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An elliptic equation under the effect of two nonlocal terms
Author(s) -
Batkam Cyril Joel
Publication year - 2016
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.3587
Subject(s) - mathematics , sign (mathematics) , bounded function , domain (mathematical analysis) , mountain pass , elliptic curve , mathematical analysis , type (biology) , energy (signal processing) , pure mathematics , mountain pass theorem , mathematical physics , nonlinear system , physics , ecology , statistics , quantum mechanics , biology
The aim of this note is to investigate the existence of signed and sign‐changing solutions to the Kirchhoff type problem 0.1− a + b ∫ Ω | ∇ u | 2Δ u =∫ Ω1 p | u | p2 p| u | p − 2 u in Ωu = 0 on ∂ Ω ,where Ω is a bounded smooth domain inR N ( N = 1,2,3), a , b > 0 and 2 < p < 2 ⋆ , with 2 ⋆ =+ ∞ if N = 1,2 and 2 ⋆ =6 if N = 3. Using variational methods, we show that (0.1) possesses three solutions of mountain pass type (one positive, one negative and one sign‐changing) and infinitely many high‐energy sign‐changing solutions. Copyright © 2015 John Wiley & Sons, Ltd.