Abstract
In this paper, we study the following nonlinear Choquard equation −ϵ2Δu+Kxu=1/8πϵ2∫ℝ3u2y/x−ydyu,x∈ℝ3, where ϵ>0 and Kx is a positive bounded continuous potential on ℝ3. By applying the reduction method, we proved that for any positive integer k, the above equation has a positive solution with k spikes near the local maximum point of Kx if ϵ>0 is sufficiently small under some suitable conditions on Kx.
Highlights
Introduction and Main ResultsIn this paper, we consider the following nonlinear Choquard equations8 >< −ε2Δu + KðxÞu = φu, x ∈ R3, >: −ε2Δφ = juj2 2, x ∈ R3, ð1Þ where ε > 0 and KðxÞ is a positive bounded continuous potential
= ðð1Þ/ð8πε2 applying the ððu2 reduction ðyÞÞ/ðjx − yjÞÞdyÞu, method, we proved that for any positive integer k, the above equation has a positive solution with k spikes near the local maximum point of KðxÞ if ε > 0 is sufficiently small under some suitable conditions on KðxÞ
Penrose [3] proposed it as a model of selfgravitating matter, in a programme in which quantum state reduction is understood as a gravitational phenomenon
Summary
X ∈ R3, ð1Þ where ε > 0 and KðxÞ is a positive bounded continuous potential. The Choquard equation first appeared as early as in 1954, in a work by Pekar describing the quantum mechanics of a polaron at rest [1]. In [14], Lions derived the existence of ground state solutions of (2) under some suitable conditions on KðxÞ if ε > 0 is small enough. For any positive integer k > 0, Wei and Winter [15] proved that there exist a solution of (2) concentrating at k points which are all local minimums or local maximums or non-degenerate critical points of KðxÞ under the conditions that inf R3 K > 0, K ∈ C2ðR3Þ provided ε is sufficiently small. Applying the existence and the nondegeneracy of ground state solutions for (5), inspired by [21, 22], we want to exploit the finite dimensional reduction method to investigate the existence of positive multi-spikes solutions for (2) under the conditions imposed on KðxÞ as follows:. In Appendix, some technical estimates and an energy expansion for the functional corresponding to problem (2) will be established
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