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has been cited by the following article:

• 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.