We are concerned with the following modified nonlinear Schrödinger system:-Δu+u-(1/2)uΔ(u2)=(2α/(α+β))|u|α-2|v|βu, x∈Ω, -Δv+v-(1/2)vΔ(v2)=(2β/(α+β))|u|α|v|β-2v, x∈Ω, u=0, v=0, x∈∂Ω, whereα>2, β>2, α+β<2·2*, 2*=2N/(N-2)is the critical Sobolev exponent, andΩ⊂ℝN (N≥3)is a bounded smooth domain. By using the perturbation method, we establish the existence of both positive and negative solutions for this system.
In this paper, we study the following autonomous nonlinear Schrödinger system (discussed in the paper), where
λ
,
μ
, and
ν
are positive parameters;
2
∗
=
2
N
/
N
−
2
is the critical Sobolev exponent; and
f
satisfies general subcritical growth conditions. With the help of the Pohožaev manifold, a ground state solution is obtained.
Abstract
We consider the following coupled fractional Schrödinger system: $$ \textstyle\begin{cases} (-\Delta )^{s}u+\lambda _{1}u=\mu _{1} \vert u \vert ^{2p-2}u+ \beta \vert v \vert ^{p} \vert u \vert ^{p-2}u, \\ (-\Delta )^{s}v+\lambda _{2}v=\mu _{2} \vert v \vert ^{2p-2}v+\beta \vert u \vert ^{p} \vert v \vert ^{p-2}v \end{cases}\displaystyle \quad \text{in } {\mathbb{R}^{N}}, $$
{
(
−
Δ
)
s
u
+
λ
1
u
=
μ
1
|
u
|
2
p
−
2
u
+
β
|
v
|
p
|
u
|
p
−
2
u
,
(
−
Δ
)
s
v
+
λ
2
v
=
μ
2
|
v
|
2
p
−
2
v
+
β
|
u
|
p
|
v
|
p
−
2
v
in
R
N
,
with $0< s<1$
0
<
s
<
1
, $2s< N\le 4s$
2
s
<
N
≤
4
s
and $1+\frac{2s}{N}< p<\frac{N}{N-2s}$
1
+
2
s
N
<
p
<
N
N
−
2
s
, under the following constraint: $$ \int _{\mathbb{R}^{N}} \vert u \vert ^{2}\,dx=a_{1}^{2} \quad \text{and}\quad \int _{ \mathbb{R}^{N}} \vert v \vert ^{2}\,dx=a_{2}^{2}. $$
∫
R
N
|
u
|
2
d
x
=
a
1
2
and
∫
R
N
|
v
|
2
d
x
=
a
2
2
.
Assuming that the parameters $\mu _{1}$
μ
1
, $\mu _{2}$
μ
2
, $a_{1}$
a
1
, $a_{2}$
a
2
are fixed quantities, we prove the existence of normalized solution for different ranges of the coupling parameter $\beta >0$
β
>
0
.
Existence of Solutions for a Modified Nonlinear Schrödinger System
