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.
(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.Search 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.Search 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
© 2022 Mathematical Institute Slovak Academy of Sciences
