Existence Of Sign-changing Solutions Research Articles

Sign Changing Solution of a Semilinear Schrödinger Equation with Constraint

The purpose of this paper is to study a semilinear Schrödinger equation with constraint in H1(RN), and prove the existence of sign changing solution. Under suitable conditions, we obtain a negative solution, a positive solution and a sign changing solution by using variational methods.

临界非线性分数阶 Schr?dinger 方程变号解的存在性

临界非线性分数阶 Schr?dinger 方程变号解的存在性

High energy sign-changing solutions for Coron's problem

We study the existence of sign changing solutions to the following problem(0.1){Δu+|u|p−1u=0inΩε;u=0on∂Ωε, where p=n+2n−2 is the critical Sobolev exponent and Ωε is a bounded smooth domain in Rn, n≥3, of the form Ωε=Ω\\B(0,ε). Here Ω is a smooth bounded domain containing the origin 0 and B(0,ε) denotes the ball centered at the origin with radius ε>0. We construct a new type of sign-changing solutions with high energy to problem (0.1), when the parameter ε is small enough.

Existence of sign-changing solutions for $p(x)$-Laplacian Kirchhoff type problem in $\mathbb{R}^N$

The $p(x)$-Laplacian Kirchhoff type equation involving the nonlocal term $b \int_{\mathbb{R}^N} (1/p(x)) \lvert\nabla u\rvert^{p(x)}dx$ is investigated. Based on the variational methods, deformation lemma and other technique of analysis, it is proved that the problem possesses one least energy sign-changing solution $u_b$ which has precisely two nodal domains. Moreover, the convergence property of $u_b$ as the parameter $b \searrow 0$ is also obtained.

The sign-changing solutions for a class of nonlocal elliptic problem in an annulus

In this paper, using the fixed point index method, we present a result on the existence of sign-changing solutions for a class of nonlocal elliptic problem in an annulus.

The existence of sign-changing solution for a class of quasilinear Schr\xf6dinger\u2013Poisson systems via perturbation method

This paper is concerned with the existence of a sign-changing solution to a class of quasilinear Schrödinger–Poisson systems. There are some technical difficulties in applying variational methods directly to the problem because the quasilinear term makes it impossible to find a suitable space in which the corresponding functional possesses both smoothness and compactness properties. In order to overcome the difficulties caused by nonlocal term and quasi-linear term, we shall apply the perturbation method by adding a 4-Laplacian operator to consider the perturbation problem. And then, by using the approximation technique, a sign-changing solution with precisely two nodal domains is derived.

On exact multiplicity for a second order equation with radiation boundary conditions

A second order ordinary differential equation with a superlinear term $g(x,u)$ under radiation boundary conditions is studied. Using a shooting argument, all the results obtained in the previous work \cite{AKR3} for a Painlevé II equation are extended. It is proved that the uniqueness or multiplicity of solutions depend on the interaction between the mapping $\frac {\partial g}{\partial u}(\cdot,0)$ and the first eigenvalue of the associated linear operator. Furthermore, two open problems posed in \cite{AKR3} regarding, on the one hand, the existence of sign-changing solutions and, on the other hand, exact multiplicity are solved.

Sign changing bubbling solutions for a critical Neumann problem in 3D

Let Ω be a smooth bounded domain in R3. We consider the problem Δu−λu+|u|4u=0inΩ,with zero Neumann boundary conditions, and investigate the existence of sign changing solutions, which concentrate simultaneously around k different points of Ω as λ approaches a special value λ0. We characterize the number λ0 in terms of the Green function of −Δ+λ. In particular, we construct a family of solutions exhibiting bubbling behavior at two different points of an annular domain as λ tends to λ0. In doing so, we also derive a representation formula of the Green function of −Δ with Neumann boundary conditions in the annulus in terms of zonal harmonics.

Multiplicity and concentration of solutions for a fractional Schrödinger equation via Nehari method and pseudo-index theory

We study a nonlinear fractional Schrödinger equation motivated by nonlocal quantum mechanics. Under suitable assumptions on the potentials, we explore the existence, concentration, convergence, and decay estimates of ground state solutions for this equation. Moreover, the multiplicity of solutions is constructed via pseudo-index theory, and the existence of sign-changing solutions is obtained via the Nehari method.

Multiple sign-changing solutions for nonlinear Schrödinger equations with potential well

We study the existence of sign-changing solutions for the following nonlinear Schrödinger equation where the potential has a potential well with bottom independent of the parameter . We show that as more and more sign-changing solutions of the nonlinear Schrödinger equation exist. The solutions lie in and are localized near the bottom of the potential well, but not near local minima of the potential.

Sign-changing solutions for fractional Schrödinger–Poisson system in ℝ3

ABSTRACTWe consider the existence of sign-changing solutions for the following fractional Schrödinger–Poisson system where , is a positive parameter, is a continuous potential function. Since multiple nonlocal terms are involving in the system, some new techniques are applied to prove the existence of a least energy sign-changing solution . Moreover, we prove that the energy of any sign-changing solution to this system is strictly larger than twice of the ground state energy. Furthermore, the asymptotic behavior of as is also analyzed.

Existence of sign-changing solutions for a class of p-Laplacian Kirchhoff-type equations

ABSTRACTIn this paper, we investigate the existence of least energy sign-changing solutions for the following p-Laplacian Kirchhoff type equation where are constants, are positive continuous functions which vanish at infinity, the nonlinearity f is a function with a subcritical growth. Using a minimization argument and the Nehari manifold method, we prove the existence of a least energy sign-changing solution to this problem.

Multiple and sign-changing solutions for discrete Robin boundary value problem with parameter dependence

Abstract In this paper, we study second-order nonlinear discrete Robin boundary value problem with parameter dependence. Applying invariant sets of descending flow and variational methods, we establish some new sufficient conditions on the existence of sign-changing solutions, positive solutions and negative solutions of the system when the parameter belongs to appropriate intervals. In addition, an example is given to illustrate our results.

Existence of Sign-Changing Solutions for a Nonlocal Problem of p-Kirchhoff Type

This work is devoted to study the existence of sign-changing solutions to nonlocal problems involving the p-Laplacian. Our approach is based on the variational method and quantitative deformation lemma.

Sign-Changing Solutions for Semilinear Elliptic Equations with Dependence on the Gradient Via the Nehari Method

In this paper, we consider a class of elliptic equation with dependence on the gradient. The existence of sign-changing solutions is established via the Nehari method and iterative technique.

Sign-Changing Solutions for Discrete Dirichlet Boundary Value Problem

Using invariant sets of descending flow and variational methods, we establish some sufficient conditions on the existence of sign-changing solutions, positive solutions and negative solutions for second-order nonlinear difference equations with Dirichlet boundary value problem. Some results in the literature are improved.

Existence of sign-changing solution with least energy for a class of Kirchhoff-type equation in ℝ N

Existence of sign-changing solution with least energy for a class of Kirchhoff-type equation in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>ℝ</mml:mi> <mml:mi>N</mml:mi> </mml:msup> </mml:mrow> </mml:math>

Study on the Existence of Sign-Changing Solutions of Case Theory Based a Class of Differential Equations Boundary-Value Problems

By using the fixed point theorem under the case structure, we study the existence of sign-changing solutions of A class of second-order differential equations three-point boundary-value problems, and a positive solution and a negative solution are obtained respectively, so as to popularize and improve some results that have been known.

Nodal solutions for a generalized quasilinear Schrödinger equation with critical exponents

This paper is concerned with constructing nodal radial solutions for generalized quasilinear Schrodinger equations in \begin{document} $\mathbb{R}^N$ \end{document} with critical growth which arise from plasma physics, fluid mechanics, as well as the self-channeling of a high-power ultashort laser in matter. We find the critical exponents for a generalized quasilinear Schrodinger equations and obtain the existence of sign-changing solution with k nodes for any given integer \begin{document} $k ≥ 0$ \end{document} .

Sign-Changing Solutions for Discrete Second-Order Periodic Boundary Value Problems

In this paper, we study the existence of sign-changing solutions and positive solutions for second-order nonlinear difference equations on a finite discrete segment with periodic boundary condition provided that the nonlinearity is asymptotically linear at infinity. The critical point theory and variational methods are employed to discuss this problem.