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Article

Dammak, Makkia ; Amor Ben Ali, Abir ; Taarabti, Said
Positive solutions for concave-convex elliptic problems involving $p(x)$-Laplacian. (English). Mathematica Bohemica, vol. 147 (2022), issue 2, pp. 155-168
MSC: 35J20, 35J60, 35J62, 35J70, 35K57 | MR 4407349 | Zbl 07547247 | DOI: 10.21136/MB.2021.0099-20
Keywords:
variable exponent Sobolev space; $p(x)$-Laplace operator; concave-convex nonlinearities; variational method
Summary:
We study the existence and nonexistence of positive solutions of the nonlinear equation $$ -\Delta _{p(x)} u = \lambda k(x) u^{q} \pm h(x) u^r\ \text {in}\ \Omega ,\quad u=0\ \text {on}\ \partial \Omega $$ where $\Omega \subset \mathbb {R}^N$, $N\geq 2$, is a regular bounded open domain in $\mathbb {R}^N$ and the $p(x)$-Laplacian $$ \Delta _{p(x)} u := \mbox {div}( |\nabla u|^{p(x)-2} \nabla u) $$ is introduced for a continuous function $p(x)>1$ defined on $\Omega $. The positive parameter $\lambda $ induces the bifurcation phenomena. The study of the equation (Q) needs generalized Lebesgue and Sobolev spaces. In this paper, under suitable assumptions, we show that some variational methods still work. We use them to prove the existence of positive solutions to the problem (Q) in $W_0^{1,p(x)}(\Omega )$. When we prove the existence of minimal solution, we use the sub-super solutions method.
References:
[1] Alves, C. O., Barreiro, J. L. P.: Existence and multiplicity of solutions for a $p(x)$-Laplacian equation with critical growth. J. Math. Anal. Appl. 403 (2013), 143-154. DOI 10.1016/j.jmaa.2013.02.025 | MR 3035079 | Zbl 1283.35043
[2] Alves, C. O., Souto, M. A. S.: Existence of solutions for a class of problems in $ \Bbb{R}^N $ involving the $p(x)$-Laplacian. Contributions to Nonlinear Analysis Progress in Nonlinear Differential Equations and their Applications 66. Birkhäuser, Basel (2006), 17-32. DOI 10.1007/3-7643-7401-2_2 | MR 2187792 | Zbl 1193.35082
[3] Ambrosetti, A., Brézis, H., Cerami, G.: Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal. 122 (1994), 519-543. DOI 10.1006/jfan.1994.1078 | MR 1276168 | Zbl 0805.35028
[4] Antontsev, S. N., Shmarev, S. I.: A model porous medium equation with variable exponent of nonlinearity: Existence, uniqueness and localisation properties of solutions. Nonlinear Anal., Theory Methods Appl., Ser. A 60 (2005), 515-545. DOI 10.1016/j.na.2004.09.026 | MR 2103951 | Zbl 1066.35045
[5] Chabrowski, J., Fu, Y.: Existence of solutions for $p(x)$-Laplacian problems on a bounded domain. J. Math. Anal. Appl. 306 (2005), 604-618. DOI 10.1016/j.jmaa.2004.10.028 | MR 2136336 | Zbl 1160.35399
[6] Chen, Y., Levine, S., Rao, M.: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66 (2006), 1383-1406. DOI 10.1137/050624522 | MR 2246061 | Zbl 1102.49010
[7] Silva, J. P. P. Da: On some multiple solutions for a $p(x)$-Laplacian equation with critical growth. J. Math. Anal. Appl. 436 (2016), 782-795. DOI 10.1016/j.jmaa.2015.11.078 | MR 3446979 | Zbl 1335.35082
[8] Edmunds, D. E., Rákosník, J.: Density of smooth functions in $W^{k,p(x)}(\Omega)$. Proc. R. Soc. Lond., Ser. A 437 (1992), 229-236. DOI 10.1098/rspa.1992.0059 | MR 1177754 | Zbl 0779.46027
[9] Edmunds, D. E., Rákosník, J.: Sobolev embedding with variable exponent. Stud. Math. 143 (2000), 267-293. DOI 10.4064/sm-143-3-267-293 | MR 1815935 | Zbl 0974.46040
[10] Fan, X.: On the sub-supersolution method for $p(x)$-Laplacian equations. J. Math. Anal. Appl. 330 (2007), 665-682. DOI 10.1016/j.jmaa.2006.07.093 | MR 2302951 | Zbl 1206.35103
[11] Fan, X., Shen, J., Zhao, D.: Sobolev embedding theorems for spaces $W^{k,p(x)}(\Omega)$. J. Math. Anal. Appl. 262 (2001), 749-760. DOI 10.1006/jmaa.2001.7618 | MR 1859337 | Zbl 0995.46023
[12] Fan, X., Zhao, D.: On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$. J. Math. Anal. Appl. 263 (2001), 424-446. DOI 10.1006/jmaa.2000.7617 | MR 1866056 | Zbl 1028.46041
[13] Kefi, K.: $p(x)$-Laplacian with indefinite weight. Proc. Am. Math. Soc. 139 (2011), 4351-4360. DOI 10.1090/S0002-9939-2011-10850-5 | MR 2823080 | Zbl 1237.35054
[14] Kováčik, O., Rákosník, J.: On spaces $L^{p(x)}$ and $W^{k,p(x)}$. Czech. Math. J. 41 (1991), 592-618. DOI 10.21136/CMJ.1991.102493 | MR 1134951 | Zbl 0784.46029
[15] Marcos, A., Abdou, A.: Existence of solutions for a nonhomogeneous Dirichlet problem involving $p(x)$-Laplacian operator and indefinite weight. Bound. Value Probl. 2019 (2019), Article ID 171, 21 pages. DOI 10.1186/s13661-019-1276-z | MR 4025568
[16] Orlicz, W.: Über konjugierte Exponentenfolgen. Stud. Math. 3 (1931), 200-211 German. DOI 10.4064/sm-3-1-200-211 | Zbl 0003.25203
[17] Rădulescu, V., Repovš, D.: Combined effects in nonlinear problems arising in the study of anisotropic continuous media. Nonlinear Anal., Theory Methods Appl., Ser. A 75 (2012), 1524-1530. DOI 10.1016/j.na.2011.01.037 | MR 2861354 | Zbl 1237.35043
[18] Růžička, M.: Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Mathematics 1748. Springer, Berlin (2000). DOI 10.1007/BFb0104029 | MR 1810360 | Zbl 0962.76001
[19] Saoudi, K.: Existence and nonexistence of positive solutions for quasilinear elliptic problem. Abstr. Appl. Anal. 2012 (2012), Article ID 275748, 9 pages. DOI 10.1155/2012/275748 | MR 2955036 | Zbl 1250.35086
[20] Saoudi, K.: Existence and multiplicity of solutions for a quasilinear equation involving the $p(x)$-Laplace operator. Complex Var. Elliptic Equ. 62 (2017), 318-332. DOI 10.1080/17476933.2016.1219999 | MR 3598980 | Zbl 06707978
[21] Silva, A.: Multiple solutions for the $p(x)$-Laplace operator with critical growth. Adv. Nonlinear Stud. 11 (2011), 63-75. DOI 10.1515/ans-2011-0103 | MR 2724542 | Zbl 1226.35049
[22] Takáč, P., Giacomoni, J.: A $p(x)$-Laplacian extension of the Díaz-Saa inequality and some applications. Proc. R. Soc. Edinb., Sect. A, Math. 150 (2020), 205-232. DOI 10.1017/prm.2018.91 | MR 4065080 | Zbl 1436.35210
[23] Yücedağ, Z.: Solutions of nonlinear problems involving $p(x)$-Laplacian operator. Adv. Nonlinear Anal. 4 (2015), 285-293. DOI 10.1515/anona-2015-0044 | MR 3420320 | Zbl 1328.35058
[24] Zhikov, V. V.: Averaging of functionals of the calculus of variations and elasticity theory. Math. USSR, Izv. 29 (1987), 33-66 translation from Izv. Akad. Nauk SSSR, Ser. Mat. 50 1986 675-710. DOI 10.1070/IM1987v029n01ABEH000958 | MR 0864171 | Zbl 0599.49031
[25] Zhikov, V. V.: On passage to the limit in nonlinear variational problems. Russian Acad. Sci. Sb. Math. 76 (1993), 427-459 translation from Mat. Sb. 183 1992 47-84. DOI 10.1070/SM1993v076n02ABEH003421 | MR 1187249 | Zbl 0767.35021
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