Existence of least energy sign-changing solution for the nonlinear Schrödinger system with two types of nonlocal terms
Li and Hao Boundary Value Problems (2016) 2016:220
DOI 10.1186/s13661-016-0728-y
RESEARCH
Open Access
Existence of least energy sign-changing
solution for the nonlinear Schrödinger system
with two types of nonlocal terms
Yuhua Li and Yawen Hao*
*
Correspondence:
School of Mathematical Sciences,
Shanxi University, Taiyuan, Shanxi
030006, P.R. China
Abstract
In the paper, we are concerned with the system of Kirchhoff-Schrödinger-Poisson
system under certain assumptions on V1 , V2 , K and f . We are interested in the
existence of least energy sign-changing
solutions to the system on RN . Because two
2
kinds of nonlocal terms φu and RN |∇u| are involved in the system, the methods are
different
from the Kirchhoff or the Schrödinger-Poisson system. The two nonlocal
terms RN |∇u|2 and φu make that the functional J(u+ + u– ) = J(u+ ) + J(u– ). Moreover,
the nonlocal term φu does not have the convergence property because of the
assumption V2 . In addition, the convergence of these two nonlocal terms are
different. In the present paper, we unify the increasing property conditions on
sign-changing solution in previous papers. We construct a new homotopy operator
and then weaken the assumption that f is C 1 to that of f being only continuous. We
prove that the system has a sign-changing solution via a constraint variational
method combining with Brouwer’s degree theory.
MSC: 47J30; 34B15
Keywords: Schrödinger-Poisson system; Kirchhoff; sign-changing solution;
Brouwer’s degree; constraint variational
1 Introduction
In this paper, we consider the nonlinear Kirchhoff-Schrödinger-Poisson system
–( + b RN |∇u| )u + V (x)u + V (x)φu = K(x)f (u), x ∈ RN ,
x ∈ RN ,
–φ = V (x)u ,
(.)
for ≤ N ≤ . System (.) with b = has been introduced while looking for the existence of standing waves for the Schrödinger equation acting with an electrostatic field.
We refer to [] and the references therein for more details as regards the physics aspects.
This problem is called the Schrödinger-Poisson system. If V = , system (.) is the Kirchhoff equation, which is the stationary problem associated to the time-dependent problem,
which models small vertical vibrations of an elastic string [].
In recent years, Schrödinger equations with nonlocal terms have attracted much attention. There are several nonlocal Schrödinger equations such as the Kirchhoff and the
Schrödinger-Poisson system, and fractional order differential equations. Because of the
© The Author(s) 2016. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License
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indicate if changes were made.
Li and Hao Boundary Value Problems (2016) 2016:220
Page 2 of 13
nonlocal effect, the convergence and energy functional in a variational reduction are different from local ones. There have been papers studying fractional order differential equations; see [–]. When b = , (.) is the Schrödinger-Poisson system. While V = , (.)
is the famous Kirchhoff equation. There has been interest in studying problem (.) with
b = or V = under various assumptions on K and V or V . For b = , K(x) = , the existence of positive solutions, ground state solutions, and multiple solutions for (.) has been
massively addressed in the mathematical literature [–]. When V = , there have been
many papers to study the existence of positive solution and infinitely many solutions; see
[–]. Especially, [] studied the existence of nonnegative solutions to critical Kirchhoff problem. In addition, the systems with two types of nonlocal terms has been studied;
see [, ]. In [], fractional p-Kirchhoff equations was considered. The sign-changing
solutions to Kirchhoff problems are also considered. For example, in [–], the authors proved the existence of sign-changing solutions for b = , V (x) = K(x) = . When
V = , [] studied the existence of the least energy sign-changing solution to (.) with
V (x) = K(x) = . However, there have appeared few papers to study the case of b = and
V = ; see []. In [], the authors considered infinitely many solutions to (.).
Motivated by the above work, the main aim of this paper is to study the existence of a
sign-change of (.) when the potentials V and K decay to zero as |x| → ∞. Precisely, we
suppose:
(V) V : RN → R is a smooth function and there exist a, c > and τ ∈ (, ) such that
a
≤ V (x) ≤ c,
+ |x|τ
∀x ∈ RN ,
and V ∈ L∞ (RN ) ∪ L(–N)/N (RN ) is nonnegative;
(K) K : RN → R is a smooth function and there exist ξ > τ , d > such that
< K(x) ≤
d
,
+ |x|ξ
∀x ∈ RN .
As regards the function f , we assume f ∈ C(R, R) and we have the following hypotheses:
(f ) f (t) = o(|t|) as t → ;
u
(f ) limt→∞ ft(t)
= +∞, where F(t) = f (t) dt;
(x)t
(f ) there exists a θ ∈ [, ) such that K(x)f (t)–θV
is nondecreasing on (–∞, ) and (, ∞),
|t|
respectively;
(f ) |f (t)| ≤ C(|t| + |t|p ), p > .
The conditions (f )-(f ) with θ = are usual for the Kirchhoff equation or the Schrödinger-Poisson system; see [, ]. The condition (f ) is equivalent to the one in [,
]
f (tτ )
|
] sign( – t) + θ V (x) |–t
≥ , t > , τ = .
(V ) K(x)[ fτ(τ ) – (tτ
)
(tτ )
In fact, we have the next remark.
Remark . Condition (f ) is equivalent to (V ).
Li and Hao Boundary Value Problems (2016) 2016:220
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Proof We only prove the case of t > . For t ∈ (, ], tτ ≤ τ , then
| – t |
f (τ ) f (tτ )
sign(
–
t)
+
θ
V
(x)
–
K(x)
τ
(tτ )
(tτ )
– t
f (τ ) f (tτ )
+
θ
V
(x)
= K(x)
–
τ
(tτ )
(tτ )
=
K(x)f (τ ) – θ V (x)τ K(x)f (tτ ) – θ V (x)tτ
–
.
τ
(tτ )
If t > , then by tτ ≥ τ , we have
| – t |
f (τ ) f (tτ )
sign(
–
t)
+
θ
V
(x)
+
K(x)
τ
(tτ )
(tτ )
– t
f (τ ) f (tτ )
– θ V (x)
= –K(x)
+
τ
(tτ )
(tτ )
=–
K(x)f (τ ) – θ V (x)τ K(x)f (tτ ) – θ V (x)tτ
+
.
τ
(tτ )
Therefore, the conclusion holds.
In addition, we must notice that the condition on V cannot ensure the nonlocal term
φun → φu if un u in LN/(N–) . The least energy sign-changing solution to the KirchhoffSchrödinger-Poisson system has not been studied under the conditions (V) and (K). We
unify the conditions θ = and θ > and generalize to the problem (.). Throughout this
q
paper, we consider the weighted space LK of measurable u : RN → R such that
|u|q,K :=
N
R
q
q
K(x)u(x) dx < ∞.
p
If K is a constant, LK is the usual space Lp and the norm is denoted by | · |p . The weighted
Sobolev spaces E defined by setting
N
V (x)u dx < +∞ ,
E = u ∈ D, RN :
R
make E a Hilbert space with the inner product and the norm
N
(u, v) =
R
∇u · ∇v + V (x)uv dx,
u = (u, u) .
Then we have the following embedding theorem, which (...truncated)
