TY - JOUR
T1 - Nontrivial solutions for p-Laplacian systems
AU - Hai, D. D.
AU - Wang, Haiyan
N1 - Copyright:
Copyright 2007 Elsevier B.V., All rights reserved.
PY - 2007/6/1
Y1 - 2007/6/1
N2 - The paper deals with the existence and nonexistence of nontrivial nonnegative solutions for the sublinear quasilinear systemdiv (| ∇ ui |p - 2 ∇ ui) + λ fi (u1, ..., un) = 0 in Ω, ui = 0 on ∂ Ω, i = 1, ..., n, where p > 1, Ω is a bounded domain in RN (N ≥ 2) with smooth boundary, and fi, i = 1, ..., n, are continuous, nonnegative functions. Let u = (u1, ..., un), {norm of matrix} u {norm of matrix} = ∑i = 1n | ui |, we prove that the problem has a nontrivial nonnegative solution for small λ > 0 if one of lim{norm of matrix} u {norm of matrix} → 0 frac(fi (u), {norm of matrix} u {norm of matrix}p - 1) is infinity. If, in addition, all lim{norm of matrix} u {norm of matrix} → ∞ frac(fi (u), {norm of matrix} u {norm of matrix}p - 1) is zero, we show that the problem has a nontrivial nonnegative solution for all λ > 0. A nonexistence result is also obtained.
AB - The paper deals with the existence and nonexistence of nontrivial nonnegative solutions for the sublinear quasilinear systemdiv (| ∇ ui |p - 2 ∇ ui) + λ fi (u1, ..., un) = 0 in Ω, ui = 0 on ∂ Ω, i = 1, ..., n, where p > 1, Ω is a bounded domain in RN (N ≥ 2) with smooth boundary, and fi, i = 1, ..., n, are continuous, nonnegative functions. Let u = (u1, ..., un), {norm of matrix} u {norm of matrix} = ∑i = 1n | ui |, we prove that the problem has a nontrivial nonnegative solution for small λ > 0 if one of lim{norm of matrix} u {norm of matrix} → 0 frac(fi (u), {norm of matrix} u {norm of matrix}p - 1) is infinity. If, in addition, all lim{norm of matrix} u {norm of matrix} → ∞ frac(fi (u), {norm of matrix} u {norm of matrix}p - 1) is zero, we show that the problem has a nontrivial nonnegative solution for all λ > 0. A nonexistence result is also obtained.
KW - Elliptic system
KW - Schauder Fixed-Point Theorem
KW - p-Laplacian
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U2 - 10.1016/j.jmaa.2006.07.072
DO - 10.1016/j.jmaa.2006.07.072
M3 - Article
AN - SCOPUS:33846914696
SN - 0022-247X
VL - 330
SP - 186
EP - 194
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
IS - 1
ER -