Variational methods for degenerate Kirchhoff equations
Sheng-Sen Lu
对于一类以$\mathbb{R}^N$为背景的退化自治Kirchhoff方程，我们分别用山路法和对称山路法以及Clark定理处理了$N=2,3$和$N\geq5$的情形.
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.
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Hyperbolic-parabolic singular perturbation for mildly degenerate Kirchhoff equations with weak dissipation
本文考虑具有小参数和弱耗散项的轻度退化Kirchhoff方程。我们证明了整体解的存在性时，参数是小的初始数据的大小。然后，我们提供全球的时间误差估计的差异，我们的问题的解决方案和相应的一阶问题的解决方案。
Asymptotic limits for mildly degenerate Kirchhoff equations
研究了一类带耗散项的轻度退化Kirchhoff方程解的渐近性态。我们得到了解的时间二阶导数的一个新的估计。此外，我们重整化的解决方案，以这样一种方式，重整化作为一个没有零的限制，因为t走向无穷大。最后，我们明确地计算了这个极限的范数，并证明了它不依赖于初始数据。
On a variational method for solving a class of $p$-Kirchhoff type problem
本文研究了一类含变号权函数的$p$ -Kirchhoff型问题.通过Nehari流形方法证明了该问题多个正解的存在性。
Mildly degenerate Kirchhoff equations with weak dissipation: global existence and time decay
考虑一类退化拟线性弱耗散Kirchhoff方程的双曲-抛物奇异扰动问题。这意味着当t趋于+无穷大时，耗散项的系数趋于零。我们证明了双曲型问题有一个唯一的整体解决方案，适当的值的参数。我们还证明了解衰减到零，随着t趋于+无穷大，与抛物型极限问题的解决方案相同的速度。
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…
Degenerate Schr{ö}dinger-Kirchhoff $(p, N)$-Laplacian problem With singular Trudinger-Moser nonlinearity in $\mathbb{R}^N$
本文研究ℝ^N中具有奇异指数非线性的退化(p, N)-Laplacian Schrödinger-Kirchhoff问题，利用山路定理、Ekeland变分原理、奇异Trudinger-Moser不等式及新的Brézis-Lieb型引理，证明了非平凡非负解的存在性。
Degenerate Elliptic PDEs on a Network with Kirchhoff Conditions
本文研究一般网络上的半线性退化椭圆型方程，结合非线性 Kirchhoff 条件与 Dirichlet 边界条件，涵盖从一阶到一致椭圆型方程，主要贡献是证明强比较原理，并据此通过 Perron 方法得出连续粘性解的存在性与唯一性。
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.
This article concerns on the existence of multiple solutions for a new Kirchhoff-type problem with negative modulus. We prove that there exist three nontrivial solutions when the parameter is enough small via the variational methods and algebraic analysis. Moreover, our fundamental technique is one of the Mountain Pass Lemma, Ekeland variational principle, and Minimax principle.
Hyperbolic-parabolic singular perturbation for mildly degenerate Kirchhoff equations: decay-error estimates
考虑退化的Kirchhoff方程在二阶时间导数前含有一个小参数ε。众所周知，当λ足够小时，这些方程有整体解，并且这些解以与极限问题（抛物型）相同的速率衰减为t -> +无穷大。在本文中，我们证明了衰减误差估计的双曲问题的解决方案和相应的抛物问题的解决方案之间的差异。这些估计同时表明，当t-> 0和t -> +无穷大时，差值趋于零。关于衰减率，结果是差衰减得比两项分别衰减得快（因为t -> +无穷大）。证明涉及一个非线性步骤，我们将傅立叶分量与最低频率分离，然后是一个线性步骤，我们利用经典能量的加权版本。
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, multiplicity and regularity of positive weak solutions for the following Kirchhoff-Choquard problem: \begin{equation*} \begin{array}{cc} \displaystyle M\left( \iint\limits_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}\,dxdy\right) (-Δ)^s u = \fracλ{u^γ} + \left( \int\limits_Ω \frac{|u(y)|^{2^{*}_{μ,s}}}{|x-y|^ μ}\, dy\right) |u|^{2^{*}_{μ,s}-2}u \;\text{in} \; Ω, %\quad \quad u > 0\quad \text{in} \; Ω, \quad \quad u = 0\quad \text{in} \; \mathbb{R}^{N}\backslashΩ, \end{array} \end{equation*} where $Ω$ is open bounded domain of $\mathbb{R}^{N}$ with …
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 $$\|u\|_{s,A}=\left(\iint_{\mathbb{R}^{2N}}\frac{|u(x)-e^{i(x-y)\cdot A(\frac{x+y}{2})}u(y)|^2}{|x-y|^{N+2s}}dx dy+\int_{\mathbb{…
A priori bounds for positive solutions of Kirchhoff type equations
本文给出了无变分结构的Kirchhoff型方程正解的先验估计。
This paper is concerned with an elliptic system of Kirchhoff type, driven by the variable-order fractional $p(x)$-operator. With the help of the direct variational method and Ekeland variational principle, we show the existence of a weak solution. This is our first attempt to study this kind of system, in the case of variable-order fractional variable exponents. Our main theorem extends in several directions previous results.
Degenerate singular Kirchhoff problems in Musielak-Orlicz spaces
本文研究了 Musielak-Orlicz 空间中由非齐次算子驱动的退化奇异 Kirchhoff 方程，利用纤维化方法和 Nehari 流形分解证明了至少两个解的存在性，结果涵盖多种二阶微分算子。
A degenerate Kirchhoff-type problem involving variable $s(\cdot)$-order fractional $p(\cdot)$-Laplacian with weights
本文针对含权的非局部变阶分数阶 $p(\cdot)$-Kirchhoff 方程，在适当结构条件下证明非平凡解的存在性与多重性，并将结论推广至退化情形。
On degenerate fractional Schrödinger-Kirchhoff-Poisson equations with upper critical nonlinearity and electromagnetic fields
该研究针对带临界非线性项与电磁场的三维退化分数阶Schrödinger-Kirchhoff-Poisson方程，利用集中紧性原理、变分法及精细分析工具，证明了参数趋于零且Kirchhoff项在零点退化时解的存在性与多重性。
In this paper, we consider Kirchhoff-Schrodinger equations with singular exponential nonlinearities in R^4,using singular Adams inequality and variational techniques, we get the existence of ground state solutions. Moreover, we also get the same results without the Ambrosetti-Rabinowitz (AR) condition.
Spectral gap global solutions for degenerate Kirchhoff equations
考虑Hilbert空间上的二阶Cauchy问题$$u''+m(|A^{1/2}u|^2)Au=0, u(0)=u_{0}, u'(0)=u_{1},$$，其中$m:[0,+\infty)\to[0,+\infty)$是连续函数，$A$是自伴非负算子.众所周知，这个问题承认当地的时间解决方案，提供$u_{0}$和$u_{1}$是足够的规则，依赖于$m$的连续性模，并在严格/弱双曲方程。我们证明了对于这样的初始数据$(u_{0},u_{1})$存在两对初始数据$(\overline{u}_{0},\overline{u}_{1})$，$(\widehat{u}_{0},\widehat{u}_{1})$，它们的解是全局的，并且使得$u_{0}=\overline{u}_{0}+\widehat{u}_{0}$，$u_{1}=\overline{u}_{1}+\widehat{u}_{1}$ .这是一个副产品的初始数据与适当的频谱间隙，扩展了以前的结果，在严格的双曲的情况下，光滑的非线性$m$的全局存在结果。
Multiple solutions for a Kirchhoff-type equation with general nonlinearity
本文研究了如下的自治Kirchhoff型方程$$-M\left(\int_{\mathbb{R}^N}|\nabla{u}|^2\right)Δ{u}= f(u),~~~~u\in H^1(\mathbb{R}^N),$$，其中$M$是连续非退化函数，$N\geq2$ .在对$M$适当的附加条件和对非线性项$f$的一般Berestycki-Lions型假设下，利用变分方法建立了多解的存在性结果，并从非变分的观点自然地解释了这些结果.
On a generalized timoshenko-kirchhoff equation
本文考虑了$\mathbb R^{N}, N\geq2$中有界区域上的广义四阶非线性Kirchhoff方程在Navier边界条件下的次线性项。我们采用了一个变量变换，这减少了问题的半线性。然后，变分和拓扑工具，以证明解的存在性。
In this paper, we study the existence of three solutions for a Kirchhoff equation involving the nonlocal fractional p-Laplacian considering Sobolev and Hardy nonlinearities at subcritical and critical growths. The proof is based on Mountain Pass Theorem, Ekeland's variational principle and constrained minimization in Nehari sets.
Kirchhoff equations in generalized Gevrey spaces: local existence, global existence, uniqueness
本文给出了广义Gevrey空间中Kirchhoff方程的一些最新结果。我们表明，这些空间是自然的框架，经典的结果可以统一和扩展。特别是，我们专注于初始数据的存在性和唯一性的结果，其规律性取决于非线性项的连续性模量，无论是在严格双曲的情况下，在退化双曲的情况下。
Three weak solutions for a $(p, q)$-Schrödinger-Kirchhoff type equation
该研究针对满足del Pino-Felmer型条件的含连续正势的(p, q)-Schrödinger-Kirchhoff方程，运用Recceri经典变分方法证明了其三个弱解的存在性。
In this paper we prove the existence of positive ground state solution for a class of linearly coupled systems involving Kirchhoff-Schrödinger equations. We study the subcritical and critical case. Our approach is variational and based on minimization technique over the Nehari manifold. We also obtain a nonexistence result using a Pohozaev identity type.
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.
Nontrivial solutions for a $(p,q)$-Kirchhoff type system with concave-convex nonlinearities on locally finite graphs
本文在局部有限图上研究带扰动项的$(p,q)$-Kirchhoff型系统，利用山路定理、Ekeland变分原理和Clark定理证明了完全非平凡解与半平凡解的存在性及多重性，并给出解不存在的条件。
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 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 existence of least energy nodal solutions for some class of Kirchhoff equations and Choquard equations
本文研究了一类Kirchhoff型问题的最小能量节点解的存在性。由于Kirchhoff方程是一个非局部方程，寻找变号解的变分设置不同于局部情况。利用变号Nehari流形上的约束极小化方法，证明了Kirchhoff问题存在能量超过最小能量两倍的最小能量节点解。作为我们方法的一个副产品，我们得到了Choquard方程最小能量变号解的存在性，并证明了变号解的能量严格大于最小能量，且小于最小能量的两倍.
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.
Well-Posedness of Fully Nonlinear PDEs on Star-Shaped Networks with Nonlinear Kirchhoff Conditions
本文建立星形网络上带非线性基尔霍夫边界条件的完全非线性偏微分方程的适定性，通过构造Eikonal方程测试函数证明不连续粘性解的比较定理，并证明广义基尔霍夫粘性解即基尔霍夫粘性解。
In this paper we establish the existence of at least two weak solutions for the following fractional Kirchhoff problem involving singular and exponential nonlinearity \begin{equation*} \left\{\begin{split} M\left(\|u\|^{\frac{n}{s}}\right)(-Δ)^s_{n/s}u & = μu^{-q}+ u^{r-1}\exp( u^β)\;\text{in}\;\Om, u&>0,\;\text{in}\; \Om, u &= 0,\;\text{in}\; \mb R^n \setminus{\Om}, \end{split} \right. \end{equation*} where $\Om$ is smooth bounded domain in $\mb R^n$, {$n\geq 1$}, $s\in (0,1)$, $μ>0$ is a real parameter, $β<\frac{n}{n-s}$ and $q\in (0,1)$. We have considered the degenerate Kirchhoff case here…
In this paper, we consider the following singularly perturbed Kirchhoff equation \begin{equation*} -(\varepsilon^2a+\varepsilon b\int_{\mathbb{R}^3}|\nabla u|^2dx)Δu+V(x)u=P(x)|u|^{p-2}u+Q(x)|u|^4u,\quad x\in\mathbb{R}^3, \end{equation*} where $\varepsilon>0$ is a small parameter, $a, b > 0$ are constants, $p\in(4,6)$ and $V, P, Q$ are potential functions satisfying some competing conditions. We prove the existence of a positive ground state solution by using variational methods, and we determine a concrete set related to the potentials $V,P$ and $Q$ as the concentration position of these grou…
Kirchhoff equations with strong damping
我们认为基尔霍夫方程的强阻尼，即摩擦项取决于权力的“弹性”运营商。我们解决当地和全球存在的解决方案，在两个不同的制度，取决于摩擦项的指数。当指数大于大于1/2时，耗散占优势，在仅假设非线性项的退化双曲性和连续性的条件下，我们得到了能量空间中的整体解的存在性.当指数小于1/2时，我们假设严格双曲性，并考虑一个依赖于非线性项的连续模和阻尼指数的相空间。在这个相空间中，我们证明了局部解的存在性，并在初始值足够小的情况下证明了全局解的存在性。我们假设的规则性的初始数据和非线性项是弱于经典的结果与标准阻尼的基尔霍夫方程。证明利用了线性化方程和适当定义的插值空间的一些最近的尖锐结果。
Uniqueness and Nondegeneracy of positive solutions to Kirchhoff equations and its applications in singular perturbation problems
本文建立了Kirchhoff方程\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*}正能量解的唯一性和非退化性，其中$a,b>0$，$1<p<5$为常数.然后，作为应用，我们在对势函数$V:\mathbb{R}^3\to \mathbb{R}$的适当假设下，得到了扰动Kirchhoff问题\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*}（$ε>0$充分小）解的存在性和局部唯一性.利用Lyapunov-Schmidt约化方法得到了解的存在性结果。这似乎是第一次研究奇摄动Kirchhoff问题的约化方法，因为以前的所有结果都是通过各种变分方法得到的。这种方法的另一个优点是它给出了所有$p\in (1,5)$的扰动问题的统一证明，这与文献中使用变分方法有很大不同。局部唯一性结果是全新的。它是利用一类局部Pohozaev恒等式得到的，这类恒等式是邓、林和严最近…
Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy
本文应用改进的势阱方法和变分方法研究一类Kirchhoff型抛物型方程解的长时间行为。对于任意的初始能量，得到了解的整体存在性和有限时间爆破。为了更精确一点，我们将给出一个阈值结果，当初始能量是亚临界和临界时，解的存在性或在有限时间内爆破，分别。在这些情况下的整体解的$L^2(Ω)$规范的衰减率也得到。此外，当初始能量为超临界时，还得到了整体解和爆破解存在的一些充分条件。
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…
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…
On the indefinite Kirchhoff type problems with local sublinearity and linearity
本文研究不定Kirchhoff型问题：\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*}，其中$N\geq 1$，$M(t)=am\left( t\right) +b$，$m\in C(\mathbb{R}^{+})$和$ f(x,u)=g(x,u)+h(x)u^{q-1}$ .我们要求$f$是\textquotedblleft local \textquotedblright \ sublinear at the origin和\textquotedblleft local \textquotedblright \ linear at the infinite。利用山路引理和Ekeland变分原理，得到了非平凡解的存在性和多解性。特别地，建立了三个非平凡解存在的判据。
Existence and multiplicity results for a class of Kirchhoff-Choquard equations with a generalized sign-changing potential
研究一类带广义变号势的Kirchhoff-Choquard方程，证明非退化情形下解的存在性与多重性，并保证退化情形下解的存在。
We study the existence and multiplicity of positive solutions for a family of fractional Kirchhoff equations with critical nonlinearity of the form \begin{equation*} M\left(\int_Ω|(-Δ)^{\fracα{2}}u|^2dx\right)(-Δ)^α u= λf(x)|u|^{q-2}u+|u|^{2^*_α-2}u\;\; \text{in}\; Ω,\;u=0\;\textrm{in}\;\mathbb R^n\setminus Ω, \end{equation*} where $Ω\subset \mathbb R^n$ is a smooth bounded domain, $ M(t)=a+\varepsilon t, \; a, \; \varepsilon>0,\; 0<α<1, \; 2α<n<4α$ and $ \; 1<q<2$. Here $2^*_α={2n}/{(n-2α)}$ is the fractional critical Sobolev exponent, $λ$ is a positive parameter and the coefficient $f(x)$ is…
Solutions to discrete nonlinear Kirchhoff-Choquard equations
该研究针对三维整数格点上的离散非线性Kirchhoff-Choquard方程，在合适的位势与非线性项假设下，借助变分法证明了方程非平凡解和基态解的存在性，丰富了离散非局部方程的解的存在性理论。
Multiple sign-changing solutions to a class of Kirchhoff type problems
本文研究了一类非局部Kirchhoff型问题变号解的存在性，其形式为\begin{equation}\label{s}\tag{S} -\Big(a+b\int_Ω|\nabla u|^2dx\Big)Δu=f(x,u)\, \text{ in }Ω,\quad\quad u=0 \text{ on }\partialΩ, \end{equation}，其中$Ω$是$\mathbb{R}^N$（$N=1,2,3$）中具有光滑边界的有界区域，$a>0$，$b>0$，$f:\overlineΩ\times\mathbb{R}\to\mathbb{R}$是连续函数.我们给出了一个肯定的答案长期存在的问题，关于存在两个以上的符号变化的解决方案\eqref{s}。更准确地说，我们在本文中，如果$f$是全球3-超线性，次临界和奇的第二个变量，那么\eqref{s}具有一个无界序列的符号变化的解决方案。我们的方法是变分的，并依赖于一个新的符号变化版本的对称山路定理建立在本文中。
Infinitely many solutions for a biharmonic-Kirchhoff system on locally finite graphs
针对局部有限图上的双调和-Kirchhoff系统，通过变分法结合对称山路引理，证明当非线性项满足超4次线性增长时存在无穷多解，且结果优于欧几里得环境下的对应系统。
In this paper we consider the following class of fractional Kirchhoff equations with critical growth: \begin{equation*} \left\{ \begin{array}{ll} \left(\varepsilon^{2s}a+\varepsilon^{4s-3}b\int_{\mathbb{R}^{3}}|(-Δ)^{\frac{s}{2}}u|^{2}dx\right)(-Δ)^{s}u+V(x)u=f(u)+|u|^{2^{*}_{s}-2}u \quad &\mbox{ in } \mathbb{R}^{3}, \\ u\in H^{s}(\mathbb{R}^{3}), \quad u>0 &\mbox{ in } \mathbb{R}^{3}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $a, b>0$ are constants, $s\in (\frac{3}{4}, 1)$, $2^{*}_{s}=\frac{6}{3-2s}$ is the fractional critical exponent, $(-Δ)^{s}$ is the …
Degenerate Kirchhoff problems with nonlinear Neumann boundary condition
针对带非线性Neumann边界的退化Kirchhoff型方程，本文基于变分方法在非局部约束集上极小化能量泛函，证明了节点基态解的存在性。
In this paper, we are concerned with the Neumann problem for a class of quasilinear stationary Kirchhoff-type potential systems, which involves general variable exponents elliptic operators with critical growth and real positive parameter. We show that the problem has at least one solution, which converges to zero in the norm of the space as the real positive parameter tends to infinity, via combining the truncation technique, variational method, and the concentration-compactness principle for variable exponent under suitable assumptions on the nonlinearities.
Least energy sign-changing solution for degenerate Kirchhoff double phase problems
本文针对含退化 Kirchhoff 系数的双相非局部 Dirichlet 问题，利用变分法、定量形变引理与 Poincaré–Miranda 定理，证明正、负定号解及最小能量变号解的存在性。
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-…
Strongly Singular Nonlocal Kirchhoff-Type Equations with Variable Exponents: Existence, Regularity, and Renormalized Solutions
本文解决非局部Kirchhoff型方程变量指数下的强奇异问题，通过五个原创定理建立完整理论，涵盖存在性、正则性和重正化解构造。
The objective of this work is to investigate a nonlocal problem involving singular and critical nonlinearities:\begin{equation*}\left\{\begin{array}{ll} ([u]_{s,p}^p)^{σ-1}(-Δ)^s_p u = \fracλ{u^γ}+u^{ p_s^{*}-1 }\quad \text{in }Ω,\\ u>0,\;\;\;\;\quad \text{in }Ω,\\ u=0,\;\;\;\;\quad \text{in }\mathbb{R}^{N}\setminus Ω,\end{array} \right. \end{equation*} where $Ω$ is a bounded domain in $\mathbb{R}^N$ with the smooth boundary $\partial Ω$, $0 < s< 1<p<\infty$, $N> sp$, $1<σ<p^*_s/p,$ with $p_s^{*}=\frac{Np}{N-ps},$ $ (- Δ)_p^s$ is the nonlocal $p$-Laplace operator and $[u]_{s,p}$ is the Gagliar…
We obtain the existence of ground state solution for the nonlocal problem $$ m\left(\int_{\mathbb{R}^2}(|\nabla u|^2 + b(x)u^2) \textrm{d}x\right)(-Δu + b(x)u) = A(x)f(u) \ \ \ \textrm{in} \ \ \ \mathbb{R}^2, $$ where $m$ is a Kirchhoff-type function, $b$ may be negative and noncoercive, $A$ is locally bounded and the function $f$ has critical exponential growth. We also obtain new results for the classical Schrödinger equation, namely the local case $m\equiv 1$. In the proofs we apply Variational Methods beside a new Trudinger-Moser type inequality.
We consider the fractional Schrödinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity $\varepsilon^{2s}M([u]_{s,A_\varepsilon}^2)(-Δ)_{A_\varepsilon}^su + V(x)u =$ $|u|^{2_s^\ast-2}u + h(x,|u|^2)u,$ $\ \ x\in \mathbb{R}^N,$ where $ u(x) \rightarrow 0$ as $|x| \rightarrow \infty,$ and $(-Δ)_{A_\varepsilon}^s$ is the fractional magnetic operator with $0<s<1$, $2_s^\ast = 2N/(N-2s),$ $M : \mathbb{R}^{+}_{0} \rightarrow \mathbb{R}^{+}$ is a continuous nondecreasing function, $V:\mathbb{R}^N \rightarrow \mathbb{R}^+_0,$ and $A: \mathbb{R}^N \rightarrow \mathbb{R}^N$ are t…
Positive solutions to an elliptic equation in $\mathbb{R}^N$ of the Kirchhoff type
本文考虑了如下Kirchhoff型问题$$\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})} $$，其中$N\geq3$，$2<p<2^*=\frac{2N}{N-2}$，$a,b>0$为参数，$V(x)$为势函数.在$V(x)$上的适当条件下，利用变分方法证明了当$b$足够小时，$(\mathcal{P}_{a,b})$存在正解，并在$N\geq4$情形下得到了正解的不存在性.我们在$N=3$情形中的结果部分改进了\cite{G15,LY14}的结果，而在$N\geq4$情形中的结果据我们所知是全新的。结合标度技术，我们还对$(\mathcal{P}_{a,b})$的自治形式$V(x)\equivλ>0$的正解结构给出了整体描述.该结果可视为\cite{A12,A13}研究的部分补充。
Multiscale analysis of a degenerate pseudoparabolic variational inequality, modelling the two-phase flow with dynamical capillary pressure in a perforated domain, is the main topic of this work. Regularisation and penalty operator methods are applied to show the existence of a solution of the nonlinear degenerate pseudoparabolic variational inequality defined in a domain with microscopic perforations, as well as to derive a priori estimates for solutions of the microscopic problem. The main challenge is the derivation of a priori estimates for solutions of the variational inequality, uniformly…
Compactons and their variational properties for degenerate KdV and NLS in dimension 1
本文分析了一类KdV型和NLS型退化色散方程的驻波解和行波解。与非退化KdV和NLS方程的标准孤子解形成鲜明对比的是，这里研究的椭圆算子的退化允许紧支撑的稳定或旅行状态。当我们在$1$维中工作时，ODE方法适用，然而所考虑的模型具有形式上守恒的哈密顿量、质量和动量泛函，这也允许变分分析。
Blowup behavior for a degenerate elliptic sinh-Poisson equation with variable intensities
在本文中，我们提供了一个完整的爆破图片的解决方案序列的椭圆sinh-泊松方程的可变强度所产生的背景下的统计力学描述的二维湍流，由Onsager。涡强度用定义在区间上的概率测度来描述。在Dirichlet边界条件下，我们建立了边界爆破点的排除，我们表明，浓度质量没有剩余的L1-条款，我们确定的基尔霍夫的哈密顿爆破点的位置。我们允许该测度是一般的Borel测度，它可以是退化的。“我们的主要结果对于标准的sinh-泊松方程也是新的。
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