Positive bounded solutions for nonlinear polyharmonic problems in the unit ball

Mâagli, Habib; Zine El Abidine, Zagharide

doi:10.1515/auom-2017-0041

Abstract

In this paper, we study the existence of positive solutions for the following nonlinear polyharmonic equation (-∆)^mu+λf(x, u) = 0 in B; subject to some boundary conditions, where m is a positive integer, λ is a nonnegative constant and B is the unit ball of ℝⁿ (n ≥ 2). Under some appropriate assumptions on the nonnegative nonlinearity term f(x, u) and by using the Schäuder fixed point theorem, the existence of positive solutions is obtained. At last, examples are given for illustration.

References

[1] I. Bachar, H. Mâagli, Positive bounded solutions for semilinear elliptic equations in smooth domains, Bull. Belg. Math. Soc. Simon Stevin 20 (2013), 707-714.10.36045/bbms/1382448190
Search in Google Scholar Back to article
[2] I. Bachar, H. Mâagli, S. Masmoudi, M. Zribi, Estimates for the Green function and singular solutions for polyharmonic nonlinear equation, Abstract and Applied Analysis 2003 (2003), 715-741.10.1155/S1085337503209039
Open DOI Search in Google Scholar Back to article
[3] S. Ben Othman, On a singular sublinear polyharmonic problem, Abstract and Applied Analysis 2006 (2006), 1-14.10.1155/AAA/2006/27969
Open DOI Search in Google Scholar Back to article
[4] S. Ben Othman, H. Mâagli, M. Zribi, Existence results for polyharmonic boundary value problems in the unit ball, Abstract and Applied Analysis 2007 (2007), 1-17.10.1155/2007/56981
Open DOI Search in Google Scholar Back to article
[5] T. Boggio, Sulle funzioni di Green d'ordine m, Rend. Circ. Math. Palermo, 20 (1905), 97-135. 10.1007/BF03014033
Search in Google Scholar Back to article
[6] A. Dhii, Z. Zine El Abidine, Asymptotic behavior of positive solutions of a semilinear polyharmonic problem in the unit ball, Nonlinear Analysis 75 (2012), 625-636.10.1016/j.na.2011.08.064
Search in Google Scholar Back to article
[7] J. Giacomoni, P. Mishra, K. Sreenadh, Fractional elliptic equations with critical exponential nonlinearity, Adv. Nonlinear Anal. 5 (2016), no. 1, 57-74.10.1515/anona-2015-0081
Search in Google Scholar Back to article
[8] S. Gontara, Z. Zine El Abidine, Existence of positive bounded solutions for some nonlinear polyharmonic elliptic systems, Electronic Journal of Differential Equations 113 (2010), 1-18.
Search in Google Scholar Back to article
[9] S. Kumar, D. Kumar, J. Singh, Fractional modelling arising in unidirectional propagation of long waves in dispersive media, Adv. Nonlinear Anal. 5 (2016), no. 4, 383-394.10.1515/anona-2013-0033
Search in Google Scholar Back to article
[10] H. Mâagli, F. Toumi, M. Zribi, Existence of positive solutions for some polyharmonic nonlinear boundary-value problems, Electronic Journal of Differential Equations 58 (2003), 1-19.
Search in Google Scholar Back to article
[11] G. Molica Bisci, V. R_adulescu, Ground state solutions of scalar field fractional Schrödinger equations, Calc. Var. Partial Differential Equations 54 (2015), no. 3, 2985-3008.10.1007/s00526-015-0891-5
Search in Google Scholar Back to article
[12] G. Molica Bisci, V. R_adulescu, R. Servadei, Variational Methods for Nonlocal Fractional Problems, Encyclopedia of Mathematics and its Applications, Vol. 162, Cambridge University Press, Cambridge, 2016.10.1017/CBO9781316282397
Search in Google Scholar Back to article
[13] G. Molica Bisci, D. Repov_s, On doubly nonlocal fractional elliptic equations, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 26 (2015), no. 2, 161-176.10.4171/rlm/700
Search in Google Scholar Back to article
[14] P. Pucci, M. Xiang, B. Zhang, Existence and multiplicity of entire solutions for fractional p-Kirchho equations, Adv. Nonlinear Anal. 5 (2016), no. 1, 27-55.10.1515/anona-2015-0102
Search in Google Scholar Back to article
[15] D. Repovš, B. Zhang, X. Zhang, Existence and symmetry of solutions for critical fractional Schrödinger equations with bounded potentials, Nonlinear Anal. 142 (2016), 48-68.10.1016/j.na.2016.04.012
Search in Google Scholar Back to article

Positive bounded solutions for nonlinear polyharmonic problems in the unit ball

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