Nontrivial complex solutions for magnetic Schrodinger equations with critical nonlinearities
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Using minimization arguments we establish the existence of a complex solution to the magnetic Schrödinger equation
-(∇ + iA(x))2u + u = ƒ(|u|2)u in ℝN,
where N ≥ 3, A:ℝN → ℝN is the magnetic potential and ƒ satisfies some critical growth assumptions. First we obtain bounds from a real Pohozaev manifold. Then relate them to Sobolev imbedding constants and to the least energy level associated with the real equation in absence of the magnetic field (i.e., with A(x) = 0). We also apply the Lions Concentration Compactness Principle to the modula of the minimizing sequences involved.
CitationBarile, S., & Figueiredo, G. M. (2018). Nontrivial complex solutions for magnetic Schrodinger equations with critical nonlinearities. Electronic Journal of Differential Equations, 2018(174), pp. 1-21.
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