Schrodinger-Poisson system with steep potential well
By Yongsheng Jiang, Huan-Song Zhou. 2009
169 citations recorded by the source corpus.
In this paper, we consider the following Schrodinger–Poisson system (Pλ){−Δu+(1+μg(x))u+λϕ(x)u=|u|p−1u,x∈R3,−Δϕ=u2,lim|x|→+∞ϕ(x)=0, where λ, μ are positive parameters, p∈(1,5), g(x)∈L∞(R3) is nonnegative and g(x)≡0 on a bounded domain in R3. In this case, μg(x) represents a potential well that steepens as μ getting large. If μ=0, (Pλ) was well studied in Ruiz (2006) [18]. If μ≠0 and g(x) is not radially symmetric, it is unknown whether (Pλ) has a nontrivial solution for p∈(1,2). By priori estimates and approximation methods we prove that (Pλ) with p∈(1,2) has a ground state if μ large and λ small. In the meantime, we prove also that (Pλ) with p∈[3,5) has a nontrivial solution for any λ>0 and μ large. Moreover, some behaviors of the solutions of (Pλ) as λ→0, μ→+∞ and |x|→+∞ are discussed.
Subfields: Applied Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/83fa9a48ea171925fa438ce706d808747cda8393
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In Applied Mathematics
In Mathematical Physics
Citations in the corpus
- The concentration behavior of ground state solutions for a fractional Schrödinger–Poisson system (2017, unknown): “See [1,25,36,44,48] and the references therein for the existence of solutions.”
- Existence of positive solutions to Schrödinger–Poisson type systems with critical exponent (2014, result): “Related results for more general f were also obtained in [1, 28, 32], and solutions of Schrödinger–Poisson systems with nonconstant potential (where b is a function of x) were considered in [17, 20, 21, 30, 31].”
- Ground state solutions for an indefinite Kirchhoff type problem with steep potential well (2014, methodology): “Very recently, Jiang and Zhou [15] first applied the steep potential well into Schrödinger-Poisson system, and obtained the existence and concentration results by combining domains approximation with priori estimates.”