AM  Vol.2 No.9 , September 2011
Positive Solutions to the Nonhomogenous p-Laplacian Problem with Nonlinearity Asymptotic to up-1at Infinity in RN
Author(s) Li Wang
ABSTRACT
In this paper, we study the following problem {-Δpu+V(x)|u|p-2u=K(x)f(u)+h(x) in□ N, uW1,p(□ N), u>0 in □ N, (*) where 1<pN,the potential V(x) is a positive bounded function, hLp'(□ N), 1/p'+1/p=1, 1<pN, 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.

Cite this paper
nullL. Wang, "Positive Solutions to the Nonhomogenous p-Laplacian Problem with Nonlinearity Asymptotic to up-1at Infinity in RN," Applied Mathematics, Vol. 2 No. 9, 2011, pp. 1068-1075. doi: 10.4236/am.2011.29148.
References
[1]   C. A. Stuart and H. S. Zhou, “A Variational Problem Related to Self-Trapping of an Electromagnetic Field,” Mathematical Methods in the Applied Sciences, Vol. 19, No. 17, 1996, pp. 1397-1407. doi:10.1002/(SICI)1099-1476(19961125)19:17<1397::AID-MMA833>3.0.CO;2-B

[2]   J. Louis, “On the Existence of Bounded Palais-Smale Sequences and Application to a Landesman-Lazer-Type Problem Set on ,” Proceedings of the Royal Society of Edinburgh: Section A, Vol. 129, No. 4, 1999, pp. 787-809.

[3]   P. L. Lions, “On the Existence of Positive Solutions of Semilinear Elliptic Equations,” SIAM Review, Vol. 24, No. 4, 1982, pp. 441-467. doi:10.1137/1024101

[4]   C. A. Stuart and H. S. Zhou, “Applying the Mountain Pass Theorem to an Asymptotically Linear Elliptic Equation on ,” Communications in Partial Differential Equations, Vol. 24, No. 9-10, 1999, pp. 1731-1758. doi:10.1080/03605309908821481

[5]   J. Louis and T. Kazunaga, “A Positive Solution for an Asymptotically Linear Elliptic Problem on Autonomous at In?nity,” ESAIM: Control, Optimisation and Calculus of Variations, Vol. 7, 2002, pp. 597-614.

[6]   A. Antonio, F. Veronica and M. Andrea, “Ground States of Nonlinear Schr?dinger Equations with Potentials Vanishing at Infinity,” Journal of the European Mathematical Society, Vol. 7, No. 1, 2005, pp. 117-144.

[7]   C. Y. Liu, Z. P. Wang and H. S. Zhou, “Asymptotically Linear Schr?dinger Equation with Potential Vanishing Atinfinity,” Journal of Differential Equations, Vol. 245, No. 1, 2008, pp. 201-222. doi:10.1016/j.jde.2008.01.006

[8]   X. P. Zhu and H. S. Zhou, “Existence of Multiple Positive Solutions of Inhomogeneous Semilinear Elliptic Problems in Unbounded Domains,” Proceedings of the Royal Society of Edinburgh: Section A, Vol. 115, No. 3-4, 1990, pp. 301-318.

[9]   Z. P. Wang and H. S. Zhou, “Positive Solutions for a Nonhomogeneous Elliptic Equation on without (A-R) Condition,” Journal of Mathematical Analysis and Applications, Vol. 353, No. 1, 2009, pp. 470-479. doi:10.1016/j.jmaa.2008.11.080

[10]   D. G. de Figueiredo, P. L. Lions and R. D. Nussbaum, “A Priori Estimates and Existence of Positive Solutions of Semilinear Elliptic Equations,” Journal de Mathématiques Pures et Appliquées, Vol. 9, No. 61, 1982, pp. 41-63.

[11]   Z. L. Liu, “Positive Solutions of Superlinear Elliptic Equations,” Journal of Functional Analysis, Vol. 167, No. 2, 1999, pp. 370-398. doi:10.1006/jfan.1999.3446

[12]   Z. L. Liu, S. J. Li and Z. Q. Wang, “Positive Solutions of Elliptic Boundary Value Problems without the (P.S.) Type Assumption,” Indiana University Mathematics Journal, Vol. 50, No. 3, 2001, pp. 1347-1369. doi:10.1512/iumj.2001.50.1941

[13]   E. Ivar, “Convexity Methods in Hamiltonian Mechanics,” Springer-Verlag, Berlin, 1990.

[14]   G. B. Li and H. S. Zhou, “The Existence of a Weak Solution of Inhomogeneous Quasilinear Elliptic Equation with Critical Growth Conditions,” Acta Mathematica Sinica, Vol. 11, No. 2, 1995, pp. 146-155.

[15]   J. L. Vzquez, “A Strong Maximum Principle for Some Quasilinear Elliptic Equations,” Applied Mathematics & Optimization, Vol. 12, No. 3, 1984, pp. 191-202. doi:10.1007/BF01449041

 
 
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