带体积约束的Van der Waals-Cahn-Hilliard方程的Lusternik-Schnirelman与Morse理论

Double Bubbles with High Constant Mean Curvatures in Riemannian Manifolds

We obtain existence of double bubbles of large and constant mean curvatures in Riemannian manifolds. These arise as perturbations of geodesic standard double bubbles centered at critical points of the ambient scalar curvature and aligned along eigen-vectors of the ambient Ricci tensor. We also obtain general multiplicity results via Lusternik-Schnirelman theory, and extra ones in case of double bubbles whose opposite boundaries have the same mean curvature.

一个双扰动Schrödinger-Bopp-Podolsky-Proca系统的多重性结果

A multiplicity result for a double perturbed Schrödinger-Bopp-Podolsky-Proca system

本文利用Lusternik-Schnirelmann范畴，证明了无边界三维紧致黎曼流形上双扰动Schrödinger-Bopp-Podolsky-Proca系统解的数量取决于流形拓扑性质的多重性结果。

Multiple solutions for a NLS equation with critical growth and magnetic field

In this paper, we are concerned with the multiplicity of nontrivial solutions for the following class of complex problems $$ (-i\nabla - A(μx))^{2}u= μ|u|^{q-2}u + |u|^{2^{*}-2}u \ \mbox{in} \ Ω, \ \ \ \ u \in H^{1}_{0}(Ω,\mathbb{C}), $$ where $Ω\subset \mathbb{R}^{N} (N \geq 4)$ is a bounded domain with smooth boundary. Using the Lusternik-Schnirelman theory, we relate the number of solutions with the topology of $Ω$.

Low energy solutions for singularly perturbed coupled nonlinear systems on a Riemannian manifold with boundary

Let (M,g) be asmooth, compact Riemannian manifold with smooth boundary, with n= dim M= 2,3. We suppose the boundary of M to be a smooth submanifold of M with dimension n-1. We consider a singularly perturbed nonlinear system, namely Klein-Gordon-Maxwell-Proca system, or Klein-Gordon-Maxwell system of Scrhoedinger-Maxwell system on M. We prove that the number of low energy solutions, when the perturbation parameter is small, depends on the topological properties of the boundary of M, by means of the Lusternik Schnirelmann category. Also, these solutions have a unique maximum point that lies on …

Multiple solutions for a quasilinear Schrödinger equation on $\mathbb{R}^{N}$}

The multiplicity of positive weak solutions for a quasilinear Schrödinger equations $-L_p u +(λA(x)+1)|u|^{p-2}u= h(u)$ in $\mathbb{R}^N$ is established, where $L_p u\doteq ε^{p}Δ_p u +ε^{p}Δ_p (u^2)u$, $A$ is a nonnegative continuous function and nonlinear term $h$ has a subcritical growth. We achieved our results by using minimax methods and Lusternik-Schnirelman theory of critical points.

Positive solutions of singularly perturbed nonlinear elliptic problem on Riemannian manifolds with boundary

Let (M,g) be a smooth connected compact Riemannian manifold of finite dimension n \geq 2 with a smooth boundary \partial M. We consider the problem -ε^2Δ_gu+u=|u|^{p-2}u, u>0 on M, \partial u/ \partialν=0 on \partial M where ν is an exterior normal to \partial M. The number of solutions of this problem depends on the topological properties of the manifold. In particular we consider the Lusternik Schnirelmann category of the boundary.

常数Q-曲率共形度量多重性结果

Multiplicity results for constant Q-curvature conformal metrics

本文证明闭流形上次临界Paneitz-Branson方程至少存在Cat(M)个正解，进而得出正爱因斯坦流形乘积共形类中常数Q-曲率度量的多重性。

Systems of coupled Schrödinger equations with sign-changing nonlinearities via classical Nehari manifold approach

We propose existence and multiplicity results for the system of Schrödinger equations with sign-changing nonlinearities in bounded domains or in the whole space $\mathbb{R}^N$. In the bounded domain we utilize the classical approach via the Nehari manifold, which is (under our assumptions) a differentiable manifold of class $\mathcal{C}^1$ and the Fountain theorem by Bartsch. In the space $\mathbb{R}^N$ we additionally need to assume the $\mathbb{Z}^N$-periodicity of potentials and our proofs are based on the concentration-compactness lemma by Lions and the Lusternik-Schnirelmann values.

Multiplicity results for a class of quasilinear equations with exponential critical growth

In this work, we prove the existence and multiplicity of positive solutions for the following class of quasilinear elliptic equations $$ \left \{ \begin{array}{lll} -ε^{N}Δ_{N} u + \left(1+μA(x) \right)\left| u\right|^{N-2}u= f(u)\,\,\,\, \mbox{ in } \, \RR^{N},\\ u > 0 \,\,\,\, \mbox{ in } \, \RR^{N}, \end{array} \right. $$ \\ where $Δ_{N}$ is the N-Laplacian operator, $N \geq 2$, $f$ is a function with exponential critical growth, $μ$ and $ε$ are positive parameters and $A$ is a nonnegative continuous function verifying some hypotheses. To obtain our results, we combine variational arguments…

闭可平行化流形上具小体积约束的多相 Allen-Cahn-Hilliard 系统解的多重性

Multiplicity of solutions to the multiphasic Allen-Cahn-Hilliard system with a small volume constraint on closed parallelizable manifolds

本文利用 Lusternik-Schnirelmann 和无穷维 Morse 理论，证明了在闭可平行化黎曼流形上具小体积约束的三相 ACH 矢量方程存在多重解，并给出依赖于流形拓扑不变量的解数量下界，同时证明了解非退化度量集合的通用性。

带边界黎曼流形上质量约束Allen-Cahn方程的多解性结果

Multiplicity results for mass constrained Allen-Cahn equations on Riemannian manifolds with boundary

该研究针对带边界黎曼流形上的质量约束Allen-Cahn方程，结合拓扑、变分方法与几何测度论工具，在小质量约束和小扩散参数假设下，依据流形或其边界的Lusternik--Schnirelmann畴数得到解数下界，还基于Morse不等式得到更强的多解性结果。

关于具有对数非线性的奇异摄动$(p, N)$-拉普拉斯薛定谔方程

On singularly perturbed $(p, N )$-Laplace Schrödinger equation with logarithmic nonlinearity

该研究针对含对数非线性与Trudinger-Moser临界指数非线性的奇异摄动$(p,N)$-拉普拉斯薛定谔方程，结合变分法、惩罚技术与畴数理论，建立了正解数量与势函数最小值集拓扑性质的关联，明确了基态的存在性、多重性等特性。