Weak solutions for double phase problem driven by the (p(x),q(x))-Laplacian operator under Dirichlet boundary conditions
DOI:
https://doi.org/10.5269/bspm.62182Resumen
In the present paper, in view of the topological degree methods and the theory of the variable exponent Sobolev spaces, we discuss a Dirichlet boundary value problem for elliptic equations involving the $(p(x),q(x))$-Laplacian operator with a reaction term depending on the gradient and on two real parameters. Under certain assumptions, we establish the existence of at least one weak solution to this problem. Our results extends some recent work in the literature.
Referencias
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22. M. A. Ragusa, A. Tachikawa, Regularity for minimizers for functionals of double phase with variable exponents, Adv. Nonlinear Anal. 9, 710-728 (2020). https://doi.org/10.1515/anona-2020-0022
23. M. A. Ragusa, A. Tachikawa, Boundary regularity of minimizers of p(x)−energy functionals, Annales de L'Institut Poincar'e Analyse non lineaire. 34 (2017), no. 6, 1633-1637. https://doi.org/10.1016/j.anihpc.2017.09.004
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25. M. Ruzicka, Electrorheological fuids: modeling and mathematical theory, Springer Science & Business Media, 2000. https://doi.org/10.1007/BFb0104030
26. B. S. Wang, G. L. Hou, B. Ge, Existence of solutions for double-phase problems by topological degree, Journal of Fixed Point Theory and Applications. 23 (2021), no. 1, 1-10. https://doi.org/10.1007/s11784-021-00847-3
27. E. Zeidler, Nonlinear Functional Analysis and its Applications II/B, Springer-Verlag, New York, 1990. https://doi.org/10.1007/978-1-4612-0981-2
28. V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 50 (1986), no. 4, 675-710.
29. V. V. Zhikov, On variational problems and nonlinear elliptic equations with nonstandard growth conditions, J. Math. Sci. 173, 463-570 (2011). https://doi.org/10.1007/s10958-011-0260-7
30. V. V. Zhikov, On some variational problems Russ. J. Math. Phys. 5 (1997), 105-116.
31. V. V. Zhikov, S. M. Kozlov, O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer, Berlin, 1994.
32. V. V. Zhikov, On Lavrentiev's Phenomenon, Russ. J. Math. Phys. 3 (1995), 249-269.
2. E. Acerbi and G. Mingione, Regularity results for a class of functionals with nonstandard growth, Arch. Ration. Mech. Anal. 156 (2001), 121-140. https://doi.org/10.1007/s002050100117
3. R. Alsaedi, Perturbed subcritical Dirichlet problems with variable exponents, Elec. J. of Dif. Equ. 295 (2016), 1-12.
4. S. Antontsev, S. Shmarev, A model porous medium equation with variable exponent of nonlinearity: Existence, uniqueness and localization properties of solutions, Nonlinear Anal. 60 (2005), 515-545. https://doi.org/10.1016/j.na.2004.09.026
5. A. Bahrouni, V. D. Radulescu, D. D. Repoves, Double phase transonic flow problems with variable growth: Nonlinear patterns and stationary waves, Nonlinearity 32 (2019), no. 7, 2481-2495. https://doi.org/10.1088/1361-6544/ab0b03
6. V. Benci, P. D'Avenia, D. Fortunato, L. Pisani, Solitons in several space dimensions: Derrick's problem and infinitely many solutions, Arch. Ration. Mech. Anal. 154 (2000), no. 4, 297-324. https://doi.org/10.1007/s002050000101
7. J. Berkovits, Extension of the Leray-Schauder degree for abstract Hammerstein type mappings, J Differ. Equ. 234 (2007), 289-310. https://doi.org/10.1016/j.jde.2006.11.012
8. Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), 1383-1406. https://doi.org/10.1137/050624522
9. L. Cherfils, Y. Il'yasov, On the stationary solutions of generalized reaction diffusion equations with p&q-Laplacian, Commun. Pure Appl. Anal. 4 (2005), 9-22. https://doi.org/10.3934/cpaa.2005.4.9
10. X. L. Fan, D. Zhao, On the Spaces Lp(x) (Ω) and W m,p(x) (Ω), J. Math. Anal, Appl. 263 (2001) 424-446. https://doi.org/10.1006/jmaa.2000.7617
11. X. L. Fan, Q. H. Zhang, Existence of solutions for p(x)-laplacian Dirichlet problem, Nonlinear Anal. 52 (2003), 1843-1852. https://doi.org/10.1016/S0362-546X(02)00150-5
12. I. S. Kim, S. J. Hong, A topological degree for operators of generalized (S+) type, Fixed Point Theory and Appl. 1 (2015), 1-16. https://doi.org/10.1186/s13663-015-0445-8
13. O. Kovacik and J. R'akosn'ık, On spaces Lp(x) and W1,p(x) , Czechoslovak Math. J. 41 (1991), no. 4, 592-618. https://doi.org/10.21136/CMJ.1991.102493
14. W. L. Liu, G. W. Dai, Existence and multiplicity results for double phase problem, J Differ Equ. 265 (2018), 4311-4334. https://doi.org/10.1016/j.jde.2018.06.006
15. P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions, Arch. Ration. Mech. Anal. 105 (1989), 267-284. https://doi.org/10.1007/BF00251503
16. M. E. Ouaarabi, A. Abbassi, C. Allalou, Existence result for a Dirichlet problem governed by nonlinear degenerate elliptic equation in weighted Sobolev spaces, J. Elliptic Parabol Equ. 7 (2021), no. 1, 221-242. https://doi.org/10.1007/s41808-021-00102-3
17. M. E. Ouaarabi, C. Allalou, A. Abbassi, On the Dirichlet Problem for some Nonlinear Degenerated Elliptic Equations with Weight, 2021 7th International Conference on Optimization and Applications (ICOA) (2021), 1-6. https://doi.org/10.1109/ICOA51614.2021.9442620
18. M. E. Ouaarabi, A. Abbassi, C. Allalou, Existence Result for a General Nonlinear Degenerate Elliptic Problems with Measure Datum in Weighted Sobolev Spaces, International Journal On Optimization and Applications. 1 (2021), no. 2, 1-9. https://doi.org/10.1007/s41808-021-00102-3
19. M. E. Ouaarabi, A. Abbassi, C. Allalou, Existence and uniqueness of weak solution in weighted Sobolev spaces for a class of nonlinear degenerate elliptic problems with measure data, International Journal of Nonlinear Analysis and Applications. 13 (2021), no. 2, 2635-2653.
20. N. S. Papageorgiou, D. D. Repoveˇs, C. Vetro, Positive solutions for singular double phase problems, J. Math. Anal. Appl. 501 (2020), 123-896. https://doi.org/10.1016/j.jmaa.2020.123896
21. N. S. Papageorgiou, V. D. Radulescu, D. D. Repoves, Ground state and nodal solutions for a class of double phase problems, Zeitschrift fur angewandte Mathematik und Physik. 71 (2020), 1-15. https://doi.org/10.1007/s00033-019-1224-x
22. M. A. Ragusa, A. Tachikawa, Regularity for minimizers for functionals of double phase with variable exponents, Adv. Nonlinear Anal. 9, 710-728 (2020). https://doi.org/10.1515/anona-2020-0022
23. M. A. Ragusa, A. Tachikawa, Boundary regularity of minimizers of p(x)−energy functionals, Annales de L'Institut Poincar'e Analyse non lineaire. 34 (2017), no. 6, 1633-1637. https://doi.org/10.1016/j.anihpc.2017.09.004
24. K. R. Rajagopal, M. R ˙uzicka, Mathematical modeling of electrorheological materials, Continuum mechanics and thermodynamics. 13 (2001), no. 1, 59-78. https://doi.org/10.1007/s001610100034
25. M. Ruzicka, Electrorheological fuids: modeling and mathematical theory, Springer Science & Business Media, 2000. https://doi.org/10.1007/BFb0104030
26. B. S. Wang, G. L. Hou, B. Ge, Existence of solutions for double-phase problems by topological degree, Journal of Fixed Point Theory and Applications. 23 (2021), no. 1, 1-10. https://doi.org/10.1007/s11784-021-00847-3
27. E. Zeidler, Nonlinear Functional Analysis and its Applications II/B, Springer-Verlag, New York, 1990. https://doi.org/10.1007/978-1-4612-0981-2
28. V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 50 (1986), no. 4, 675-710.
29. V. V. Zhikov, On variational problems and nonlinear elliptic equations with nonstandard growth conditions, J. Math. Sci. 173, 463-570 (2011). https://doi.org/10.1007/s10958-011-0260-7
30. V. V. Zhikov, On some variational problems Russ. J. Math. Phys. 5 (1997), 105-116.
31. V. V. Zhikov, S. M. Kozlov, O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer, Berlin, 1994.
32. V. V. Zhikov, On Lavrentiev's Phenomenon, Russ. J. Math. Phys. 3 (1995), 249-269.
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2022-12-27
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