TITLE:
Positive Solutions to the Nonhomogenous p-Laplacian Problem with Nonlinearity Asymptotic to up-1at Infinity in RN
AUTHORS:
Li Wang
KEYWORDS:
Positive Solutions, p-Laplacian, Nonlinearity Asymptotic, Mountain Pass Theorem
JOURNAL NAME:
Applied Mathematics,
Vol.2 No.9,
September
19,
2011
ABSTRACT: In this paper, we study the following problem
{-Δpu+V(x)|u|p-2u=K(x)f(u)+h(x) in□ N,
u∈W1,p(□ N), u＞0 in □ N, (*)
where 1pN,the potential V(x) is a positive bounded function, h∈Lp＇(□ N), 1/p＇+1/p=1, 1pN, h≥0, h≠0f(s) is nonlinearity asymptotical to sp-1at infinity, that is, f(s)~O(sp-1) as s→+∞. The aim of this paper is to discuss how to use the Mountain Pass theorem to show the existence of positive solutions of the present problem. Under appropriate assumptions on V, K, h and f, we prove that problem (*) has at least two positive solutions even if the nonlinearity f(s) does not satisfy the Ambrosetti-Rabinowitz type condition:
0≤F(u)≤∫uo f(s)ds≤1/p+θ f(u)u, u＞0, θ＞0.