In this paper we consider the problem {-Δu ± φu + W' (x, u) = 0 in ℝ2, Δ φ = u2 in ℝ2 where W is assumed nonnegative. In dimension three, the problem with the sign + (we call it (P+)) was considered and solved in [22], whereas in the same paper it was showed that no nontrivial solution exists if we consider the sign - (say it (P-)). We provide a general existence result for (P+) and two examples falling in the case (P-) for which there exists at least a nontrivial solution.
The planar Schrödinger-Poisson system with a positive potential
Azzollini A.
2021-01-01
Abstract
In this paper we consider the problem {-Δu ± φu + W' (x, u) = 0 in ℝ2, Δ φ = u2 in ℝ2 where W is assumed nonnegative. In dimension three, the problem with the sign + (we call it (P+)) was considered and solved in [22], whereas in the same paper it was showed that no nontrivial solution exists if we consider the sign - (say it (P-)). We provide a general existence result for (P+) and two examples falling in the case (P-) for which there exists at least a nontrivial solution.File in questo prodotto:
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