Existence of Solutions for a Quasilinear Schrödinger Equation with Potential Vanishing

Acta Mathematicae Applicatae Sinica, English Series - We study the following quasilinear Schrödinger equation $$ - \Delta u + V(x)u - \Delta ({u^2})u = K(x)g(u),\,\,\,\,\,\,\,\,x \in...     

On the Chern-Simons-Schrödinger Equation with Critical Exponential Growth

Acta Mathematica Sinica, English Series - This paper is concerned with the following Chern-Simons-Schrödinger equation $$- Delta u + V(left| x right|)u + left({int_{left| x...     

On the Existence of Ground State Solutions to a Quasilinear Schr?dinger Equation involving p-Laplacian

We consider the following quasilinear Schr?dinger equation involving p-Laplacian ■ in R^N,where N > p > 1, η ≥p/(2(p-1)), p < q < 2ηp*(μ), p*(s) =(p(N-s))/(N-p), and λ, μ, ν are parameters with λ > 0,μ, ν ∈ [0, p). Via the Mountain Pass Theorem and the Concentration Compactness Principle, we establish the existence of nontrivial ground state solutions for the above problem.     

Ground State Solutions for a Quasilinear Schrödinger Equation

In this paper, we study the following quasilinear Schrödinger equation of the form

$$\begin{aligned} -\Delta u+V(x)u-\Delta (u^{2})u= g(x,u),~~~ x\in \mathbb {R}^N \end{aligned}$$

where V and g are 1-periodic in \(x_{1},\ldots ,x_{N}\), and g is a superlinear but subcritical growth as \(|u|\rightarrow \infty \). We develop a more direct and simpler approach to prove the existence of ground state solutions.

     

Sharp Condition for Global Existence of Supercritical Nonlinear Schr?dinger Equation with a Partial Confinement

We study the L²-supercritical nonlinear Schr?dinger equation(NLS) with a partial confinement,which is the limit case of the cigar-shaped model in Bose-Einstein condensate(BEC). By constructing a cross constrained variational problem and establishing the invariant manifolds of the evolution fow, we show a sharp condition for global existence.     

Existence and Multiplicity of Solutions for a Class of Quasilinear Elliptic Equations with Exponential Growth北大核心CSCD

本文研究一类具有次临界多项式增长或次临界指数型(临界指数型)增长的(p,2)-拉普拉斯方程一个正解及无穷多非平凡解的存在性,运用山路定理及喷泉定理,得到了拉普拉斯方程非平凡解的一些存在性结果.     

Semiclassical Asymptotics for a Difference Schrödinger Equation with Two Coalescent Turning Points

Mathematical Notes -     

Effective Approximation for the Semiclassical Schrödinger Equation

The computation of the semiclassical Schrödinger equation presents major challenges because of the presence of a small parameter. Assuming periodic boundary conditions, the standard approach consists of semi-discretisation with a spectral method, followed by an exponential splitting. In this paper we sketch an alternative strategy. Our analysis commences with the investigation of the free Lie algebra generated by differentiation and by multiplication with the interaction potential: it turns out that this algebra possesses a structure which renders it amenable to a very effective form of asymptotic splitting: exponential splitting where consecutive terms are scaled by increasing powers of the small parameter. This leads to methods which attain high spatial and temporal accuracy and whose cost scales as \({\mathcal {O}}\!\left( M\log M\right) \) , where \(M\) is the number of degrees of freedom in the discretisation.     

Multi-bump Bound States of Schrödinger Equations with a Critical Frequency

We consider existence and qualitative properties of standing wave solutions $\Psi(x,t) = e^{-iEt/h}u(x)We consider existence and qualitative properties of standing wave solutions to the nonlinear Schr?dinger equation with E being a critical frequency in the sense that inf . We verify that if the zero set of W − E has several isolated points x _i () near which W − E is almost exponentially flat with approximately the same behavior, then for h > 0 small enough, there exists, for any integer k, , a standing wave solution which concentrates simultaneously on , where is any given subset of . This generalizes the result of Byeon and Wang in 3 (Arch Rat Mech Anal 165: 295–316, 2002).Supported by the Alexander von Humboldt foundation and NSFC(No:10571069).     

Infinitely Many Sign-changing Solutions for a Schrdinger Equation in R~N

We consider the existence of infinitely many sign-changing solutions for the nonlinear time-independent schrodinger equations of the form where Vλ(x) =λa(x) 1. This problem originates from various problems in physics and mathematical physics. In constructive field theory, (1.1) is called a nonlinear Euclidean scalar field equation. In chemical dynamics, a solution of (1.1) is a stationary state of the reaction diffusion equation     

Existence and Asymptotic Behavior of Ground State Solutions for Quasilinear Schr?dinger Equations with Unbounded Potential

The authors study the existence of standing wave solutions for the quasilinear Schr?dinger equation with the critical exponent and singular coefficients. By applying the mountain pass theorem and the concentration compactness principle, they get a ground state solution. Moreover, the asymptotic behavior of the ground state solution is also obtained.     

Soliton Solutions for a Singular Schrödinger Equation with Any Growth Exponents

This paper is concerned with a kind of quasilinear Schrödinger equation with combined nonlinearities, a convex term with any growth and a singular term, in a bounded smooth domain. Multiplicity results are obtained by critical point theory together with truncation arguments and the method of upper and lower solutions.     

Semiclassical symmetric Schr?dinger equations: Existence of solutions concentrating simultaneously on several spheres

We study the radially symmetric Schr?dinger equation

On the Growth Properties of Solutions for a Generalized Bi-Axially Symmetric Schr?dinger Equation

In this paper, we have considered the generalized bi-axially symmetric Schr?dinger equation ?~2φ/?x~2+?~2φ/?y~2+(2ν/x)?φ/?x+(2μ/y)?φ/?y+ {K~2- V(r)}φ = 0,where μ, ν≥ 0, and r V(r) is an entire function of r = +(x~2+ y~2)~(1/2) corresponding to a scattering potential V(r). Growth parameters of entire function solutions in terms of their expansion coefficients, which are analogous to the formulas for order and type occurring in classical function theory, have been obtained. Our results are applicable for the scattering of particles in quantum mechanics.     

Single Peak Solutions for a Schr?dinger Equation with Variable Exponent

We study the following Schr?dinger equation with variable exponent■ where■ . Under certain assumptions on a vector field related to a(x), we use the Lyapunov–Schmidt reduction to show the existence of single peak solutions to the above problem. We also obtain local uniqueness and exact multiplicity results for this problem by the Pohozaev type identity.     

Existence of Positive Solutions for a Class of Critical Fractional Schrödinger Equations with Potential Vanishing at Infinity

In this paper, we investigate the following critical fractional Schrödinger equation

$$\begin{aligned} (-\Delta )^su+V(x)u=|u|^{2_s^*-2}u+\lambda K(x)f(u), \ x \in \mathbb {R}^N, \end{aligned}$$

where \(\lambda >0\), \(0<s<1\), \((-\Delta )^s\) denotes the fractional Laplacian of order s, \(V, \ K\) are nonnegative continuous functions satisfying some conditions and f is a continuous function, \(N>2s\) and \(2_s^*=\frac{2N}{N-2s}\). We prove that the equation has a positive solution for large \(\lambda \) by the variational method.

     

Existence of Multi-bump Solutions for a Class of Quasilinear Schrödinger Equations in $${\mathbb{R}^{N}}$$ Involving Critical Growth

In this paper we prove the existence of multi-bump solutions for a class of quasilinear Schrödinger equations of the form \({-\Delta{u} + (\lambda{V} (x) + Z(x))u - \Delta(u^{2})u = \beta{h}(u) + u^{22*-1}}\) in the whole space, where h is a continuous function, \({V, Z : \mathbb{R}^{N} \rightarrow \mathbb{R}}\) are continuous functions. We assume that V(x) is nonnegative and has a potential well \({\Omega : = {\rm int} V^{-1}(0)}\) consisting of k components \({\Omega_{1}, \ldots , \Omega{k}}\) such that the interior of Ω _i is not empty and \({\partial\Omega_{i}}\) is smooth. By using a change of variables, the quasilinear equations are reduced to a semilinear one, whose associated functionals are well defined in the usual Sobolev space and satisfy the geometric conditions of the mountain pass theorem for suitable assumptions. We show that for any given non-empty subset. \({\Gamma \subset \{1, \ldots ,k\}}\), a bump solution is trapped in a neighborhood of \({\cup_{{j}\in\Gamma}\Omega_{j}}\) for\({\lambda > 0}\) large enough.