Ismail Aydin, Cihan Unal
This paper is concerned with a nonlinear Steklov boundary-value problem involving weighted $p(.)$-Laplacian. Using the Ricceri's variational principle, we obtain the existence of at least three weak solutions in double weighted variable exponent Sobolev space.
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A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle
该文研究带内部固定球形障碍物的第一Steklov-拉普拉斯特征值问题，证明内球和体积固定时球壳在近球形集合中局部最大化第一特征值。
A monotonicity result for the first Steklov-Dirichlet Laplacian eigenvalue
本文研究带球形洞的环形区域中拉普拉斯算子的第一Steklov-Dirichlet特征值，证明了外部区域中心对称时特征值关于洞的单调性结果。
In this paper we consider in a bounded domain $Ω\subset \mathbb{R}^N$ with smooth boundary an eigenvalue problem for the negative $(p,q)$-Laplacian with a Steklov type boundary condition, where $p\in (1,\infty)$, $q\in (2,\infty)$ and $p\neq q$. A full description of the set of eigenvalues of this problem is provided, thus essentially extending a recent result by Abreu and Madeira [1] related to the $(p,2)$-Laplacian.
In this paper we study double phase problems with nonlinear boundary condition and gradient dependence. Under quite general assumptions we prove existence results for such problems where the perturbations satisfy a suitable behavior in the origin and at infinity. Our proofs make use of variational tools, truncation techniques and comparison methods. The existence of the obtained solutions depends on the first eigenvalues of the Robin and Steklov eigenvalue problems for the $p$-Laplacian.
Weighted Neumann-to-Steklov limits for nonlinear eigenvalues and trace constants
本文研究由边界集中内权生成的非线性Neumann到Steklov极限，证明当参数趋于零时，加权(p,q)-Neumann特征值收敛到对应的加权(p,q)-Steklov特征值，同时极小解收敛到Steklov极小解。
In this paper we consider in a bounded domain $Ω\subset \mathbb{R}^N$ with smooth boundary an eigenvalue problem for the negative $(p,q)$-Laplacian with a Steklov-like boundary condition, where $p,\, q\in (1,\infty)$, $p\neq q$, including the open case $p\in (1,\infty)$, $q\in (1, 2)$, $p\neq q$. A full description of the set of eigenvalues of this problem is provided. Our results complement those previously obtained by Abreu and Madeira \cite{AM}, Barbu and Moroşanu \cite{BM}, Fărcăşeanu, Mihăilescu and Stancu-Dumitru \cite{FMS}, Mihăilescu \cite{MMih}, Mihăilescu and Moroşanu \cite{MM}.
For a given bounded Lipschitz set $Ω$, we consider a Steklov--type eigenvalue problem for the Laplacian operator whose solutions provide extremal functions for the compact embedding $H^1(Ω)\hookrightarrow L^2(\partial Ω)$. We prove that a conjectured reverse Faber--Krahn inequality holds true at least in the class of Lipschitz sets which are "close" to a ball in a Hausdorff metric sense. The result implies that among sets of prescribed measure, balls are local minimizers of the embedding constant.
Brock-type isoperimetric inequality for Steklov eigenvalues of the Witten-Laplacian
本文在加权体积固定约束下证明了欧氏与双曲空间中有界域上Witten-Laplacian的Steklov型特征值的Brock型等周不等式，拓展了F. Brock的经典结果，并提出相关开放问题。
On the weighted Steklov eigenvalue problems in outward cuspidal domains
本文研究外尖点区域中加权Steklov及加权Schrödinger--Steklov特征值问题，证明其在线性与非线性情形下的可解性。
In this paper we prove the existence of a maximum for the first Steklov-Dirichlet eigenvalue in the class of convex sets with a fixed spherical hole under volume constraint. More precisely, if $Ω=Ω_0 \setminus \bar{B}_{R_1}$, where $B_{R_1}$ is the ball centered at the origin with radius $R_1>0$ and $Ω_0\subset\mathbb{R}^n$, $n\geq 2$, is an open bounded and convex set such that $B_{R_1}\Subset Ω_0$, then the first Steklov-Dirichlet eigenvalue $σ_1(Ω)$ has a maximum when $R_1$ and the measure of $Ω$ are fixed. Moreover, if $Ω_0$ is contained in a suitable ball, we prove that the spherical shel…
In this note we present upper bounds for the variational eigenvalues of the Steklov $p$-Laplacian on domains of $\mathbb R^n$, $n\geq 2$. We show that for $1<p\leq n$ the variational eigenvalues $σ_{p,k}$ are bounded above in terms of $k,p,n$ and $|\partialΩ|$ only. In the case $p>n$ upper bounds depend on a geometric constant $D(Ω)$, the $(n-1)$-distortion of $Ω$ which quantifies the concentration of the boundary measure. We prove that the presence of this constant is necessary in the upper estimates for $p>n$ and that the corresponding inequality is sharp, providing examples of domains with …
We study the Steklov eigenvalue problem for the $\infty-$orthotropic Laplace operator defined on convex sets of $\mathbb{R}^N$, with $N\geq2$, considering the limit for $p\to+\infty$ of the Steklov problem for the $p-$orthotropic Laplacian. We find a limit problem that is satisfied in the viscosity sense and a geometric characterization of the first non trivial eigenvalue. Moreover, we prove Brock-Weinstock and Weinstock type inequalities among convex sets, stating that the ball in a suitable norm maximizes the first non trivial eigenvalue for the Steklov $\infty-$orthotropic Laplacian, once w…
Exponential Decays of Steklov Eigenfunctions for the Magnetic Laplacian
针对有界Lipschitz区域上带有限型磁场的磁薛定谔算子对应的Dirichlet-to-Neumann映射，证明场强足够大时其基态在磁场最大阶消失边界子集邻域外呈指数衰减。
Compactness of weighted Sobolev trace operators and non-linear Steklov problems
研究带权Sobolev迹算子在向外尖突域的紧性，通过Sobolev空间上的复合算子方法，为非线性Steklov问题提供正确的函数空间设定，并证明非平凡解的存在性。
Isocapacitary constants for the $p$-Laplacian on compact manifolds
本文研究了$ p $-Laplacian在紧致流形上的Steklov和Neumann等容常数，并推导了与Sobolev常数及特征值相关的上下界。
Nonresonance for problems involving $(p,q)$-Laplacian equations with nonlinear perturbations
研究带非线性反应项和非齐次Neumann边界条件的$(p,q)$-Laplacian问题的可解性，通过变分方法与临界点理论，证明非线性项渐近低于第一特征值或Steklov-Neumann特征值线时弱解的存在性。
In the framework of the Laplacian transport, described by a Robin boundary value problem in an exterior domain in $\mathbb{R}^n$, we generalize the definition of the Poincaré-Steklov operator to $d$-set boundaries, $n-2< d<n$, and give its spectral properties to compare to the spectra of the interior domain and also of a truncated domain, considered as an approximation of the exterior case. The well-posedness of the Robin boundary value problems for the truncated and exterior domains is given in the general framework of $n$-sets. The results are obtained thanks to a generalization of the conti…
Let $M^n$ be a smooth compact manifolds with smooth boundary. We show that for a generic $C^k$ metic on $\bar{M^n}$ with $k>n-1$, the nonzero Steklov eigenvalues are simple. Moreover, we also prove that the non-constant Steklov eigenfunctions have zero as a regular value and are Morse functions on the boundary for such generic metric. These results generalize the celebrated results on Laplacians by Uhlenbeck to the Steklov setting.
Geometric bounds for Steklov and weighted Neumann eigenvalues on Euclidean domains
研究在欧几里得空间中维数d≥3的有界域的前两个非零Steklov特征值的上界，并证明当d≥7时这些界是最佳的。
The second eigenvalue of the Robin Laplacian is shown to be maximal for the disk among simply-connected planar domains of fixed area when the Robin parameter is scaled by perimeter in the form $α/L(Ω)$, and $α$ lies between $-2π$ and $2π$. Corollaries include Szegő's sharp upper bound on the second eigenvalue of the Neumann Laplacian under area normalization, and Weinstock's inequality for the first nonzero Steklov eigenvalue for simply-connected domains of given perimeter. The first Robin eigenvalue is maximal, under the same conditions, for the degenerate rectangle. When area normalization o…
A weighted Reilly formula for differential forms and sharp Steklov eigenvalue estimates
本文建立带边黎曼流形上微分形式的加权Reilly公式，并利用它得到微分形式Steklov特征值第一正特征值的尖锐下界及与边界Hodge Laplacian谱的比较结果，同时提出一个类比Escobar猜想的公开问题。
The nonlinear Steklov problem in outward cuspidal domains
本文研究了在外向尖点域中非线性Steklov特征值问题，通过加权迹嵌入的紧致性得到第一个非平凡特征值的变分刻画并证明了弱解的存在性。
We extend some classical inequalities between the Dirichlet and Neumann eigenvalues of the Laplacian to the context of mixed Steklov--Dirichlet and Steklov--Neumann eigenvalue problems. The latter one is also known as the sloshing problem, and has been actively studied for more than a century due to its importance in hydrodynamics. The main results of the paper are applied to obtain certain geometric information about nodal sets of sloshing eigenfunctions. The key ideas of the proofs include domain monotonicity for eigenvalues of mixed Steklov problems, as well as an adaptation of Filonov's me…
We discuss isoperimetric inequalities for the magnetic Laplacian on bounded domains of $\mathbb R^2$ endowed with an Aharonov-Bohm potential. When the flux of the potential around the pole is not an integer, the lowest eigenvalue for the Neumann and the Steklov problems is positive. We establish isoperimetric inequalitites for the lowest eigenvalue in the spirit of the classical inequalities of Szegö-Weinberger, Brock and Weinstock, the model domain being a disk with the pole at its center. We consider more generally domains in the plane endowed with a rotationally invariant metric, which incl…
In this paper we study a Steklov-Robin eigenvalue problem for the Laplacian in annular domains. More precisely, we consider $Ω=Ω_0 \setminus \overline{B}_{r}$, where $B_{r}$ is the ball centered at the origin with radius $r>0$ and $Ω_0\subset\mathbb{R}^n$, $n\geq 2$, is an open, bounded set with Lipschitz boundary, such that $\overline{B}_{r}\subset Ω_0$. We impose a Steklov condition on the outer boundary and a Robin condition involving a positive $L^{\infty}$-function $β(x)$ on the inner boundary. Then, we study the first eigenvalue $σ_β(Ω)$ and its main properties. In particular, we investi…
The paper is concerned with the Steklov eigenvalue problem on cuboids of arbitrary dimension. We prove a two-term asymptotic formula for the counting function of Steklov eigenvalues on cuboids in dimension d greater or equal to 3. Apart from the standard Weyl term, we calculate explicitly the second term in the asymptotics, capturing the contribution of the (d-2)-dimensional facets of a cuboid. Our approach is based on lattice counting techniques. While this strategy is similar to the one used for the Dirichlet Laplacian, the Steklov case carries additional complications. In particular, it is …
Isoperimetric Bounds for Weighted Steklov Eigenvalues with Radial Weights
研究具有径向权重的Steklov特征值问题，建立低阶特征值的等周不等式，利用显式谱计算、变分原理和双密度加权等周不等式，提出新的加权等周不等式。
For a given bounded domain $Ω$ with smooth boundary in a smooth Riemannian manifold $(\mathcal{M},g)$, we show that the Poisson type upper-estimate of the heat kernel associated to the Dirichlet-to-Neumann operator, the Sobolev trace inequality, the Log-Sobolev trace inequality, the Nash trace inequality, and the Rozenblum-Lieb-Cwikel type inequality are all equivalent. Upon decomposing the Dirichlet-to-Neumann operator into a sum of the square root of the Laplacian and a pseudodifferntial operator and by applying Grubb's method of symbolic calculus for the corresponding pseudodifferential hea…
In this paper, we consider the numerical approximation of the Steklov eigenvalue problem that arises in inverse acoustic scattering. The underlying scattering problem is for an inhomogeneous isotropic medium. These eigenvalues have been proposed to be used as a target signature since they can be recovered from the scattering data. A Galerkin method is studied where the basis functions are the Neumann eigenfunctions of the Laplacian. Error estimates for the eigenvalues and eigenfunctions are proven by appealing to Weyl's Law. We will test this method against separation of variables in order to …
Characterizations of eigenvalues and eigenfunctions of the Laplacian on a product domain are obtained. When zero Dirichlet, Robin or Neumann boundary conditions are specified on each factor, then the eigenfunctions on the product domain are precisely the products of the eigenfunctions on the individual factors. There is a related result when Steklov boundary conditions are specified on the second factor. These results enable the characterization of certain Hilbert-Sobolev spaces as tensor products and descriptions of some orthogonal bases of the spaces. A different characterization of the trac…
An indefinite Coulomb interaction from the Steklov spectrum of perforated manifolds
研究在穿孔闭流形上斯捷克洛夫特征值及相关函数向加权拉普拉斯 - 贝尔特拉米对应物的收敛速率，推导其高阶展开，确定二维和三维的校正尺度，通过辅助函数估计量化差异，弥合谱几何与点系统文献差距。
Using methods in the spirit of deterministic homogenisation theory we obtain convergence of the Steklov eigenvalues of a sequence of domains in a Riemannian manifold to weighted Laplace eigenvalues of that manifold. The domains are obtained by removing small geodesic balls that are asymptotically densely uniformly distributed as their radius tends to zero. We use this relationship to construct manifolds that have large Steklov eigenvalues. In dimension two, and with constant weight equal to 1, we prove that Kokarev's upper bound of $8π$ for the first nonzero normalised Steklov eigenvalue on or…
Isoperimetric bounds for the weighted Neumann and Steklov eigenvalues in dimensions three through six
该论文确定了三维至六维有界欧几里得域上质量归一化第一加权 Neumann 特征值和 Steklov 特征值的尖锐等周界，并解决了 Vinokurov 提出的问题 1.12。
Let $M$ be a closed hypersurface in $\mathbb{R}^{n}$ and $Ω$ be a bounded domain such that $M= \partialΩ$. In this article, we obtain an upper bound for the first non-zero eigenvalue of the following problems. \begin{itemize} \item Closed eigenvalue problem: \begin{align*} %\label{eqn:closedep} Δ_p u = λ_{p} \ |u|^{p-2} \ u \qquad \mbox{ on } \quad {M}. \end{align*} \item Steklov eigenvalue problem: \begin{align*} \begin{array}{rcll} Δ_{p}u &=& 0 & \mbox{ in } Ω,\\ |\nabla u|^{p-2} \frac{\partial u}{\partial ν} &=& μ_{p} \ |u|^{p-2} \ u &\mbox{ on } M . \end{array} \end{align*} \end{itemize}
We associate a sequence of variational eigenvalues to any Radon measure on a compact Riemannian manifold. For particular choices of measures, we recover the Laplace, Steklov and other classical eigenvalue problems. In the first part of the paper we study the properties variational eigenvalues and establish a general continuity result, which shows for a sequence of measures converging in the dual of an appropriate Sobolev space, that the associated eigenvalues converge as well. The second part of the paper is devoted to various applications to shape optimization. The main theme is studying shar…
In this study, we define double weighted variable exponent Sobolev spaces $W^{1,q(.),p(.)}\left( Ω,\vartheta _{0},\vartheta \right) $ with respect to two different weight functions. Also, we investigate the basic properties of this spaces. Moreover, we discuss the existence of weak solutions for weighted Dirichlet problem of $p(.)$-Laplacian equation \begin{equation*} \left\{ \begin{array}{cc} -\text{div}\left( \vartheta (x)\left\vert \nabla f\right\vert ^{p(x)-2}\nabla f\right) =\vartheta _{0}(x)\left\vert f\right\vert ^{q(x)-2}f & x\in Ω\\ f=0 & x\in \partial Ω\end{array} \right. \end{equati…
In this paper we study an eigenvalue problem for the so called $(p,2)$-Laplace operator on a smooth bounded domain under a nonlinear Steklov type boundary condition, namely \begin{equation} \left\{ \begin{aligned} -Δ_pu-Δu & =λa(x)u \ \ \text{in}\ Ω,\\ (|\nabla u|^{p-2}+1)\dfrac{\partial u}{\partialν} & =λb(x)u \ \ \text{on}\ \partialΩ. \end{aligned} \right. \end{equation} Under suitable integrability and boundedness assumptions on the positive weight functions $a$ and $b$, we show that, for all $p>1$, the eigenvalue set consists of an isolated null eigenvalue plus a continuous family of eigen…
It was conjectured by Escobar [J. Funct. Anal. 165 (1999), 101--116] that for an $n$-dimensional ($n\geq 3$) smooth compact Riemannian manifold with boundary, which has nonnegative Ricci curvature and boundary principal curvatures bounded below by $c>0$, the first nonzero Steklov eigenvalue is greater than or equal to $c$ with equality holding only on isometrically Euclidean balls with radius $1/c$. In this paper, we confirm this conjecture in the case of nonnegative sectional curvature. The proof is based on a combination of Qiu--Xia's weighted Reilly-type formula with a special choice of the…
Surface-to-Volume Homogenization of NonLinear Steklov Eigencouples with NonHomogeneous Orlicz Growth in Perforated Domains
该研究针对穿孔域中具有非齐次Orlicz增长的非线性Steklov特征对，通过强逼近微观迹的方法，建立了其均匀化极限的相关结果。
Sharp upper bounds for Steklov eigenvalues on warped products
该研究针对基为带边界紧致黎曼流形、纤维为闭黎曼流形的扭曲乘积，推导了Steklov特征值的等周型尖锐上界，运用分离变量与谱近似方法完成证明。
We prove the existence of at least three solutions for a weighted $p$-Laplacian operator involving Dirichlet boundary condition in a weighted Sobolev space. The main tool we use here is a three solution theorem in reflexive Banach spaces due to G. Bonanno and B. Ricceri.
We characterize one-sided weighted Sobolev spaces $W^{1,p}(\mathbb{R},ω)$, where $ω$ is a one-sided Sawyer weight, in terms of a.e.~and weighted $L^p$ limits as $α\to1^-$ of Marchaud fractional derivatives of order $α$. Similar results for weighted Sobolev spaces $W^{2,p}(\mathbb{R}^n,ν)$, where $ν$ is an $A_p$-Muckenhoupt weight, are proved in terms of limits as $s\to1^-$ of fractional Laplacians $(-Δ)^s$. These are Bourgain--Brezis--Mironescu-type characterizations for weighted Sobolev spaces. We also complement their work by studying a.e.~and weighted $L^p$ limits as $α,s\to0^+$.
Infinitely many solutions for three quasilinear Laplacian systems on weighted graphs
该研究利用无紧性条件的抽象临界点定理，证明了加权图上三类带参数的拟线性拉普拉斯方程组在参数处于确定范围时存在无穷多范数无界的非平凡解。
We will find Green's function for the standard weighted Laplacian and use the corresponding Green's potential to solve Poisson's equation in the unit disc with zero boundary values, in the sense of radial $L^1$-means, for complex Borel measures $μ$ satisfying the condition $\int_\mathbb{D} (1-|w|^2)^{α+1} d|μ|(w) < \infty$ for $α>-1$.
In this paper, we prove some qualitative properties for the positive solutions to some degenerate elliptic equation given by \[-\operatorname{div}(w|\nabla u|^{p-2}\nabla u)=f(x,u);\;\;w\in \mathcal{A}_p\] on smooth domain and for varying nonlinearity $f$.
Multiplicity Results for the Nonlinear Weighted Laplacian Equation on the Sierpiński Gasket
研究 Sierpiński 垫片上加权拉普拉斯方程 Dirichlet 边值问题，通过直接法、山路定理及相关多重性定理，在非线性项满足适当增长条件下给出解的存在性与多重性结果。
We obtain the global weighted $W^{1,p}$ estimates for weak solutions of nonlinear elliptic equations over Reifenberg flat domains. Where nonlinearity $A(x,z,ξ)$ is assumed to be local uniform continuous in $z$ and have small BMO semi-norm in $x$. Moreover, we derive Besov regularity for solutions of a class of special harmonic equations by making use of $W^{1,p}$ estimate. Keywords: global weighted $W^{1,p}$ estimates; quasilinear equations; Besov regularity
This paper considers the inverse boundary value problem for the equation $\nabla\cdot(σ\nabla u+a|\nabla u|^{p-2}\nabla u)=0$. We give a procedure for the recovery of the coefficients $σ$ and $a$ from the corresponding Dirichlet-to-Neumann map, under suitable regularity and ellipticity assumptions.
On the Aleksandrov--Bakelman--Pucci estimates for the weighted $1$-Laplacian
该研究针对加权1-拉普拉斯算子及相关完全非线性算子的粘性下解，建立了 Aleksandrov-Bakelman-Pucci 型估计，证明中以拟凹包络替代经典 ABP 估计中的凹包络。
This paper describes the singular value decomposition (SVD) of the Poisson kernel for the Dirichlet problem for the Laplacian on bounded regions in R^N, N >=2. This operator is a compact linear transformation from L^2 of the boundary to L^2 of the region. These singular values and functions are related to the eigenvalues and eigenfunctions of the Dirichlet Biharmonic Steklov eigenproblem. The Bergman harmonic projection on L^2 is characterized and the Reproducing kernel for the real harmonic Bergman space is described. Optimal finite rank approximations of the Poisson kernel, with error estima…
We consider the fractional Laplacian operator $(-Δ)^s$ (let $ s \in (0,1) $) on Euclidean space and investigate the validity of the classical integration-by-parts formula that connects the $ L^2(\mathbb{R}^d) $ scalar product between a function and its fractional Laplacian to the nonlocal norm of the fractional Sobolev space $ \dot{H}^s(\mathbb{R}^d) $. More precisely, we focus on functions belonging to some weighted $ L^2 $ space whose fractional Laplacian belongs to another weighted $ L^2 $ space: we prove and disprove the validity of the integration-by-parts formula depending on the behavio…
In this note we prove horizontal weighted Rellich inequalities for the sub-Laplacian and for sub-Laplacians with drift on general stratified groups. We show how the presence of a drift improves the known inequalities. Moreover, we obtain several versions of weighted Rellich inequalities for the sub-Laplacian with drift on the polarizable Carnot groups, also with the weights associated with the fundamental solution of the sub-Laplacian. The obtained results are already new for the Laplacian in the usual Euclidean setting of ${\mathbb R}^n$, embedding the classical Rellich inequality into a fami…
In this article we obtain improved versions of Stein-Weiss and Caffarelli-Kohn-Nirenberg inequalities, involving Besov norms of negative smoothness. As an application of the former, we derive the existence of extremals of the Stein-Weiss inequality in certain cases, some of which are not contained in the celebrated theorem of E. Lieb.
In this paper we prove the fractional Gagliardo-Nirenberg inequality on homogeneous Lie groups. Also, we establish weighted fractional Caffarelli-Kohn-Nirenberg inequality and Lyapunov-type inequality for the Riesz potential on homogeneous Lie groups. The obtained Lyapunov inequality for the Riesz potential is new already in the classical setting of $\mathbb{R}^{N}$. As an application, we give two-sided estimate for the first eigenvalue of the Riesz potential. Also, we obtain Lyapunov inequality for the system of the fractional $p$-sub-Laplacian equations and give an application to estimate it…
Infinitely many solutions for two generalized poly-Laplacian systems on weighted graphs
本研究利用基于山路定理和拓扑度理论的变分方法，证明了加权有限图及带Dirichlet边值的加权局部有限图上的两类广义多拉普拉斯系统，均存在能量分别趋于正负无穷的两列无穷多解。
Sharp weighted Sobolev trace inequalities and fractional powers of the Laplacian
本文建立一族尖锐Sobolev迹不等式，将Laplacian分数次幂实现为广义Dirichlet-to-Neumann算子，推广了Caffarelli--Silvestre和Yang的结果。
An isoperimetric inequality for the second Robin eigenvalue of the Weighted Laplacian
本文证明在特定条件下，球体是最大化加权拉普拉斯算子第二罗宾特征值的最优形状。
Edge asymptotics for weighted Robin equations on cylinders with applications to spectral fractional Laplacians
该研究针对柱体上加权退化/奇异问题，用Almgren型方法推导Robin方程边缘渐近性，将其应用于谱分数阶拉普拉斯算子，得到局部渐近性、唯一延拓及统一的局部衰减率量化结果。
Combined with our previous work \cite{LW19eigenvalue}, we prove sharp lower bound estimates for the first nonzero eigenvalue of the weighted $p$-Laplacian with $1< p< \infty$ on a compact Bakry-Émery manifold $(M^n,g,f)$, without boundary or with a convex boundary and Neumann boundary condition, satisfying $\text{Ric}+\nabla^2 f \geq κ\, g$ for some $κ\in \mathbb{R}$.
Elliptic $p$-Laplacian systems with nonlinear boundary condition
本文研究具有非线性边界条件的椭圆 p-Laplacian 方程组，通过上下解方法证明正极小解和负极大解的存在性，并利用第二特征值的变分刻画及非光滑截断能量泛函证明第三个非平凡解的存在性。
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