Bound states of nonlinear Schrödinger equations with magnetic fields

We study bound states of the following nonlinear Schr?dinger equation in the presence of a magnetic field: $$ \left\{\begin{array}{l} \left(-i\hbar\nabla+A(x)\right)^2u+V(x)u=g(x,|u|)u \\ |u|\in H^1(\mathbb{R}^N) \end{array} \right. $$ where ${A: \mathbb{R}^N\to\mathbb{R}^N, V: \mathbb{R}^N\to\mathbb{R}}$ and ${g: \mathbb{R}^N\times\mathbb{R}\to [0,\infty)}$ . We prove that if V is bounded below with the set ${\{x\in\mathbb{R}^N: V(x) < b\}\not=\emptyset}$ having finite measure for some b?>?0, inf V???0, and g satisfies some growth conditions, then for any integer m when ${\hbar >0 }$ is sufficiently small the problem has m geometrically different solutions.     

Gain of regularity for semilinear Schrödinger equations

We discuss local existence and gain of regularity for semilinear Schr?dinger equations which generally cause loss of derivatives. We prove our results by advanced energy estimates. More precisely, block diagonalization and Doi's transformation, together with symbol smoothing for pseudodifferential operators with nonsmooth coefficients, apply to systems of Schr?dinger-type equations. In particular, the sharp G?rding inequality for pseudodifferential operators whose coefficients are twice continuously differentiable, plays a crucial role in our proof. Received: 14 December 1998     

Bound states for a coupled Schrödinger system

We consider the existence of bound states for the coupled elliptic system

Ground states for quasilinear Schrödinger equations with critical growth

For a class of quasilinear Schrödinger equations with critical exponent we establish the existence of both one-sign and nodal ground states of soliton type solutions by the Nehari method. The method is to analyze the behavior of solutions for subcritical problems from our earlier work (Liu et al. Commun Partial Differ Equ 29:879–901, 2004) and to pass limit as the exponent approaches to the critical exponent.     

Bound states to critical quasilinear Schr?dinger equations

In this paper, we consider the critical quasilinear Schr?dinger equations of the form

Standing waves for discrete nonlinear Schrödinger equations: sign-changing nonlinearities

We consider the discrete nonlinear Schrödinger equation with infinitely growing potential and sign-changing power nonlinearity. Making use of critical point theory, we prove an existence and multiplicity result for standing wave solutions.     

Existence and concentration behavior of sign-changing solutions for quasilinear Schrödinger equations

We consider the quasilinear Schrödinger equations of the form ?ε²Δu + V(x)u ? ε²Δ(u²)u = g(u), x∈ R^N, where ε > 0 is a small parameter, the nonlinearity g(u) ∈ C¹(R) is an odd function with subcritical growth and V(x) is a positive Hölder continuous function which is bounded from below, away from zero, and inf_ΛV(x) < inf?_ΛV(x) for some open bounded subset Λ of R^N. 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⁺.     

Multiple solutions to logarithmic Schrödinger equations with periodic potential

We study a class of logarithmic Schrödinger equations with periodic potential which come from physically relevant situations and obtain the existence of infinitely many geometrically distinct solutions.     

Bound states of schrödinger equations with electromagnetic fields and vanishing potentials

We study the bound states to nonlinear Schrödinger equations with electro-magnetic fields $\text{i}h\frac{?\psi }{?t}={(\frac{h}{i}?-A(x))}^2\psi +V(x)\psi -K(x){|}^{p-1}\psi =0,\text{on}{?}_{+}\times {?}^N$. Let $G(x)={[V(x)]}^{\frac{p+1}{p-1}-\frac{N}{2}}{[K(x)]}^{-\frac{2}{p-1}}$ and suppose that G(x) has k local minimum points. For h > 0 small, we find multi-bump bound states ψh(x,t) = _e^?lEt/^h_Uh(χ) with Uh concentrating at the local minimum points of G(x) simultaneously as h → 0. The potentials V(x) and K(x) are allowed to be either compactly supported or unbounded at infinity.