Positive solutions for concave-convex elliptic problems involving p ( x ) -Laplacian

Makkia Dammak; Abir Amor Ben Ali; Said Taarabti

Mathematica Bohemica (2022)

  • Volume: 147, Issue: 2, page 155-168
  • ISSN: 0862-7959

Abstract

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We study the existence and nonexistence of positive solutions of the nonlinear equation - Δ p ( x ) u = λ k ( x ) u q ± h ( x ) u r in Ω , u = 0 on ∂ Ω where Ω ⊂ ℝ N , N ≥ 2 , is a regular bounded open domain in ℝ N and the p ( x ) -Laplacian Δ p ( x ) u : = div ( | ∇ u | p ( x ) - 2 ∇ u ) is introduced for a continuous function p ( x ) > 1 defined on Ω . The positive parameter λ 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 ) ( Ω ) . When we prove the existence of minimal solution, we use the sub-super solutions method.

How to cite

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Dammak, Makkia, Amor Ben Ali, Abir, and Taarabti, Said. "Positive solutions for concave-convex elliptic problems involving $p(x)$-Laplacian." Mathematica Bohemica 147.2 (2022): 155-168. <http://eudml.org/doc/298236>.

@article{Dammak2022,
abstract = {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 $\Omega \subset \mathbb \{R\}^N$, $N\ge 2$, 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.},
author = {Dammak, Makkia, Amor Ben Ali, Abir, Taarabti, Said},
journal = {Mathematica Bohemica},
keywords = {variable exponent Sobolev space; $p(x)$-Laplace operator; concave-convex nonlinearities; variational method},
language = {eng},
number = {2},
pages = {155-168},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Positive solutions for concave-convex elliptic problems involving $p(x)$-Laplacian},
url = {http://eudml.org/doc/298236},
volume = {147},
year = {2022},
}

TY - JOUR
AU - Dammak, Makkia
AU - Amor Ben Ali, Abir
AU - Taarabti, Said
TI - Positive solutions for concave-convex elliptic problems involving $p(x)$-Laplacian
JO - Mathematica Bohemica
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 147
IS - 2
SP - 155
EP - 168
AB - 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 $\Omega \subset \mathbb {R}^N$, $N\ge 2$, 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.
LA - eng
KW - variable exponent Sobolev space; $p(x)$-Laplace operator; concave-convex nonlinearities; variational method
UR - http://eudml.org/doc/298236
ER -

References

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  1. Alves, C. O., Barreiro, J. L. P., 10.1016/j.jmaa.2013.02.025, J. Math. Anal. Appl. 403 (2013), 143-154. (2013) Zbl1283.35043MR3035079DOI10.1016/j.jmaa.2013.02.025
  2. Alves, C. O., Souto, M. A. S., 10.1007/3-7643-7401-2_2, Contributions to Nonlinear Analysis Progress in Nonlinear Differential Equations and their Applications 66. Birkhäuser, Basel (2006), 17-32. (2006) Zbl1193.35082MR2187792DOI10.1007/3-7643-7401-2_2
  3. Ambrosetti, A., Brézis, H., Cerami, G., 10.1006/jfan.1994.1078, J. Funct. Anal. 122 (1994), 519-543. (1994) Zbl0805.35028MR1276168DOI10.1006/jfan.1994.1078
  4. Antontsev, S. N., Shmarev, S. I., 10.1016/j.na.2004.09.026, Nonlinear Anal., Theory Methods Appl., Ser. A 60 (2005), 515-545. (2005) Zbl1066.35045MR2103951DOI10.1016/j.na.2004.09.026
  5. Chabrowski, J., Fu, Y., 10.1016/j.jmaa.2004.10.028, J. Math. Anal. Appl. 306 (2005), 604-618. (2005) Zbl1160.35399MR2136336DOI10.1016/j.jmaa.2004.10.028
  6. Chen, Y., Levine, S., Rao, M., 10.1137/050624522, SIAM J. Appl. Math. 66 (2006), 1383-1406. (2006) Zbl1102.49010MR2246061DOI10.1137/050624522
  7. Silva, J. P. P. Da, 10.1016/j.jmaa.2015.11.078, J. Math. Anal. Appl. 436 (2016), 782-795. (2016) Zbl1335.35082MR3446979DOI10.1016/j.jmaa.2015.11.078
  8. Edmunds, D. E., Rákosník, J., 10.1098/rspa.1992.0059, Proc. R. Soc. Lond., Ser. A 437 (1992), 229-236. (1992) Zbl0779.46027MR1177754DOI10.1098/rspa.1992.0059
  9. Edmunds, D. E., Rákosník, J., 10.4064/sm-143-3-267-293, Stud. Math. 143 (2000), 267-293. (2000) Zbl0974.46040MR1815935DOI10.4064/sm-143-3-267-293
  10. Fan, X., 10.1016/j.jmaa.2006.07.093, J. Math. Anal. Appl. 330 (2007), 665-682. (2007) Zbl1206.35103MR2302951DOI10.1016/j.jmaa.2006.07.093
  11. Fan, X., Shen, J., Zhao, D., 10.1006/jmaa.2001.7618, J. Math. Anal. Appl. 262 (2001), 749-760. (2001) Zbl0995.46023MR1859337DOI10.1006/jmaa.2001.7618
  12. Fan, X., Zhao, D., 10.1006/jmaa.2000.7617, J. Math. Anal. Appl. 263 (2001), 424-446. (2001) Zbl1028.46041MR1866056DOI10.1006/jmaa.2000.7617
  13. Kefi, K., 10.1090/S0002-9939-2011-10850-5, Proc. Am. Math. Soc. 139 (2011), 4351-4360. (2011) Zbl1237.35054MR2823080DOI10.1090/S0002-9939-2011-10850-5
  14. Kováčik, O., Rákosník, J., 10.21136/CMJ.1991.102493, Czech. Math. J. 41 (1991), 592-618. (1991) Zbl0784.46029MR1134951DOI10.21136/CMJ.1991.102493
  15. Marcos, A., Abdou, A., 10.1186/s13661-019-1276-z, Bound. Value Probl. 2019 (2019), Article ID 171, 21 pages. (2019) MR4025568DOI10.1186/s13661-019-1276-z
  16. Orlicz, W., 10.4064/sm-3-1-200-211, Stud. Math. 3 (1931), 200-211 German. (1931) Zbl0003.25203DOI10.4064/sm-3-1-200-211
  17. Rădulescu, V., Repovš, D., 10.1016/j.na.2011.01.037, Nonlinear Anal., Theory Methods Appl., Ser. A 75 (2012), 1524-1530. (2012) Zbl1237.35043MR2861354DOI10.1016/j.na.2011.01.037
  18. Růžička, M., 10.1007/BFb0104029, Lecture Notes in Mathematics 1748. Springer, Berlin (2000). (2000) Zbl0962.76001MR1810360DOI10.1007/BFb0104029
  19. Saoudi, K., 10.1155/2012/275748, Abstr. Appl. Anal. 2012 (2012), Article ID 275748, 9 pages. (2012) Zbl1250.35086MR2955036DOI10.1155/2012/275748
  20. Saoudi, K., 10.1080/17476933.2016.1219999, Complex Var. Elliptic Equ. 62 (2017), 318-332. (2017) Zbl06707978MR3598980DOI10.1080/17476933.2016.1219999
  21. Silva, A., 10.1515/ans-2011-0103, Adv. Nonlinear Stud. 11 (2011), 63-75. (2011) Zbl1226.35049MR2724542DOI10.1515/ans-2011-0103
  22. Takáč, P., Giacomoni, J., 10.1017/prm.2018.91, Proc. R. Soc. Edinb., Sect. A, Math. 150 (2020), 205-232. (2020) Zbl1436.35210MR4065080DOI10.1017/prm.2018.91
  23. Yücedağ, Z., 10.1515/anona-2015-0044, Adv. Nonlinear Anal. 4 (2015), 285-293. (2015) Zbl1328.35058MR3420320DOI10.1515/anona-2015-0044
  24. Zhikov, V. V., 10.1070/IM1987v029n01ABEH000958, Math. USSR, Izv. 29 (1987), 33-66 translation from Izv. Akad. Nauk SSSR, Ser. Mat. 50 1986 675-710. (1987) Zbl0599.49031MR0864171DOI10.1070/IM1987v029n01ABEH000958
  25. Zhikov, V. V., 10.1070/SM1993v076n02ABEH003421, Russian Acad. Sci. Sb. Math. 76 (1993), 427-459 translation from Mat. Sb. 183 1992 47-84. (1993) Zbl0767.35021MR1187249DOI10.1070/SM1993v076n02ABEH003421

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