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

Author(s)
Li Wang

ABSTRACT

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＇}(□* *^{N}), 1/*p＇*+1/*p*=1, 1＜*p*＜*N*, *h*≥0, *h*≠0*f(s)* is nonlinearity asymptotical to *s*^{p-1}at infinity, that is, *f(s)*~*O(s*^{p-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)*≤∫^{u}_{o}* f(s)*ds≤1/*p+θ* *f(u)u*, *u*＞0, *θ*＞0.

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

Cite this paper

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

nullL. Wang, "Positive Solutions to the Nonhomogenous

References

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[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.

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[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

[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