Existence and concentration behavior of sign-changing solutions for quasilinear Schr?dinger equations equations

We consider the quasilinear Schrdinger equations of the form-ε~2?u + V(x)u- ε~2?(u2)u = g(u), x ∈ R~N,where ε 0 is a small parameter, the nonlinearity g(u) ∈ C~1(R) is an odd function with subcritical growth and V(x) is a positive Hlder continuous function which is bounded from below, away from zero, and infΛV(x) inf ?ΛV(x) for some open bounded subset Λ of RN. We prove that there is an ε0 0 such that for all ε∈(0, ε0],the above mentioned problem possesses a sign-changing solution uε which exhibits concentration profile around the local minimum point of V(x) as ε→ 0~+.     

Existence and concentration of positive solutions for coupled Schrödinger equations

In this paper, we study the existence and concentration of positive solution of a class of coupled Schrödinger equations. We admit that the potentials may not be non-negative and suppose that the intersection of the sets has positive Lebesgue measure. By studying the modified functional of the associated functional carefully, we establish the existence of positive least energy solutions for the coupled Schrödinger system. Moreover, we prove the concentration phenomenon of the positive solution when the parameter goes to infinity.     

Existence of ground state solutions for quasilinear Schrödinger equations with super-quadratic condition

In this paper, we study the following quasilinear Schrödinger equation $-\Delta u+u-\Delta \left(u^2\right)u=h\left(u\right),x\in {\mathbb{R}}^N,$where $N\ge 3$, $2^1=\frac{2N}{N-2}$, $h$ is a continuous function. By using a change of variable, we obtain the existence of ground state solutions. Unlike the condition ${lim}_{\left|u\right|\to \infty }\frac{{\int }_0^{u}h\left(s\right)ds}{{\left|u\right|}^4}=\infty$, we only need to assume that ${lim}_{\left|u\right|\to \infty }\frac{{\int }_0^{u}h\left(s\right)ds}{{\left|u\right|}^2}=\infty$.     

Soliton solutions for generalized quasilinear Schrödinger equations

By introducing a new variable replacement, we study the existence of nontrivial solutions for generalized quasilinear Schrödinger equations which appear from plasma physics, as well as high-power ultrashort laser in matter.     

Infinitely many solutions for a class of quasilinear Schrödinger equations involving sign-changing weight functions

Using a change of variables and the constrained critical point theory, we first prove the existence and multiplicity of solutions for a class of quasilinear Schrödinger equations. Next, we consider a quasilinear equation related to the superfluid film in plasma physics with a sign-changing weight function. Using a new natural constraint, we establish the existence of infinitely many solutions for the equation.     

Multiple positive solutions for a quasilinear system of Schrödinger equations

We consider the quasilinear system

Normalized solutions and mass concentration for supercritical nonlinear Schrödinger equations

Science China Mathematics - In this paper, we deal with the existence and concentration of normalized solutions to the supercritical nonlinear Schrödinger equation $$left{ {matrix{ { -...     

Existence of nontrivial solutions and high energy solutions for Schrödinger–Kirchhoff-type equations in RN

In the present paper, the following Schrödinger–Kirchhoff-type problem: (1.1)$?(a+b{\int }_{R^N}{\left|?u\right|}^{2}dx)\Delta u+V\left(x\right)u=f(x,u)\text{,}\text{in}{R}^N$ is studied and four new existence results for nontrivial solutions and a sequence of high energy solutions for problem (1.1) are obtained by using a symmetric Mountain Pass Theorem.     

Existence of breathers for discrete nonlinear Schrödinger equations

In this paper, we obtain a new sufficient condition on the existence of breathers for the discrete nonlinear Schrödinger equations by using critical point theory in combination with periodic approximations. The classical Ambrosetti–Rabinowitz superlinear condition is improved.