自然增长条件下含Hardy位势的椭圆型方程解的存在性

Abstract

Abstract:

In this paper, the authors study a quasilinear elliptic equation with Hardy potential under natural growth condition. The authors extend the sign condition from nonnegative to a minus part, prove that at this time the corresponding functional of the equation still satisfies the (PS) condition, and then prove the existence of two nontrivial weak solutions via the nonsmooth critical point theory.

Key words: Elliptic equation, Natural growth condition, Hardy potential, Nonsmooth critical point theory

CLC Number:

35J35

LI Zhou-Xin, SHEN Yao-Tian. Existence of Solutions for Elliptic Equations Involving Hardy Potential under Natural Growth Condition[J].Acta mathematica scientia,Series A, 2011, 31(6): 1470-1478.

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References

[1] Chen Z C, Luo T. The eigenvalue problem for p-Laplacian-Like equations. Acta Math Sinica (in Chinese), 2003, 46: 631--638

[2] Mitidieri E. A simple approach to Hardy inequalities. Mat Zametki, 2000, 67(4): 563--572; Translation in Math Notes, 2000, 67(4): 479--486

[3] Brézis H, Marcus M. Hardy's inequalities revisited. Ann Scuola Normale Sup Pisa Cl Scienze, 1997, 4: 217--237

[4] Vázquez J L, Zuazua E. The Hardy inequality and the asymptotic behavior of the heat equation with an inverse-square potential. J Funct Analysis, 2000, 173: 103--153

[5] Shen Y T, Yao Y X. Nonlinear elliptic equations with critical potential and critical parameter. Proceedings of the Royal Society of Edinburgh, 2006, 136A: 1041--1051

[6] Pellacci B, Squassina M. Unbounded critical points for a class of lower semicontinuous functionals. J Differential Equations, 2004, 201: 25--62

[7] Pellacci B. Multiple critical points for nondifferentiable functionals involving Hardy potentials. Nonlinear Anal, 2005, 61: 517--542

[8] Squassina M. Existence, multiplicity, perturbation, and concentration results for a class of quasi-linear elliptic problems. Electronic Journal of Differential Equations, 2006, Monograph: 7

[9] Corvellec J N, Degiovanni M, Marzocchi M. Deformation properties for continuous functionals and critical point theory. Topol Methods Nonlinear Anal, 1993, 1: 151--171

[10] Degiovanni M, Marzocchi M. A critical point theory for nonsmooth functional. Ann Mat Pura Appl, 1994, 167: 73--100

[11] Canino A. On a jumping problem for quasilinear elliptic equations. Math Z, 1997, 226: 193--210

[12] Ladyzhenskaya O A, Uraltseva N N. Linear and Quasilinear Elliptic Equations. New York: Academic Press, 1968

[13] Arcoya D, Boccardo L. Critical point for multiple integrals of the calculus of variations. Arch Ration Mech Anal, 1996, 134: 249--274

[14] Canino A, Degiovanni M. Nonsmooth Critical Point Theory and Quasilinear Elliptic Equations//A. Granas, M. Frigon, G. Sabidussi (Eds.). Topological Methods in Differential Equations and Inclusions. Montreal: NATO ASI Series-Kluwe Academic Press, 1995

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Existence of Solutions for Elliptic Equations Involving Hardy Potential under Natural Growth Condition

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