Multi-spikes solutions for a system of coupled elliptic equations with quadratic nonlinearity
Abstract: This paper is devoted to study the following systems of coupled elliptic equations with quadratic nonlinearity −ε 2 ∆v + P (x)v = µvw, x ∈ R N , −ε 2 ∆w + Q(x)w = µ 2 v 2 + γw 2 , x ∈ R N , which arises from second-harmonic generation in quadratic optical media. We assume that the potential functions P (x) and Q(x) are positive functions and have a strict local maxima at x 0. Applying the finite dimensional reduction method, for any integer 1 ≤ k ≤ N + 1, we prove the existence of positive solutions which have… Show more
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Cited by 3 publications
References 31 publications
A new type of solutions for a nonlinear Schrödinger system with $\chi ^{(2)}$ nonlinearities
We are concerned with the question of constructing a new type of solution to the problem with $\chi ^{(2)}$ χ ( 2 ) nonlinearities { − Δ u + P ( x ) u = α u v , in R N , − Δ v + Q ( x ) v = α 2 u 2 + β v 2 , in R N , where $P(x)=P(|x|)$ P ( x ) = P ( | x | ) and $Q(x)=Q(|x|)$ Q ( x ) = Q ( | x | ) are positive bounded radial potentials, $3\leq N<6$ 3 ≤ N < 6 , $\alpha >0$ α > 0 and $\alpha >\beta $ α > β . Assuming that the potentials $P(x)$ P ( x ) and $Q(x)$ Q ( x ) satisfy certain conditions, the existence of a new type of solutions is proved.
A new type of solutions for a nonlinear Schrödinger system with $\chi ^{(2)}$ nonlinearities
We are concerned with the question of constructing a new type of solution to the problem with $\chi ^{(2)}$ χ ( 2 ) nonlinearities { − Δ u + P ( x ) u = α u v , in R N , − Δ v + Q ( x ) v = α 2 u 2 + β v 2 , in R N , where $P(x)=P(|x|)$ P ( x ) = P ( | x | ) and $Q(x)=Q(|x|)$ Q ( x ) = Q ( | x | ) are positive bounded radial potentials, $3\leq N<6$ 3 ≤ N < 6 , $\alpha >0$ α > 0 and $\alpha >\beta $ α > β . Assuming that the potentials $P(x)$ P ( x ) and $Q(x)$ Q ( x ) satisfy certain conditions, the existence of a new type of solutions is proved.
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.
Existence of normalized peak solutions for a coupled nonlinear Schrödinger system
In this article, we study the following nonlinear Schrödinger system − Δ u 1 + V 1 ( x ) u 1 = α u 1 u 2 + μ u 1 , x ∈ R 4 , − Δ u 2 + V 2 ( x ) u 2 = α 2 u 1 2 + β u 2 2 + μ u 2 , x ∈ R 4 , \left\{\begin{array}{ll}-\Delta {u}_{1}+{V}_{1}\left(x){u}_{1}=\alpha {u}_{1}{u}_{2}+\mu {u}_{1},& x\in {{\mathbb{R}}}^{4},\\ -\Delta {u}_{2}+{V}_{2}\left(x){u}_{2}=\frac{\alpha }{2}{u}_{1}^{2}+\beta {u}_{2}^{2}+\mu {u}_{2},& x\in {{\mathbb{R}}}^{4},\end{array}\right. with the constraint ∫ R 4 ( u 1 2 + u 2 2 ) d x = 1 {\int }_{{{\mathbb{R}}}^{4}}\left({u}_{1}^{2}+{u}_{2}^{2}){\rm{d}}x=1 , where α > 0 \alpha \gt 0 and α > β \alpha \gt \beta , μ ∈ R \mu \in {\mathbb{R}} , V 1 ( x ) {V}_{1}\left(x) , and V 2 ( x ) {V}_{2}\left(x) are bounded functions. Under some mild assumptions on V 1 ( x ) {V}_{1}\left(x) and V 2 ( x ) {V}_{2}\left(x) , we prove the existence of normalized peak solutions by using the finite dimensional reduction method, combined with the local Pohozaev identities. Because of the interspecies interaction between the components, we aim to obtain some new technical estimates.
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