2021/02/22 by Marcos L. M. Carvalho, Carvalho, Marcos L. M., Edcarlos D. Silva +3 · 2 citations
Mathematics · #35A01 #35A15 #35A23 #35A25 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Fractional Differential Equations Solutions #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2102.11335
openalex publication_date 2021/02/22 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
It is established existence of ground and bound state solutions for Choquard\nequation considering concave-convex nonlinearities in the following form\n \n
n -
Delta u +V(x) u · amp;= · amp; (I_
alpha* |u|p)|u|p-2u+
lambda |u|q-2u,
, u\n
in H1(
mathbbRN),\n
n where \λ > 0, N \≥ 3, \α \∈ (0, N). The potential V is a\ncontinuous function and I_\α denotes the standard Riesz potential. Assume\nalso that 1 < q < 2,~2\α < p < 2^*_\α where\n2_\α=(N+\α)/N, 2_\α=(N+\α)/(N-2). Our main contribution is\nto consider a specific condition on the parameter \λ > 0 taking into\naccount the nonlinear Rayleigh quotient. More precisely, there exists\n\λn > 0 such that our main problem admits at least two positive\nsolutions for each \λ \∈ (0, \λn]. In order to do that we combine\nNehari method with a fine analysis on the nonlinear Rayleigh quotient. The\nparameter \λn > 0 is optimal in some sense which allow us to apply the\nNehari method.\n