Biagio Ricceri
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.
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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.
Exact solutions and bifurcation curves of nonlocal elliptic equations with convolutional Kirchhoff functions
本文研究了具有卷积Kirchhoff函数的一维Kirchhoff型非局部椭圆方程，建立了其精确解 $u_λ$ 及以解的无穷范数定义的分岔曲线 $λ(α)$。
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.
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。
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…
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…
Existence of nontrivial solutions to a fourth-order Kirchhoff type elliptic equation with critical exponent
本文研究了带有次临界扰动的临界四阶 Kirchhoff 型椭圆方程，通过结合 Mountain Pass Lemma 和对 Talenti's functions 的精细估计解决了低维（≤7）下弱解的存在性，并在高维（≥8）下通过对非局部系数引入截断证明了非平凡解的存在。
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…
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-…
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.
In this paper we fully characterize the sequentially weakly lower semicontinuity of the parameter-depending energy functional associated with the critical Kirchhoff problem. We also establish sufficient criteria with respect to the parameters for the convexity and validity of the Palais-Smale condition of the same energy functional. We then apply these regularity properties in the study of some elliptic problems involving the critical Kirchhoff term.
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 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 …
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.
$p$-Kirchhoff type equation with Neumann boundary conditions
本文利用Brézis和Nirenberg的抽象环绕引理研究具齐次Neumann边界条件的$p$-Kirchhoff型椭圆方程，在次临界和临界情形下得到多重性结果，补充完善了已有Kirchhoff方程理论。
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.
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.
Infinitely many solutions for a class of elliptic boundary value problems with $(p,q)$-Kirchhoff type
本文研究了一类具有$(p,q)$-Kirchhoff型的椭圆边值问题，通过应用Ding的临界点定理，在次-$(p,q)$条件下证明了该系统存在无穷多解。
High energy solutions for $p$-Kirchhoff elliptic problems with Hardy-Littlewood-Sobolev nonlinearity
本文通过Pohožaev流形和linking定理，证明了带Hardy-Littlewood-Sobolev非线性的p-Kirchhoff椭圆问题在参数范围内存在正高能量解，并研究了极限问题解的径向对称性。
Infinitely many solutions to Kirchhoff double phase problems with variable exponents
本文结合 Kajikiya 的抽象临界点定理与作者此前的广义双相问题先验界，证明右端仅局部且弱假设定义的变指数 Kirchhoff 双相椭圆方程存在一列 $L^\infty$-范数趋于零的非平凡解。
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 study the following Brezis-Nirenberg problem of Kirchhoff type $$ \left\{\aligned &-(a+b\int_Ω|\nabla u|^2dx)Δu = λ|u|^{q-2}u + δ|u|^{2}u, &\quad \text{in}\ Ω, \\ &u=0,& \text{on}\ \partialΩ, \endaligned \right. $$ where $Ω\subset \bbr^4$ is a bounded domain with the smooth boundary $\partialΩ$, $2\leq q<4$ and $a$, $b$, $λ$, $δ$ are positive parameters. We obtain some new existence and nonexistence results, depending on the values of the above parameters, which improves some known results. The asymptotical behaviors of the solutions are also considered in this paper.
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。
We consider Kirchhoff equations with a small parameter epsilon in front of the second-order time-derivative, and a dissipative term whose coefficient may tend to 0 as t -> + infinity (weak dissipation). In this note we present some recent results concerning existence of global solutions, and their asymptotic behavior both as t -> + infinity and as epsilon -> 0. Since the limit equation is of parabolic type, this is usually referred to as a hyperbolic-parabolic singular perturbation problem. We show in particular that the equation exhibits hyperbolic or parabolic behavior depending on the value…
On a Perturbed Critical p-Kirchhoff-Type Problem
本文研究带临界非线性和次临界扰动的p-Kirchhoff型问题，通过变分法和纤维化分析获得基态解和山路解，并在极值参数外得到非平凡解。
In this paper, we investigate the existence of weak solution for a Kirchhoff type problem driven by a nonlocal operator of elliptic type in a fractional Orlicz-Sobolev space, with homogeneous Dirichlet boundary conditions {\small$$ (D_{K,A}) \hspace*{0.5cm} \left\{ \begin{array}{clclc} M\left( \displaystyle \int_{\R^{2N}}A\left( [u(x)-u(y)] K(x,y)\right) dxdy\right) \mathcal{L}^K_A u & = & f(x,u) & \text{ in }& Ω, \hspace*{7cm} u & = & 0 \hspace*{0.2cm} \hspace*{0.2cm} & \text{ in } & \R^N\setminus Ω. \label{eq1} \end{array} \right. $$ } Where $\mathcal{L}^K_A$ is a nonlocal operator with sing…
Three solutions for a double phase variable exponent Kirchhoff problem
本文研究双相变指数Kirchhoff问题，通过推导新的n维向量不等式得到能量泛函正则性，结合Bonanno和Marano临界点定理证明该问题至少存在三个解。
Motivated by recent interest on Kirchhoff-type equations, in this short note we utilize a classical, yet very powerful, tool of nonlinear functional analysis in order to investigate the existence of positive eigenvalues of systems of elliptic functional differential equations. An example is presented to illustrate the theory.
In the present paper, we consider the nonlocal Kirchhoff problem \begin{eqnarray*} -\left(ε^2a+εb\int_{\mathbb{R}^{3}}|\nabla u|^{2}\right)Δu+V(x)u=u^{p},\,\,\,u>0 & & \text{in }\mathbb{R}^{3}, \end{eqnarray*} where $a,b>0$, $1<p<5$ are constants, $ε>0$ is a parameter. Under some mild assumptions on the function $V$, we obtain multi-peak solutions for $ε$ sufficiently small by Lyapunov-Schmidt reduction method. Even though many results on single peak solutions to singularly perturbed Kirchhoff problems have been derived in the literature by various methods, there exist no results on multi-peak…
Existence and multiplicity of solutions for a critical Kirchhoff type elliptic equation with a logarithmic perturbation
本文利用变分方法研究具有对数扰动的临界 Kirchhoff 型椭圆方程，通过分析随空间维数变化的能量泛函结构，获得了不同维数下解的存在性与多重性结果。
Qualitative analysis on the critical points of the Kirchhoff-Routh function
本文研究了Kirchhoff-Routh函数的临界点问题，揭示了小孔位置对临界点分布的影响，并在椭圆问题中证明了多峰解的存在性。
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。
We prove an abstract critical point theorem based on a cohomological index theory that produces pairs of nontrivial critical points with nontrivial higher critical groups. This theorem yields pairs of nontrivial solutions that are neither local minimizers nor of mountain pass type for problems with combined nonlinearities. Applications are given to subcritical and critical $p$-Laplacian problems, Kirchhoff type nonlocal problems, and critical fractional $p$-Laplacian problems.
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。
We study the ground states of the following generalization of the Kirchhoff-Love functional, $$J_σ(u)=\int_Ω\dfrac{(Δu)^2}{2} - (1-σ)\int_Ωdet(\nabla^2u)-\int_ΩF(x,u),$$ where $Ω$ is a bounded convex domain in $\mathbb{R}^2$ with $C^{1,1}$ boundary and the nonlinearities involved are of sublinear type or superlinear with power growth. These critical points correspond to least-energy weak solutions to a fourth-order semilinear boundary value problem with Steklov boundary conditions depending on $σ$. Positivity of ground states is proved with different techniques according to the range of the pa…
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.
We consider the second order Cauchy problem $$u''+\m{u}Au=0, u(0)=u_{0}, u'(0)=u_{1},$$ where $m:[0,+\infty)\to[0,+\infty)$ is a continuous function, and $A$ is a self-adjoint nonnegative operator with dense domain on a Hilbert space. It is well known that this problem admits local-in-time solutions provided that $u_{0}$ and $u_{1}$ are regular enough, depending on the continuity modulus of $m$. It is also well known that the solution is unique when $m$ is locally Lipschitz continuous. In this paper we prove that if either $<Au_{0},u_{1}>\neq 0$, or $|A^{1/2}u_{1}|^{2}\neq\m{u_{0}}|Au_{0}|^{2}…
Kirchhoff type elliptic equations with double criticality in Musielak-Sobolev spaces
本文研究了Musielak-Sobolev空间中具有双临界性的Kirchhoff型椭圆方程，利用基于山路定理的变分技术证明了其弱解的存在性。
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 $$ \mathcal{L}_{\varepsilon} u = M \left(\frac{1}{\varepsilon} \int_{\mathbb{R}^3} |\nabla u|^2 + \frac{1}{…
In this paper, we consider a class of quasilinear stationary Kirchhoff type potential systems with Neumann Boundary conditions, which involves a general variable exponent elliptic operator with critical growth. Under some suitable conditions on the nonlinearities, we establish existence and multiplicity of solutions for the problem by using the concentration-compactness principle of Lions for variable exponents found in [5, 7] and the Mountain Pass Theorem without the Palais-Smale condition given in [43].
Strongly Singular Nonlocal Kirchhoff-Type Equations with Variable Exponents: Existence, Regularity, and Renormalized Solutions
本文解决非局部Kirchhoff型方程变量指数下的强奇异问题，通过五个原创定理建立完整理论，涵盖存在性、正则性和重正化解构造。
Billiards, Refraction, and Blow-Up for Kirchhoff Equations
本文通过几何光学中的椭圆台球折射反射机制构造双模态Kirchhoff系统的横向异宿轨道，结合路线图定理，证明受迫抽象Kirchhoff方程存在有限时间爆破解。
Generalized noncooperative Schrödinger-Kirchhoff-type systems in $\mathbb{R}^N$
本文研究含一般变指数临界增长椭圆算子的非合作 Schrödinger-Kirchhoff 型系统，结合极限指标理论、加权变指数 Sobolev 集中紧性原理与对称临界性原理，证明其存在无穷多解。
In this paper, we study the following nonlinear Kirchhoff problem involving critical growth: $$ \left\{% \begin{array}{ll} -(a+b\int_Ω|\nabla u|^2dx)Δu=|u|^4u+λ|u|^{q-2}u, u=0\ \ \text{on}\ \ \partialΩ, \end{array}% \right. $$ where $1<q<2$, $λ,\ a,\ b>0$ are parameters and $Ω$ is a bounded domain in $\R^3$. We prove that there exists $λ_1=λ_1(q,Ω)>0$ such that for any $λ\in(0,λ_1)$ and $a,\ b>0$, the above Kirchhoff problem possesses at least two positive solutions and one of them is a positive ground state solution. We also establish the convergence property of the ground state solution as t…
We consider the second order Cauchy problem $$u''+m(|A^{1/2}u|^2)Au=0, u(0)=u_{0}, u'(0)=u_{1},$$ where $m:[0,+\infty)\to[0,+\infty)$ is a continuous function, and $A$ is a self-adjoint nonnegative operator with dense domain on a Hilbert space. It is well known that this problem admits local-in-time solutions provided that $u_{0}$ and $u_{1}$ are regular enough, depending on the continuity modulus of $m$, and on the strict/weak hyperbolicity of the equation. We prove that for such initial data $(u_{0},u_{1})$ there exist two pairs of initial data $(\overline{u}_{0},\overline{u}_{1})$, $(\wideh…
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. } …
Existence of nontrivial solutions to a critical Kirchhoff equation with a logarithmic type perturbation in dimension four
本文研究了四维有界区域中带对数型次临界项的临界 Kirchhoff 方程，利用变分原理和收敛技巧证明了在适当参数下局部极小解和最小能量解的存在性，并表明非局部项扩大了弱解存在的参数范围。
A unified treatment of degenerate nonlocal elliptic problems
本文通过辅助问题和固定点技术，研究非局部椭圆问题的存在性、不存在性和多重性，揭示参数阈值并描述解集。
In this paper we investigate a class of elliptic problems involving a nonlocal Kirchhoff type operator with variable coefficients and data changing its sign. Under appropriated conditions on the coefficients, we have shown existence and uniqueness of solution.
The inf-sup condition, also called the Ladyzhenskaya--Babu\v ska--Brezzi (LBB) condition, ensures the existence, uniqueness and well-posedness of a saddle point problem, relative to a partial differential equation. Discretization by the finite element method gives the discrete problem which must satisfy the discrete inf-sup condition. But, depending on the choice of finite elements, the discrete condition may fail. This paper attempts to explain why it fails from an engineer's perspective, and reviews current methods to work around this failure. The last part recalls the mathematical bases.
The Modified Energy Method for Quasilinear Wave Equations of Kirchhoff Type
本文用修正能量方法研究 Kirchhoff 型拟线性波动方程，通过 Taylor 级数构造能量，证明小初始数据的五次能量估计、延长寿命及弱解存在性。
We prove new results on the existence of positive radial solutions of the elliptic equation $-Δu= λh(|x|,u)$ in an annular domain in $\mathbb{R}^{N}, N\geq 2$. Existence of positive radial solutions are determined under the conditions that the nonlinearity function $h(t,u)$ is either superlinear or sublinear growth in $u$ or satisfies some upper and lower inequalities on $h$. Our discussion is based on a fixed point theorem due to a revised version of a fixed point theorem of Gustafson and Schmitt.
Existence of at least $k$ solutions to a fractional $p$-Kirchhoff problem involving singularity and critical exponent
本文研究含奇异项和临界 Sobolev 指数的分数阶 p-Kirchhoff 问题，通过截断论证和对称山路定理证明任意多个小能量非负解的存在性，并用 Moser 迭代得到解的一致有界性。
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