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Positive Solutions to the Nonhomogenous p-Laplacian Problem with Nonlinearity Asymptotic to up-1at Infinity in RN

Applied Mathematics·Volume 02 (2011)·Pages 1068–1075·Published 29 September 2011·DOI10.4236/am.2011.29148

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 1＜p＜N,the potential V(x) is a positive bounded function, h∈Lp＇(□ N), 1/p＇+1/p=1, 1＜p＜N, 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.

KeywordsPositive Solutions&lti&gtp&lt/i&gt-LaplacianNonlinearity AsymptoticMountain Pass Theorem

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Positive Solutions to the Nonhomogenous <i>p</i>-Laplacian Problem with Nonlinearity Asymptotic to <i>u<sup>p</i>-1</sup>at Infinity in R<i><sup>N</sup></i>

Abstract

References (15)