On the existence of solutions for nonhomogeneous Schrödinger-Poisson system
Wang et al. Boundary Value Problems
On the existence of solutions for nonhomogeneous Schrödinger-Poisson system
Lixia Wang 0 1 4
Shiwang Ma 0 1 3
Xiaoming Wang 0 1 2
Computer Science 0 1
Shangrao 0 1
0 334001 , China
1 Normal University , Shangrao, Jiangxi
2 School of Mathematics and Computer Science, Shangrao Normal University , Shangrao, Jiangxi 334001 , China
3 School of Mathematical Sciences and LPMC, Nankai University , Tianjin, 300071 , China
4 School of Sciences, Tianjin Chengjian University , Tianjin, 300384 , China
In this paper, we study the existence of solutions for the following nonhomogeneous Schrödinger-Poisson systems: - u + V(x)u + K(x)φ(x)u = f (x, u) + g(x), x ∈ R3,
Schrödinger-Poisson systems; sublinear nonlinearities; concave and
-
– φ = K(x)u2,
lim|x|→+∞ φ(x) = 0, x ∈ R3,
where f (x, u) is either sublinear in u as |u| →
∞ or a combination of concave and
convex terms. Under some suitable assumptions, the existence of solutions is proved
by using critical point theory.
MSC: 35B33; 35J65; 35Q55
convex nonlinearities; variational methods
1 Introduction
The following Schrödinger-Poisson system:
⎨⎧ – u + V (x)u + K (x)φ(x)u = f (x, u) + g(x), x ∈ R,
⎩ – φ = K (x)u,
lim|x|→+∞ φ(x) = ,
x ∈ R,
arises in several interesting physical contexts. It is well known that (.) has a strong
physical meaning since it appears in quantum mechanical models (see [, ]) and in
semiconductor theory (see [–]). From the point view of quantum mechanics, the system (.)
describes the mutual interactions of many particles []. Indeed, if the terms f (x, u) and g(x)
are replaced with , then problem (.) becomes the Schrödinger-Poisson system. In some
recent work (see [–]), different nonlinearities are added to the Schrödinger-Poisson
equation, giving rise to the so-called nonlinear Schrödinger-Poisson system. These
nonlinear terms have been traditionally used in the Schrödinger equation to model the
interaction among particles.
Many mathematicians have devoted their efforts to the study of (.) with various
nonlinearities f (x, u). We recall some of them as follows.
The case of g ≡ , that is, the homogeneous case, has been studied widely in [, , , –
]. In , Cerami and Vaira [] study system (.) in the case of f (x, u) = a(x)|u|p–u
with < p < and a(x) > . In order to recover the compactness of the embedding of
H(R) into the Lebesgue space Ls(R), s ∈ [, ), they establish a global compactness
lemma. They prove the existence of positive ground state and bound state solutions
without requiring any symmetry property on a(x) and K (x).
In , Sun et al. [] consider a more general case, that is, f (x, u) = a(x)f˜(u) where f˜
is asymptotically linear at infinity, i.e. f˜(s)/s → c as s → +∞ with a suitable constant c.
They establish a compactness lemma different from that in [] and prove the existence
of ground state solutions. In [], Ye and Tang study the existence and multiplicity of
solutions for homogeneous system of (.) when the potential V may change sign and the
nonlinear term f is superlinear or sublinear in u as |u| → ∞. For the Schrödinger-Poisson
system with sign-changing potential, see [, ].
Huang et al. [] study the case that f (x, u) is a combination of a superlinear term and
a linear term. More precisely, f (x, u) = k(x)|u|p–u + μh(x)u, where < p < and μ > ,
k ∈ C(R), k changes sign in R and lim|x|→+∞ k(x) = k∞ < . They prove the existence
of at least two positive solutions in the case that μ > μ and near μ, where μ is the
first eigenvalue of – + id in H(R) and with weight function h. In [, ], the authors
consider the critical case of p = ; in [] one studies the case of p = .
Sun et al. [] get infinitely many solutions for (.), where we have the nonlinearity
f (x, u) = k(x)|u|q–u – h(x)|u|l–u, < q < < l < ∞, i.e. the nonlinearity involving a
combination of concave and convex terms. For more results on the effect of concave and convex
terms of elliptic equations, see [, ] and the references therein.
Next, we consider the nonhomogeneous case of (.), that is, g ≡ . The existence of
radially symmetric solutions is obtained for above nonhomogeneous system in []. Chen
and Tang [] obtain two solutions for the nonhomogeneous system with f (x, u) satisfying
Amborosetti-Rabinowitz type condition and V being non-radially symmetric. In [, ],
the system with asymptotically linear and -linear nonlinearity is considered. For more
results on the nonhomogeneous case, see [, ] and the references therein.
Motivated by the work mentioned above, in the present paper, we first handle the
sublinear case, and hence make the following assumptions:
(V) V (x) ∈ C(R, R) satisfies infx∈R V (x) = a > ;
(V) for any M > , meas{x ∈ R : V (x) < M} < +∞, where meas denotes the Lebesgue
measures;
(K) K (x) ∈ L(R) ∪ L∞(R) and K (x) ≥ for all x ∈ R;
(F) there exist constants σ , γ ∈ (, ) and functions A (...truncated)