The Nehari manifold for a semilinear elliptic equation involving a sublinear term

The Nehari manifold for the equation for together with Dirichlet boundary conditions is investigated in the case where . Exploiting the relationship between the Nehari manifold and fibrering maps (i.e., maps of the form where J is the Euler functional associated with the equation), we discuss how the Nehari manifold changes as changes and show how this is linked to results on bifurcation from infinity which are associated with the problem.Received: 8 December 2003, Accepted: 10 May 2004, Published online: 16 July 2004Mathematics Subject Classification (2000): 35J20, 35J25     

The Nehari manifold for a semilinear elliptic system involving sign-changing weight functions

In this paper we study the combined effect of concave and convex nonlinearities on the number of solutions for a semilinear elliptic system (_Eλ,μ)$\left(E_{\lambda ,\mu }\right)$ with sign-changing weight function. With the help of the Nehari manifold, we prove that the system has at least two nontrivial nonnegative solutions when the pair of the parameters (λ,μ)$(\lambda ,\mu )$ belongs to a certain subset of R²${\mathbb{R}}^2$.     

The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function

The Nehari manifold for the equation −Δu(x)=λa(x)u(x)+b(x)|u(x)|^ν−1u(x) for x∈Ω together with Dirichlet boundary conditions is investigated. Exploiting the relationship between the Nehari manifold and fibrering maps (i.e., maps of the form t→J(tu) where J is the Euler functional associated with the equation) we discuss how the Nehari manifold changes as λ changes and show how existence and non-existence results for positive solutions of the equation are linked to properties of the manifold.     

On a Nirenberg-type problem involving the square root of the Laplacian

Through an Euler–Hopf-type formula, we establish existence result to a Nirenberg-type problem involving the square root of the Laplacian in sphere S²$S^2$.     

Nehari流形在一个拟线性方程组中的应用

主要研究带有非齐次边界条件的拟线性椭圆方程组的正解问题,在合适参数条件下,用变分方法和流形方法得到该椭圆方程组正解的存在性和多解性.结论推广了近期发表的结果.     

Positive solutions of nonlinear problems involving the square root of the Laplacian

We consider nonlinear elliptic problems involving a nonlocal operator: the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions. For positive solutions to problems with power nonlinearities, we establish existence and regularity results, as well as a priori estimates of Gidas-Spruck type. In addition, among other results, we prove a symmetry theorem of Gidas-Ni-Nirenberg type.     

The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions

This paper examines a class of Kirchhoff type equations that involve sign-changing weight functions. Using Nehari manifold and fibering map, the existence of multiple positive solutions is established.     

The Nehari manifold approach for a class of n\times n nonlinear elliptic systems

In this paper, by analysis of the relationship between the Nehari manifold and fibering maps, we discuss the existence, multiplicity and nonexistence of positive weak solution for the $(p_1,p_2,\ldots ,p_n)$ -Laplacian systems with sign-changing weight functions.     

Symplectic flow for the square root of the negative Laplacian

The paper is concerned with symplectic flow for the square root of the negative Laplacian, which is motivated when studying a one-dimensional model about ferromagnetic thin film without Gilbert damping term (only Gyromagnetic term remained) derived from the stationary model of A. DeSimone, R.V. Kohn, S. Müller and F. Otto. Existence of weak solution to the model is proved by viscosity approximation and commutator estimate.     

一个包含平方补数的方程

对任意正整数n,我们定义a(n)为n的平方补数,即a(n)表示能够使na(n)为完全平方数的最小正整数.本文的主要目的是利用初等方法研究方程a(n1)+a(n2)+…+a(nk)=m·a(n1+n2+…+nk)的可解性,并证明对某些特殊的正整数m及任意正整数k>1,该方程有无穷多组正整数解(n1,n2,…,nk).     

Semilinear elliptic equations for the fractional Laplacian involving critical exponential growth

We establish the existence and multiplicity of weak solutions for a class of nonlocal equations involving the fractional Laplacian operator, nonlinearities with critical exponential growth, and potentials that may change sign. The proofs of our existence results rely on minimization methods and the mountain pass theorem. Copyright © 2016 John Wiley & Sons, Ltd.     

Heat flow for the square root of the negative Laplacian for unit length vectors

The paper is concerned with heat flow for the square root of the negative Laplacian for unit length vectors, motivated by the model of DeSimone, Kohn, Otto and Müller. We show the existence of weak solutions for the periodic case by means of the penalty approximation and commutator estimate.     

Inverse source problem for a diffusion equation involving the fractional spectral Laplacian

In this paper, we consider an inverse problem related to a fractional diffusion equation. The model problem is governed by a nonlinear partial differential equation involving the fractional spectral Laplacian. This study is focused on the reconstruction of an unknown source term from a partial internal measured data. The considered ill‐posed inverse problem is formulated as a minimization one. The existence, uniqueness, and stability of the solution are discussed. Some theoretical results are established. The numerical reconstruction of the unknown source term is investigated using an iterative process. The proposed method involves a denoising procedure at each iteration step and provides a sequence of source term approximations converging in norm to the actual solution of the minimization problem. Some numerical results are presented to show the efficiency and the accuracy of the proposed approach.     

Stability of a functional equation for square root spirals

By using an idea of Heuvers, Moak and Boursaw [1], we will prove a Hyers-Ulam-Rassias stability (or a general Hyers-Ulam stability) of the functional equation (1), which is closely related to the square root spiral.