Существование слабых решений для $p(x)$-уравнения Лапласа через топологическую степень
Ключевые слова:
слабое решение, граничные условия Дирихле, пространство Соболева с переменной экспонентой, топологическая степень, $p(x)$-лапласиан
Аннотация
Мы рассматриваем уравнение Лапласа с $p(x)$-лапласианом с граничным условием Дирихле $$ \begin{cases} -\Delta_{p(x)}(u)+|u|^{p(x)-2}u= g(x,u,\nabla u), &x\in\Omega,\\ u=0, &x\in\partial\Omega. \end{cases} $$ Используя топологическую степень, предложенную Берковицем, мы доказываем, при соответствующих предположениях, существование слабых решений для этого уравнения.
Литература
1. Ait Hammou M., Azroul E. Nonlinear elliptic problems in weighted variable exponent Sobolev spaces by topological degree // Proyecciones. 2019. Vol. 38. No. 4. P. 733-751.
https://doi.org/10.22199/issn.0717-6279-2019-04-0048
2. Ait Hammou M., Azroul E. Nonlinear elliptic boundary value problems by topological degree // Recent Advances in Modeling, Analysis and Systems Control: Theoretical Aspects and Applications. Cham: Springer, 2020. P. 1-13.
https://doi.org/10.1007/978-3-030-26149-8_1
3. Ait Hammou M., Azroul E., Lahmi B. Existence of solutions for $p(x)$-Laplacian Dirichlet problem by topological degree // Bulletin of the Transilvania University of Braşov. Ser. III: Mathematics, Informatics, Physic. 2018. Vol. 11 (60). No. 2. P. 29-38.
4. Allaoui M. Continuous spectrum of Steklov nonhomogeneous elliptic problem // Opuscula Mathematica. 2015. Vol. 35. No. 6. P. 853-866.
https://doi.org/10.7494/OpMath.2015.35.6.853
5. Alsaedi R. Perturbed subcritical Dirichlet problems with variable exponents // Electronic Journal of Differential Equations. 2016. Vol. 2016. No. 295. P. 1-12.
6. Afrouzi G.A., Hadjian A., Heidarkhani S. Steklov problrms involving the $p(x)$-Laplacian // Electronic Journal of Differential Equations. 2014. Vol. 2014. No. 134. P. 1-11.
7. Antontsev S.N., Rodrigues J.F. On stationary thermo-rheological viscous flows // Annali dell' Università di Ferrara. Sez. VII. Scienze Matematiche. 2006. Vol. 52. P. 19-36.
https://doi.org/10.1007/s11565-006-0002-9
8. Berkovits J. Extension of the Leray-Schauder degree for abstract Hammerstein type mappings // Journal of Differential Equations. 2007. Vol. 234. Issue 1. P. 289-310.
https://doi.org/10.1016/j.jde.2006.11.012
9. Chen Y., Levine S., Rao R. Variable exponent, linear growth functionals in image restoration // SIAM Journal of Applied Mathematics. 2006. Vol. 66. No. 4. P. 1383-1406.
https://doi.org/10.1137/050624522
10. Deng S.-G. Positive solutions for Robin problem involving the $p(x)$-Laplacian // Journal of Mathematical Analysis and Applications. 2009. Vol. 360. Issue 2. P. 548-560.
https://doi.org/10.1016/j.jmaa.2009.06.032
11. Diening L., Harjulehto P., Hästö P., Ruzicka M. Lebesgue and Sobolev spaces with variable exponents. Berlin: Springer, 2011.
https://doi.org/10.1007/978-3-642-18363-8
12. Fan X., Han X. Existence and multiplicity of solutions for $p(x)$-Laplacian equations in $\mathbb{R}^N$ // Nonlinear Analysis: Theory, Methods and Applications. 2004. Vol. 59. Issues 1-2. P. 173-188.
https://doi.org/10.1016/j.na.2004.07.009
13. Fan X.-L., Zhang Q.-H. Existence of solutions for $p(x)$-Laplacian Dirichlet problem // Nonlinear Analysis: Theory, Methods and Applications. 2003. Vol. 52. Issue 8. P. 1843-1852.
https://doi.org/10.1016/S0362-546X(02)00150-5
14. Fan X., Zhao D. On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$ // Journal of Mathematical Analysis and Applications. 2001. Vol. 263. No. 2. P. 424-446.
https://doi.org/10.1006/jmaa.2000.7617
15. Ge G., Lv D.-J. Superlinear elliptic equations with variable exponent via perturbation method // Acta Applicandae Mathematicae. 2020. Vol. 166. P. 85-109.
https://doi.org/10.1007/s10440-019-00256-2
16. Iliaş P.S. Dirichlet problem with $p(x)$-Laplacian // Math. Reports. 2008. Vol. 10 (60). No. 1. P. 43-56.
17. Kováčik O., Rákosník J. On spaces $L^{p(x)}$ and $W^{1,p(x)}$ // Czechoslovak Mathematical Journal. 1991. Vol. 41. P. 592-618.
18. Liang Y., Wu X., Zhang Q., Zhao C. Multiple solutions of a $p(x)$-Laplacian equation involving critical nonlinearites // Taiwanese Journal of Mathematics. 2013. Vol. 17. No. 6. P. 2055-2082.
https://doi.org/10.11650/tjm.17.2013.3074
19. Marcos A., Abdou A. Existence of solutions for a nonhomogeneous Dirichlet problem involving $p(x)$-Laplacian operator and indefinite weight // Boundary Value Problems. 2019. Article number: 171.
https://doi.org/10.1186/s13661-019-1276-z
20. Nhan L.C., Chuong Q.V., Truong L.X. Potential well method for $p(x)$-Laplacian equations with variable exponent sources // Nonlinear Analysis: Real World Applications. 2020. Vol. 56. 103155.
https://doi.org/10.1016/j.nonrwa.2020.103155
21. Růžička M. Electrorheological fluids: modeling and mathematical theory. Berlin: Springer, 2000.
https://doi.org/10.1007/BFb0104029
22. Stanway R., Sproston J.L., El-Wahed A.K. Applications of electro-rheological fluids in vibration control: a survey // Smart Materials and Structures. 1996. Vol. 5. No. 4. P. 464-482.
https://doi.org/10.1088/0964-1726/5/4/011
23. Wang B.-S., Hou G.-L., Ge B. Existence and uniqueness of solutions for the $p(x)$-Laplacian equation with convection term // Mathematics. 2020. Vol. 8. No. 10. Article number: 1768.
https://doi.org/10.3390/math8101768
24. Xie W., Chen H. Existence and multiplicity of solutions for $p(x)$-Laplacian equations in $\mathbb{R}^N$ // Mathematische Nachrichten. 2018. Vol. 291. No. 16. P. 2476-2488.
https://doi.org/10.1002/mana.201700059
25. Fan X., Zhang Q., Zhao D. Eigenvalues of $p(x)$-Laplacian Dirichlet problem // Journal of Mathematical Analysis and Applications. 2005. Vol. 302. Issue 2. P. 306-317.
https://doi.org/10.1016/j.jmaa.2003.11.020
26. Zeidler E. Nonlinear functional analysis and its applications. II/B: Nonlinear monotone operators. New York: Springer, 1985.
https://doi.org/10.1007/978-1-4612-0981-2
27. Zhikov V.V. Averaging of functionals of the calculus of variations and elasticity theory // Mathematics of the USSR - Izvestiya. 1987. Vol. 29. No. 1. P. 33-66.
https://doi.org/10.1070/IM1987v029n01ABEH000958
https://doi.org/10.22199/issn.0717-6279-2019-04-0048
2. Ait Hammou M., Azroul E. Nonlinear elliptic boundary value problems by topological degree // Recent Advances in Modeling, Analysis and Systems Control: Theoretical Aspects and Applications. Cham: Springer, 2020. P. 1-13.
https://doi.org/10.1007/978-3-030-26149-8_1
3. Ait Hammou M., Azroul E., Lahmi B. Existence of solutions for $p(x)$-Laplacian Dirichlet problem by topological degree // Bulletin of the Transilvania University of Braşov. Ser. III: Mathematics, Informatics, Physic. 2018. Vol. 11 (60). No. 2. P. 29-38.
4. Allaoui M. Continuous spectrum of Steklov nonhomogeneous elliptic problem // Opuscula Mathematica. 2015. Vol. 35. No. 6. P. 853-866.
https://doi.org/10.7494/OpMath.2015.35.6.853
5. Alsaedi R. Perturbed subcritical Dirichlet problems with variable exponents // Electronic Journal of Differential Equations. 2016. Vol. 2016. No. 295. P. 1-12.
6. Afrouzi G.A., Hadjian A., Heidarkhani S. Steklov problrms involving the $p(x)$-Laplacian // Electronic Journal of Differential Equations. 2014. Vol. 2014. No. 134. P. 1-11.
7. Antontsev S.N., Rodrigues J.F. On stationary thermo-rheological viscous flows // Annali dell' Università di Ferrara. Sez. VII. Scienze Matematiche. 2006. Vol. 52. P. 19-36.
https://doi.org/10.1007/s11565-006-0002-9
8. Berkovits J. Extension of the Leray-Schauder degree for abstract Hammerstein type mappings // Journal of Differential Equations. 2007. Vol. 234. Issue 1. P. 289-310.
https://doi.org/10.1016/j.jde.2006.11.012
9. Chen Y., Levine S., Rao R. Variable exponent, linear growth functionals in image restoration // SIAM Journal of Applied Mathematics. 2006. Vol. 66. No. 4. P. 1383-1406.
https://doi.org/10.1137/050624522
10. Deng S.-G. Positive solutions for Robin problem involving the $p(x)$-Laplacian // Journal of Mathematical Analysis and Applications. 2009. Vol. 360. Issue 2. P. 548-560.
https://doi.org/10.1016/j.jmaa.2009.06.032
11. Diening L., Harjulehto P., Hästö P., Ruzicka M. Lebesgue and Sobolev spaces with variable exponents. Berlin: Springer, 2011.
https://doi.org/10.1007/978-3-642-18363-8
12. Fan X., Han X. Existence and multiplicity of solutions for $p(x)$-Laplacian equations in $\mathbb{R}^N$ // Nonlinear Analysis: Theory, Methods and Applications. 2004. Vol. 59. Issues 1-2. P. 173-188.
https://doi.org/10.1016/j.na.2004.07.009
13. Fan X.-L., Zhang Q.-H. Existence of solutions for $p(x)$-Laplacian Dirichlet problem // Nonlinear Analysis: Theory, Methods and Applications. 2003. Vol. 52. Issue 8. P. 1843-1852.
https://doi.org/10.1016/S0362-546X(02)00150-5
14. Fan X., Zhao D. On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$ // Journal of Mathematical Analysis and Applications. 2001. Vol. 263. No. 2. P. 424-446.
https://doi.org/10.1006/jmaa.2000.7617
15. Ge G., Lv D.-J. Superlinear elliptic equations with variable exponent via perturbation method // Acta Applicandae Mathematicae. 2020. Vol. 166. P. 85-109.
https://doi.org/10.1007/s10440-019-00256-2
16. Iliaş P.S. Dirichlet problem with $p(x)$-Laplacian // Math. Reports. 2008. Vol. 10 (60). No. 1. P. 43-56.
17. Kováčik O., Rákosník J. On spaces $L^{p(x)}$ and $W^{1,p(x)}$ // Czechoslovak Mathematical Journal. 1991. Vol. 41. P. 592-618.
18. Liang Y., Wu X., Zhang Q., Zhao C. Multiple solutions of a $p(x)$-Laplacian equation involving critical nonlinearites // Taiwanese Journal of Mathematics. 2013. Vol. 17. No. 6. P. 2055-2082.
https://doi.org/10.11650/tjm.17.2013.3074
19. Marcos A., Abdou A. Existence of solutions for a nonhomogeneous Dirichlet problem involving $p(x)$-Laplacian operator and indefinite weight // Boundary Value Problems. 2019. Article number: 171.
https://doi.org/10.1186/s13661-019-1276-z
20. Nhan L.C., Chuong Q.V., Truong L.X. Potential well method for $p(x)$-Laplacian equations with variable exponent sources // Nonlinear Analysis: Real World Applications. 2020. Vol. 56. 103155.
https://doi.org/10.1016/j.nonrwa.2020.103155
21. Růžička M. Electrorheological fluids: modeling and mathematical theory. Berlin: Springer, 2000.
https://doi.org/10.1007/BFb0104029
22. Stanway R., Sproston J.L., El-Wahed A.K. Applications of electro-rheological fluids in vibration control: a survey // Smart Materials and Structures. 1996. Vol. 5. No. 4. P. 464-482.
https://doi.org/10.1088/0964-1726/5/4/011
23. Wang B.-S., Hou G.-L., Ge B. Existence and uniqueness of solutions for the $p(x)$-Laplacian equation with convection term // Mathematics. 2020. Vol. 8. No. 10. Article number: 1768.
https://doi.org/10.3390/math8101768
24. Xie W., Chen H. Existence and multiplicity of solutions for $p(x)$-Laplacian equations in $\mathbb{R}^N$ // Mathematische Nachrichten. 2018. Vol. 291. No. 16. P. 2476-2488.
https://doi.org/10.1002/mana.201700059
25. Fan X., Zhang Q., Zhao D. Eigenvalues of $p(x)$-Laplacian Dirichlet problem // Journal of Mathematical Analysis and Applications. 2005. Vol. 302. Issue 2. P. 306-317.
https://doi.org/10.1016/j.jmaa.2003.11.020
26. Zeidler E. Nonlinear functional analysis and its applications. II/B: Nonlinear monotone operators. New York: Springer, 1985.
https://doi.org/10.1007/978-1-4612-0981-2
27. Zhikov V.V. Averaging of functionals of the calculus of variations and elasticity theory // Mathematics of the USSR - Izvestiya. 1987. Vol. 29. No. 1. P. 33-66.
https://doi.org/10.1070/IM1987v029n01ABEH000958
Поступила в редакцию
2021-12-28
Опубликована 2022-05-20
Опубликована 2022-05-20