Eigencurves of the p(·)-Biharmonic operator with a Hardy-type term

Laghzal, Mohamed; Khalil, Abdelouahed El; Alaoui, My Driss Morchid; Touzani, Abdelfattah

doi:10.2478/mjpaa-2020-0015

Abstract

This paper is devoted to the study of the homogeneous Dirichlet problem for a singular nonlinear equation which involves the p(·)-biharmonic operator and a Hardy-type term that depend on the solution and with a parameter λ. By using a variational approach and min-max argument based on Ljusternik-Schnirelmann theory on C¹-manifolds [13], we prove that the considered problem admits at least one nondecreasing sequence of positive eigencurves with a characterization of the principal curve μ₁(λ) and also show that, the smallest curve μ₁(λ) is positive for all 0 ≤ λ < C_H, with C_H is the optimal constant of Hardy type inequality.

References

[1] A. Ayoujil, A. EI Amrouss, On the spectrum of a fourth order elliptic equation with variable exponent, Nonlinear Anal. Theory Methods., 71(2009), 4916-4926.10.1016/j.na.2009.03.074
Search in Google Scholar Back to article
[2] P.A. Binding, Y.X. Huang, Bifurcation from eigencurves of the p-Laplacian, Differential and Integral Equations., 8(2)(1995), 405-414.
Search in Google Scholar Back to article
[3] L. Diening, P. Harjulehto, P. Hästö, M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Springer-Verlag, Heidelberg., (2011).10.1007/978-3-642-18363-8
Search in Google Scholar Back to article
[4] A. El khalil, On the spectrum of Robin boundary p-Laplacian problem, Moroccan J. of Pure and Appl. Anal.(MJPAA)., 5(2) (2019), 279-293.10.2478/mjpaa-2019-0020
Search in Google Scholar Back to article
[5] A. El khalil, S. Kellati, A. Touzani, On the principal frequency curve of the p-biharmonic operator, Arab journal of Mathematical Sciences., 17 (2011), 89-99.10.1016/j.ajmsc.2011.01.002
Search in Google Scholar Back to article
[6] A. El khalil, M. El Moumni, M.D. Morchid Alaoui, A. Touzani, p(x)-Biharmonic operator involving p(x)-Hardy’s inequality, Georgian Mathematical Journal., DOI: https://doi.org/10.1515/gmj-2018-001310.1515/gmj-2018-0013
Search in Google Scholar Back to article
[7] Abdelouahed El khalil, Mohamed Laghzal, Moulay Driss Morchid Alaoui, Abdelfattah Touzani, Eigenvalues for a class of singular problems involving p(x)-Bihamonic operator and q(x)-Hardy potential, Adv. Nonlinear Anal., 9 (2020), 1130-1144.10.1515/anona-2020-0042
Search in Google Scholar Back to article
[8] A. El khalil, M.D. Morchid Alaoui, M. Laghzal, A. Touzani, On a nonlinear PDE involving weighted p-Laplacian, Bol. Soc. Paran. Mat., 38 (2020), 131-145.10.5269/bspm.v38i5.33978
Search in Google Scholar Back to article
[9] A. El khalil, M.D. Morchid Alaoui, A. Touzani, On the spectrum of the p-biharmonic operator involving p-Hardy’s inequality, Appl. Math. Warsaw., 41 (2014), 239-246.10.4064/am41-2-11
Search in Google Scholar Back to article
[10] X.L. Fan, X. Fan, A Knobloch-type result for p(t)-Laplacian systems, J. Math. Anal. Appl., 282 (2003), 453-464.10.1016/S0022-247X(02)00376-1
Search in Google Scholar Back to article
[11] X.L. Fan, D. Zhao, On the spaces Lp(x)(Ω) and Wm,p(x)(Ω), J. Math. Anal. Appl., 263 (2001), 424-446.10.1006/jmaa.2000.7617
Search in Google Scholar Back to article
[12] M. Mihăilescu, Existence and multiplicity of solutions for a Neumann problem involving the p(x)-Laplace operator, Nonlinear Anal., 67 (2007), 1419-1425.10.1016/j.na.2006.07.027
Search in Google Scholar Back to article
[13] A. Szulkin, Ljusternik-Schnirelmann theory on C1-manifolds, Ann. Inst. Henri Poincaré Anal. Nonl., 5(2) (1988), 119-139.10.1016/s0294-1449(16)30348-1
Search in Google Scholar Back to article
[14] A. Zang, Y. Fu, Interpolation inequalities for derivatives in variable exponent Lebesgue-Sobolev spaces, Nonlinear Anal., 69 (2008), 3629-3636.10.1016/j.na.2007.10.001
Search in Google Scholar Back to article

Eigencurves of the p(·)-Biharmonic operator with a Hardy-type term

Abstract

Paradigm

My account