Normalized solutions for nonlinear Schrödinger systems with critical exponents
In this paper, we consider the following nonlocal Schrödinger system−a+b∫R3|∇u1|2dxΔu1=λ1u1+μ1|u1|p1−2u1+βr1|u1|r1−2u1|u2|r2,−a+b∫R3|∇u2|2dxΔu2=λ2u2+μ2|u2|p2−2u2+βr2|u1|r1|u2|r2−2u2,∫R3|u1|2dx=c1,∫R3|u2|2dx=c2. $$\begin{cases}-\left(a+b{\int }_{{\mathbb{R}}^{3}}\vert \nabla {u}_{1}{\vert }^{2}\mathr...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2025-02-01
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| Series: | Advanced Nonlinear Studies |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/ans-2023-0175 |
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| Summary: | In this paper, we consider the following nonlocal Schrödinger system−a+b∫R3|∇u1|2dxΔu1=λ1u1+μ1|u1|p1−2u1+βr1|u1|r1−2u1|u2|r2,−a+b∫R3|∇u2|2dxΔu2=λ2u2+μ2|u2|p2−2u2+βr2|u1|r1|u2|r2−2u2,∫R3|u1|2dx=c1,∫R3|u2|2dx=c2.
$$\begin{cases}-\left(a+b{\int }_{{\mathbb{R}}^{3}}\vert \nabla {u}_{1}{\vert }^{2}\mathrm{d}x\right){\Delta}{u}_{1}={\lambda }_{1}{u}_{1}+{\mu }_{1}\vert {u}_{1}{\vert }^{{p}_{1}-2}{u}_{1}+\beta {r}_{1}\vert {u}_{1}{\vert }^{{r}_{1}-2}{u}_{1}\vert {u}_{2}{\vert }^{{r}_{2}},\quad \hfill \\ -\left(a+b{\int }_{{\mathbb{R}}^{3}}\vert \nabla {u}_{2}{\vert }^{2}\mathrm{d}x\right){\Delta}{u}_{2}={\lambda }_{2}{u}_{2}+{\mu }_{2}\vert {u}_{2}{\vert }^{{p}_{2}-2}{u}_{2}+\beta {r}_{2}\vert {u}_{1}{\vert }^{{r}_{1}}\vert {u}_{2}{\vert }^{{r}_{2}-2}{u}_{2},\quad \hfill \\ {\int }_{{\mathbb{R}}^{3}}\vert {u}_{1}{\vert }^{2}\mathrm{d}x={c}_{1}, {\int }_{{\mathbb{R}}^{3}}\vert {u}_{2}{\vert }^{2}\mathrm{d}x={c}_{2}.\quad \hfill \end{cases}$$
In the case of b = 0, 2 < p
1, p
2 < 2*, the existence of a Mountain Pass solution and a global minimizer to the above problem was obtained (see T. Bartsch, L. Jeanjean [Proc. R. Soc. Edinburgh, Sect. A, 148 (2018), pp. 225–242.]). However, in the case of b > 0, p
1 = p
2 = 2*, 2<r1+r2<p̂
$2{< }{r}_{1}+{r}_{2}{< }\hat{p}$
or p̄≤r1+r2<2*
$\bar{p}\le {r}_{1}+{r}_{2}{< }{2}^{{\ast}}$
, similar existence results to the above problem is still unknown, here 2* is Sobolev critical exponent and p̂=103
$\hat{p}=\frac{10}{3}$
, p̄=143
$\bar{p}=\frac{14}{3}$
are the L
2-critical exponent for the system when b = 0 and b > 0 respectively. Here we focus on these unknown case. We investigate the existence of positive normalized solutions under different assumptions on β > 0. Figure 1, Tables 1 and 2 will illustrate our main results and the relationship between our work and some related works in the literature. |
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| ISSN: | 2169-0375 |