On the existence of solutions for nonhomogeneous Schrödinger-Poisson system

Boundary Value Problems, Apr 2016

In this paper, we study the existence of solutions for the following nonhomogeneous Schrödinger-Poisson systems: ( ∗ ) { − Δ u + V ( x ) u + K ( x ) ϕ ( x ) u = f ( x , u ) + g ( x ) , x ∈ R 3 , − Δ ϕ = K ( x ) u 2 , lim | x | → + ∞ ϕ ( x ) = 0 , x ∈ R 3 , 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.

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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)


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Lixia Wang, Shiwang Ma, Xiaoming Wang. On the existence of solutions for nonhomogeneous Schrödinger-Poisson system, Boundary Value Problems, 2016, pp. 76, 2016, DOI: 10.1186/s13661-016-0584-9