Schrödinger-Poisson system with steep potential well

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Abstract

In this paper, we consider the following Schrödinger-Poisson system. (Pλ){-Δu+(1+μg(x))u+λφ(x)u=|u|p-1u, x∈R{double-strcuk}3, -Δφ=u2,lim|x|→+∞Φ(x)=0, where λ, μ are positive parameters, p∈(1,5), g(x)∈L∞(R{double-struck}3) is nonnegative and g(x)≡0 on a bounded domain in R{double-struck}3. 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. © 2011 Elsevier Inc.

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Jiang, Y., & Zhou, H. S. (2011). Schrödinger-Poisson system with steep potential well. Journal of Differential Equations, 251(3), 582–608. https://doi.org/10.1016/j.jde.2011.05.006

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