Boosted Ground States for a Pseudo-Relativistic Schrödinger Equation with a double power nonlinearity

Abstract

In this paper, we investigate the existence and limit behaviours of travelling solitary waves of the form $ψ(t,x)=e^{iλt}\varphi\left(x-vt\right)$ to the nonlinear pseudo-relativistic Schrödinger equation \[ i\partial_t ψ=(\sqrt{-Δ+m^2})ψ- |ψ|^{\frac{2}{N}}ψ-μ|ψ|^{q}ψ~~\text{ on }\mathbb{R}^N, \] for $m\ge 0$ and $|v|<1$. To this end, we introduce and analyse an associated constrained variational problem, whose minimizers are termed boosted ground states and the parameter $λ$ is obtained as a Lagrangian multiplier. We first provide a complete classification for the existence and nonexistence of such boosted ground states. Based on this classification, we then study several limiting profiles, for which the exact blow-up rate is also established.

Theorems & Definitions (53)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Remark 1.7
• Lemma 2.1
• Lemma 2.2
• Remark 2.3