COMBINED EFFECTS IN A SEMILINEAR POLYHARMONIC PROBLEM IN THE UNIT BALL

doi:10.1016/S0252-9602(14)60092-8

摘要/Abstract

摘要：

Let m be a positive integer and B be the unit ball of Rⁿ (n ≥ 2). We investigate the existence, uniqueness and the asymptotic behavior of a positive continuous solution to the following semilinear polyharmonic boundary value problem
(−Δ)^mu = a₁(x)u^α^₁ + a₂(x)u^α₂ , lim_|x|→1u(x)/(1 − |x|)^m−1 = 0,
where α₁, α₂ ∈(−1, 1) and a₁, a₂are two nonnegative measurable functions on B satisfying some appropriate assumptions related to Karamata regular variation theory.

关键词: Kato class, positive solution, nonlinear polyharmonic equation, asymptotic behavior, schauder fixed point theorem

Abstract:

Key words: Kato class, positive solution, nonlinear polyharmonic equation, asymptotic behavior, schauder fixed point theorem

中图分类号:

34B27

Zagharide Zine EL ABIDINE. COMBINED EFFECTS IN A SEMILINEAR POLYHARMONIC PROBLEM IN THE UNIT BALL[J]. 数学物理学报(英文版), 2014, 34(5): 1404-1416.

Zagharide Zine EL ABIDINE. COMBINED EFFECTS IN A SEMILINEAR POLYHARMONIC PROBLEM IN THE UNIT BALL[J]. Acta mathematica scientia,Series B, 2014, 34(5): 1404-1416.

参考文献

[1] Bachar I, Mˆaagli H, Masmoudi S, Zribi M. Estimates for the Green function and singular solutions for polyharmonic nonlinear equation. Abstract and Applied Analysis, 2003, 12: 715–741

[2] Ben Othman S. On a singular sublinear polyharmonic problem. Abstract and Applied Analysis, 2006, 2006: 1–14

[3] Ben Othman S, Mˆaagli H, Masmoudi S, Zribi M. Exact asymptotic behavior near the boundary to the solution for singular nonlinear Dirichlet problems. Nonlinear Anal, 2009, 71: 4137–4150

[4] Boggio T. Sulle funzioni di Green d’ordine m. Rend Circ Math Palermo, 1905, 20: 97–135

[5] Brezis H, Oswald L. Remarks on sublinear elliptic equations. Nonlinear Anal, 1986, 10: 55–64

[6] Chemmam R, Mˆaagli H, Masmoudi S, Zribi M. Combined effects in nonlinear singular elliptic problems in a bounded domain. Adv Nonlinear Anal, 2012, 1: 301–318

[7] Cˆ?rstea F C, Radulescu V D. Extremal singular solutions for degenerate logistic-type equations in anisotropic media. C R Acad Sci Paris S´er I, 2004, 339: 119–124

[8] Coclite M M, Palmieri G. On a singular nonlinear Dirichlet problem. Comm Partial Differential Equations, 1989, 14: 1315–1327

[9] Crandall M G, Rabinowitz P H, Tartar L. On a Dirichlet problem with a singular nonlinearity. Comm Partial Differential Equations, 1977, 2: 193–222

[10] Dhifli A, Zine El Abidine Z. Asymptotic behavior of positive solutions of a semilinear polyharmonic problem in the unit ball. Nonlinear Anal, 2012, 75: 625–636

[11] Edelson A. Entire solutions of singular elliptic equations. J Math Anal Appl, 1989, 139: 523–532

[12] Ghergu M, Radulescu V D. Singular elliptic problems. Bifurcation and Asymptotic Analysis. Oxford Lec-tures Series in Mathematics and Applications, Vol 37. Oxford University Press, 2008: 320

[13] Ghergu M, Radulescu V D. Nonlinear PDEs: Mathematical Models in Biology, Chemistry and Population Genetics, Springer Monographs in Mathematics. Heidelberg: Springer-Verlag, 2012

[14] Gontara S, Mˆaagli H, Masmoudi S, Turki S. Asymptotic behavior of positive solutions of a singular nonlinear Dirichlet problems. J Math Anal Appl, 2010, 369: 719–729

[15] Gontara S, Zine El Abidine Z. Existence of positive bounded solutions for some nonlinear polyharmonic elliptic systems. Electronic J Diff Eq, 2010, 113: 1–18

[16] Grunau H -Ch, Sweers G. The role of positive boundary data in the clamped plate equation, perturbation results and other generalizations. Z Angew Math Phys, 1998, 49: 420–435

[17] Lazer A C, McKenna P J. On a singular nonlinear elliptic boundary-value problem. Proc Amer Math Soc, 1991, 111: 721–730

[18] Mˆaagli H. Asymptotic behavior of positive solutions of a semilinear Dirichlet problems. Nonlinear Anal, 2011, 74: 2941–2947

[19] Mˆaagli H, Zribi M. Existence and estimates of solutions for singular nonlinear elliptic problems. J Math Anal Appl, 2001, 263: 522–542

[20] Radulescu V D. Combined effects in nonlinear singular elliptic problems with convection. Rev Roumaine Math Pures Appl, 2008, 53(5/6): 543–553

[21] Radulescu V D. Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations: Monotonicity, Analytic and Variational Methods, Contemporary Mathematics and Its Applications Vol 6. Hindawi Publ Corp, 2008: 210

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

[23] Seneta R. Regular Varying Functions. Lectures Notes in Math 508. Berlin: Springer-Verlag, 1976

[24] Shi J P, Yao M X. On a singular semilinear elliptic problem. Proc Roy Soc Edinburgh Sect A, 1998, 128: 1389–1401

[25] Sun Y L, Li S J. Structure of ground state solutions of singular semilinear elliptic equations. Nonlinear Anal, 2003, 55: 399–417

[26] Sun Y L, Li S J. Some remarks on a superlinear-singular problem: Estimates for . Nonlinear Anal, 2008, 69: 2636–2650

[27] Sun Y L, Wu S P, Long Y M. Combined effects of singular and superlinear nonlinearities in some singular boundary value problems. J Differential Equations, 2001, 176: 511–531

[28] Zhang Z. On a Dirichlet problem with a singular nonlinearity. J Math Anal Appl, 1995, 194: 103–113

[29] Zhang Z. The asymptotic behavior of the unique solution for the singular Lane-Emden-Fowler equation. J Math Anal Appl, 2005, 312: 33–43