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Coercive and noncoercive elliptic problems with variable exponent Laplacian under Robin boundary conditions

  • Makkia Dammak and Abir Amor Ben Ali EMAIL logo
Published/Copyright: December 4, 2022
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 72 Issue 6

Abstract

In the present paper, we study the existence as well as the non-existence of some positive solutions for the equation −Δp(x) u = λ k(x) uq ± h(x) ur under Robin boundary condition in a regular open bounded domain Ω of ℝN, N ≥ 2. Δp(x) is the p(x)-Laplacian operator where p ∈ C1(Ω) and p > 1. Our proofs are based on the sub solution-super solution method and also on variational arguments.

MSC 2010: Primary 35J20; 35J60; 35K57; 35J70
Keywords: p(x)-Laplace operator; quasilinear equation; sub solution-super solution method; generalised Sobolev spaces; variational methods

  1. (Communicated by Alberto Lastra)

[1] Alsaedi, A.—Ahmad, B.: Anisotropic problems with unbalanced growth, Adv. Nonlinear Anal. 9 (2020), 1504–1515.10.1515/anona-2020-0063Search in Google Scholar

[2] Chandrasekhar, S.: An Introduction to the Study of Stellar Structure, Dover Books on Astronomy, 1985.Search in Google Scholar

[3] Deng, S. G.: Positive solutions for Robin problem involving the p(x)-Laplacian, J. Math. Anal. Appl. 360 (2009), 548–560.10.1016/j.jmaa.2009.06.032Search in Google Scholar

[4] Deng, S. G.—Wang, Q.—Cheng, S.: On the p(x)-Laplacian Robin eigenvalue problem, Appl. Math. Comput. 15(17) (2011), 5643–5649.10.1016/j.amc.2010.12.042Search in Google Scholar

[5] Fan, X. L.—Zhao, D: On the spaces Wk,p(x)(Ω) and Lp(x)(Ω), J. Math. Anal. Appl. 263 (2001), 424–446.10.1006/jmaa.2000.7617Search in Google Scholar

[6] Gelfand, I. M.: Some problems in the theory of quasi-linear equations, Section 15, due to G. I. Barenblatt, Amer. Math. Soc. Transl. 29 (1963), 295–381. Russian original: Uspekhi Mat. Nauk 14 (1959), 87–158.10.1090/trans2/029/12Search in Google Scholar

[7] Joseph, D.—Sparrow, E. M.: Nonlinear diffusion induced by nonlinear sources, Quart. Appl. Math. 28 (1970), 327–342.10.1090/qam/272272Search in Google Scholar

[8] Kefi, K.: On the robin problem with indefinite weight in Sobolev spaces with variable exponent, Z. Anal. Anwend. 37(1) (2018).10.4171/ZAA/1600Search in Google Scholar

[9] Keller, H. B.—Cohen, D. S.: Some positone problems suggested by nonlinear heat generation, J. Math. Mech. 16 (1967), 1361–1376.10.1512/iumj.1967.16.16087Search in Google Scholar

[10] Kováčik, O.—Rákosník, J.: On spaces Lp(x) and Wk,p(x), Czechoslovak Math. J. 41 (1991), 592–628.10.21136/CMJ.1991.102493Search in Google Scholar

[11] Mottin, S.: An analytical solution of the Laplace equation with Robin conditions by applying Legendre transform, Integral Transforms Spec. Funct. 27(4) (2016), 289–306.10.1080/10652469.2015.1121255Search in Google Scholar

[12] Mingione, G.—Rădulescu, V.: Recent developments in problems with nonstandard growth and nonuniform ellipticity, J. Math. Anal. Appl. 501(1) (2021), 125–197.10.1016/j.jmaa.2021.125197Search in Google Scholar

[13] Orlicz, W.: Über konjugierte Exponentenfolgen, Studia Math. 3 (1931), 200–211.10.4064/sm-3-1-200-211Search in Google Scholar

[14] Otared, K.: Introduction à la Théorie des Points Critiques et Applications aux Problèmes Elliptiques, Paris, Berlin, Heidelbreg etc., Springer-Verlag, 1993.Search in Google Scholar

[15] Rădulescu, V.—Repovš, D.: Combined effects in nonlinear problems arising in the study of anisotropic continuous media, Nonlinear Anal. 75 (2012), 1524–1530.10.1016/j.na.2011.01.037Search in Google Scholar

[16] Rădulescu, V. D.—Repovš, D. D.: Partial Differential Equations with Variable Exponents. Variational Methods and Qualitative Analysis. Monogr. Res. Notes Math., CRC Press, Boca Raton, FL, 2015.Search in Google Scholar

[17] Ragusa, M. A.—Tachikawa, A.: Regularity for minimizers for functionals of double phase with variable exponents, Adv. Nonlinear Anal. 9(1) (2020), 710–728.10.1515/anona-2020-0022Search in Google Scholar

[18] Saoudi, K.: Existence and nonexistence of positive solutions for quasilinear elliptic problem, Abstract Appl. Anal. (2012), Art. ID 275748.10.1155/2012/275748Search in Google Scholar

[19] Shi, X.—Rădulescu, V. D.—Repovš, D. D.—Zhang, Q.: Multiple solutions of double phase variational problems with variable exponent, Adv. Calc. Var. 13(4) (2020), 385–401.10.1515/acv-2018-0003Search in Google Scholar
Received: 2021-04-29
Accepted: 2021-12-07
Published Online: 2022-12-04
Published in Print: 2022-12-16

© 2022 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2022-0105
Keywords for this article
p(x)-Laplace operator; quasilinear equation; sub solution-super solution method; generalised Sobolev spaces; variational methods

Articles in the same Issue

  1. Prof. RNDr. Július korbaš, CSC. passed away
  2. Kalmbach measurability In d0-algebras
  3. Quantifiers on L-algebras
  4. Ideals of functions with compact support in the integer-valued case
  5. An algebraic study of the logic S5’(BL)
  6. Triangular numbers and generalized fibonacci polynomial
  7. A general matrix series inversion pair and associated polynomials
  8. Quantum ostrowski type inequalities for pre-invex functions
  9. Hermite-Hadamard type inequalities for interval-valued fractional integrals with respect to another function
  10. Approximating families for lattice outer measures on unsharp quantum logics
  11. On a generalized Lamé-Navier system in ℝ3
  12. Coercive and noncoercive elliptic problems with variable exponent Laplacian under Robin boundary conditions
  13. On some classical properties of normed spaces via generalized vector valued almost convergence
  14. Fan-Hemicontinuity for the gradient of the norm in Hilbert space
  15. Solvability of mixed problems for heat equations with two nonlocal conditions
  16. The global harnack estimates for a nonlinear heat equation with potential under finsler-geometric flow
  17. A note on set-star-K-Menger spaces
  18. A bivariate extension of the Omega distribution for two-dimensional proportional data
  19. Bernstein polynomials based iterative method for solving fractional integral equations
  20. The symmetric 4-Player gambler’s problem with unequal initial stakes
  21. Enveloping action: Convergence spaces
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