MULTIPLE POSITIVE SOLUTIONS FOR QUASILINEAR ELLIPTIC PROBLEMS INVOLVING CONCAVE-CONVEX NONLINEARITIES AND MULTIPLE HARDY-TYPE TERMS

doi:10.1016/S0252-9602(13)60084-3

Abstract

Abstract:

In this paper, we deal with the existence and multiplicity of positive solutions for the quasilinear elliptic problem
−Δ_pu −∑^k_i₌₁μ_i|u|^p−2/|x − a_i|_p u = |u|^p*−2u + λ|u|^q−2u, x ∈Ω,
where Ω （R^N(N ≥3) is a smooth bounded domain such that the different points a_i∈ Ω, i = 1, 2, · · · , k, 0 ≤ μ_i < ¯μ = (N−p/p )^p, λ > 0, 1 ≤ q < p, and p^* = p^N/N−p . The results depend crucially on the parameters λ, q and μi for i = 1, 2, · · · , k.

Key words: multiple positive solutions, concave-convex nonlinearities, multiple Hardy-type terms

CLC Number:

35J61

Tsing-San HSU. MULTIPLE POSITIVE SOLUTIONS FOR QUASILINEAR ELLIPTIC PROBLEMS INVOLVING CONCAVE-CONVEX NONLINEARITIES AND MULTIPLE HARDY-TYPE TERMS[J].Acta mathematica scientia,Series B, 2013, 33(5): 1314-1328.

Trendmd

References

[1] Abdellaoui B, Felli V, Peral I. Existence and nonexistence for quasilinear equations involving the p-Laplacian. Boll Unione Mat Ital Sez, 2006, B9: 445–484

[2] Ambrosetti A, Brezis H, Cerami G. Combined effects of concave and convex nonlinearities in some elliptic problems. J Funct Anal, 1994, 122: 519–543

[3] Br´ezis H, Lieb E. A relation between pointwise convergence of functions and convergence of functionals. Proc Amer Math Soc, 1983, 88: 486–490

[4] Br´ezis H, Marcus M. Hardy’s inequalities revisited. Ann Scuola Norm Sup Pisa Cl Sci (4), 1997, 25(1/2): 217–237

[5] Br´ezis K J, Zhang Y. The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function. J Differ Equ, 2003, 193: 481–499

[6] Caffarelli L, Kohn R, Nirenberg L. First order interpolation inequality with weights. Compos Math, 1984, 53: 259–275

[7] Dr´abek P, Pohozaev S I. Positive solutions for the p-Laplacian: Application of the fibering method. Proc Roy Soc Edinburgh Sect A, 1997, 127: 703–726

[8] Ekeland I. On the variational principle. J Math Anal Appl, 1974, 17: 324–353

[9] Garcia Azorero J P, Peral Alonso I. Hardy inequalities and some critical elliptic and parabolic problems. J Differ Equ, 1988, 144: 441–476

[10] Ghoussoub N, Yuan C. Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents. Trans Amer Math Soc, 2000, 352: 5703–5743

[11] Han P. Quasilinear elliptic problems with critical exponents and Hardy terms. Nonlinear Anal, 2005, 61: 735–758

[12] Hsu T S. Multiple positive solutions for semilinear elliptic equations involving multi-singular inverse square potentials and concave-convex nonlinearities. Nonlinear Anal, 2011, 74: 3703–3715

[13] Kang D. Positive solutions to the weighted critical quasilinear problems. Appl Math Com, 2009, 213: 432–439

[14] Kristaly A, Radulescu V, Varga C. Variational Principles in Mathematical Physics, Geometry, and Eco-nomics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems, Encyclopedia of Mathe-matics and its Applications, No 136. Cambridge: Cambridge University Press, 2010

[15] Lions P L. The concentration compactness principle in the calculus of variations, the limit case (I). Rev Mat Iberoamericana, 1985, 1(1): 145–201

[16] Lions P L. The concentration compactness principle in the calculus of variations, the limit case (II). Rev Mat Iberoamericana, 1985, 1(2): 45–121

[17] Radulescu V, Repovs D. Combined effects in nonlinear problems arising in the study of anisotropic con-tinuous media. Nonlinear Anal, 2012, 75: 1524–1530

[18] Radulescu V, Smets D, Willem M. Hardy-Sobolev inequalities with remainder terms. Topol Meth Nonlin Anal, 2002, 20: 145–149

[19] Trudinger N S. On Harnack type inequalities and their application to quasilinear elliptic equations. Comm Pure Appl Math, 1967, 20: 721–747

[20] Wu T F. Multiple positive solutions for a class of concave-convex elliptic problems in RN involving sign-changing weight. J Funct Anal, 2010, 258: 99–131

[21] Wu T F. Three positive solutions for Dirichlet problems involving critical Sobolev exponent and sign-changing weight. J Differ Equ, 2010, 249: 1549-1578.

[22] Wu T F. On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight
function. J Math Anal Appl, 2006, 318: 253–270