Existence of single peak solutions for a nonlinear Schrödinger system with coupled quadratic nonlinearity

Abstract

Abstract We are concerned with the following Schrödinger system with coupled quadratic nonlinearity − ε 2 Δ v + P ( x ) v = μ v w , x ∈ R N , − ε 2 Δ w + Q ( x ) w = μ 2 v 2 + γ w 2 , x ∈ R N , v > 0 , w > 0 , v , w ∈ H 1 R N , $$\begin{equation}\left\{\begin{array}{ll}-\varepsilon^{2} \Delta v+P(x) v=\mu v w, & x \in \mathbb{R}^{N}, \\ -\varepsilon^{2} \Delta w+Q(x) w=\frac{\mu}{2} v^{2}+\gamma w^{2}, & x \in \mathbb{R}^{N}, \\ v>0, \quad w>0, & v, w \in H^{1}\left(\mathbb{R}^{N}\right),\end{array}\right. \end{equation}$$ which arises from second-harmonic generation in quadratic media. Here ε > 0 is a small parameter, 2 ≤ N < 6, μ > 0 and μ > γ, P(x), Q(x) are positive function potentials. By applying reduction method, we prove that if x 0 is a non-degenerate critical point of Δ(P + Q) on some closed N − 1 dimensional hypersurface, then the system above has a single peak solution (vε , wε ) concentrating at x 0 for ε small enough.