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Positive Solutions to the Nonhomogenous *p*-Laplacian Problem with Nonlinearity Asymptotic to *u ^{p}*-1at Infinity in R

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In this paper, we study the following problem
{-Δ

*+*_{p}u*V(x)|u|*=^{p-2}u*K(x)f(u)*+*h(x)*in□^{N},*u*∈*W*(□^{1,p}*),*^{N}*u*＞0 in □^{N}, (*) where 1＜*p*＜*N*,the potential*V(x)*is a positive bounded function,*h*∈*L*(□^{p＇}*), 1/*^{N}*p＇*+1/*p*=1, 1＜*p*＜*N*,*h*≥0,*h*≠0*f(s)*is nonlinearity asymptotical to*s*at infinity, that is,^{p-1}*f(s)*~*O(s*as^{p-1})*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)*≤∫^{u}_{o}*f(s)*ds≤1/*p+θ**f(u)u*,*u*＞0,*θ*＞0.Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

L. Wang, "Positive Solutions to the Nonhomogenous

*p*-Laplacian Problem with Nonlinearity Asymptotic to*u*-1at Infinity in R^{p}*,"*^{N}*Applied Mathematics*, Vol. 2 No. 9, 2011, pp. 1068-1075. doi: 10.4236/am.2011.29148.

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