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We study the constrained minimizing problem of the energy functional related to attractive Schrödinger‐Poisson systems with periodic potentials: I(m)=infE(ϕ):ϕ∈H1(R3),‖ϕ‖L22=m, where E(ϕ):=12∫R3|∇ϕ(x)|2dx+12∫R3V(x)|ϕ(x)|2dx−14∬R3×R3|ϕ(x)|2|ϕ(y)|2|x−y|dxdy−1α+2∫R3|ϕ(x)|α+2dx, with m>0 , α>0 , and V is a… Click to show full abstract
We study the constrained minimizing problem of the energy functional related to attractive Schrödinger‐Poisson systems with periodic potentials: I(m)=infE(ϕ):ϕ∈H1(R3),‖ϕ‖L22=m, where E(ϕ):=12∫R3|∇ϕ(x)|2dx+12∫R3V(x)|ϕ(x)|2dx−14∬R3×R3|ϕ(x)|2|ϕ(y)|2|x−y|dxdy−1α+2∫R3|ϕ(x)|α+2dx, with m>0 , α>0 , and V is a continuous periodic potential. We first give a complete classification of existence and nonexistence of minimizers for the problem. In the mass‐critical case α=43 , we give a detailed description of the limiting behavior of minimizers as the mass tends to a critical value.
Journal Title: Mathematical Methods in the Applied Sciences
Year Published: 2020
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