad u =0\quad\text{in} \quad \mathbb R^n \setminus Ω, \end{array} \right. $$ where $(-Δ)^{s}_{n/s}$ is the $n/s$-fractional Laplace operator, $n\geq 1$, $s\in(0,1)$ such that $n/s\geq 2$, $Ω\subset \mathbb R^n$ is a bounded …
For a degenerate autonomous Kirchhoff equation which is set on $\mathbb{R}^N$ and involves the Berestycki-Lions type nonlinearity, we cope with the cases $N=2,3$ and $N\geq5$ by using mountain pass and symmetric mountain pass approaches and by using Clark theorem respectively.
In the present paper, we establish the uniqueness and nondegeneracy of positive energy solutions to the Kirchhoff equation \begin{eqnarray*} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\right)Δu+u=|u|^{p-1}u & & \text{in }\mathbb{R}^{3}, \end{eqnarray*} where $a,b>0$, $1<p<5$ are constants. Then, as applications, we derive the existence and local uniqueness of solutions to the perturbed Kirchhoff problem \begin{eqnarray*} -\left(ε^2a+εb\int_{\mathbb{R}^{3}}|\nabla u|^{2}\right)Δu+V(x)u=|u|^{p-1}u & & \text{in }\mathbb{R}^{3} \end{eqnarray*} for $ε>0$ sufficiently small, under some mild assump…
This paper deals with the following degenerate fractional Kirchhoff-type system with magnetic fields and critical growth: $$ \left\{ \begin{array}{lll} -\mathfrak{M}(\|u\|_{s,A}^2)[(-Δ)^s_Au+u] = G_u(|x|,|u|^2,|v|^2) + \left(\mathcal{I}_μ*|u|^{p^*}\right)|u|^{p^*-2}u \ &\mbox{in}\,\,\mathbb{R}^N,\\ \mathfrak{M}(\|v\|_{s,A})[(-Δ)^s_Av+v] = G_v(|x|,|u|^2,|v|^2) + \left(\mathcal{I}_μ*|v|^{p^*}\right)|v|^{p^*-2}v \ &\mbox{in}\,\,\mathbb{R}^N, \end{array}\right. $$ where $$\|u\|_{s,A}=\left(\iint_{\mathbb{R}^{2N}}\frac{|u(x)-e^{i(x-y)\cdot A(\frac{x+y}{2})}u(y)|^2}{|x-y|^{N+2s}}dx dy+\int_{\mathbb{…
In this paper we consider the following critical nonlocal problem $$ \left\{\begin{array}{ll} M\left(\displaystyle\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\right)(-Δ)^s u = \displaystyle\fracλ{u^γ}+u^{2^*_s-1}&\quad\mbox{in } Ω,\\ u>0&\quad\mbox{in } Ω,\\ u=0&\quad\mbox{in } \mathbb{R}^N\setminusΩ, \end{array}\right. $$ where $Ω$ is an open bounded subset of $\mathbb R^N$ with continuous boundary, dimension $N>2s$ with parameter $s\in (0,1)$, $2^*_s=2N/(N-2s)$ is the fractional critical Sobolev exponent, $λ>0$ is a real parameter, exponent $γ\in(0,1)$, $M$ models a Kirchhof…
For a bounded domain $Ω\subset\mathbb{R}^n$ let $H_Ω:Ω\timesΩ\to\mathbb{R}$ be the regular part of the Dirichlet Green function for the Laplace operator. Given a fixed arbitrary ${\mathcal C}^2$ function $f:{\mathcal D}\to\mathbb{R}$, defined on an open subset ${\mathcal D}\subset\mathbb{R}^{nN}$, and fixed coefficients $λ_1,\dots,λ_N\in\mathbb{R}\setminus\{0\}$ we consider the function $f_Ω:{\mathcal D}\capΩ^N\to\mathbb{R}$ defined as \[ f_Ω(x_1,\dots,x_N) = f(x_1,\dots,x_N) - \sum_{j,k=1}^N λ_jλ_k H_Ω(x_j,x_k). \] We prove that $f_Ω$ is a Morse function for most domains $Ω$ of class ${\mathc…
