Normalized Solutions to the Kirchhoff-Choquard Equations with Combined Growth

2024/12/09 by Divya Goel, Goel, Divya, Shilpa Gupta +1
Computer Science · #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.48550/arxiv.2412.06722

Abstract

This paper is devoted to the study of the following nonlocal equation: -(a+b‖∇ u‖₂^2(θ-1)) Δu =λu+α(I_μ∗|u|^q)|u|^q-2u+(I_μ∗|u|^p)|u|^p-2u \hboxin ℝ^N, with the prescribed norm ∫_ℝ^N |u|²= c², where N≥ 3, 0<μ0, 1<θ<(2N-μ)/(N-2), (2N-μ)/(N)0 is a suitably small real parameter, λ∈ℝ is the unknown parameter which appears as the Lagrange's multiplier and I_μ is the Riesz potential. We establish existence and multiplicity results and further demonstrate the existence of ground state solutions under the suitable range of α. We demonstrate the existence of solution in the case of q is L²-supercritical and p= (2N-μ)/(N-2), which is not investigated in the literature till now. In addition, we present certain asymptotic properties of the solutions. To establish the existence results, we rely on variational methods, with a particular focus on the mountain pass theorem, the min-max principle, and Ekeland's variational principle.

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