matmessМатематические заметки СВФУMathematical notes of NEFU2411-93262587-876XСеверо-Восточный федеральный университет имени М.К. Аммосова10.25587/2411-9326-2024-4-64-81matmess-138Research ArticleМАТЕМАТИКАО радиально-симметричных решениях третьей краевой задачи для эллиптического уравнения с p-лапласианомOn radially symmetric solutions of the third boundary value problem for a p-Laplace equationТерсеновАр. С.TersenovA. S.

Терсенов Арис Саввич 

пр. Коптюга, 4, Новосибирск 630090 

Aris S. Tersenov 

4 Acad. Koptyug Avenue, Novosibirsk 630090 

aterseno@math.nsc.ruСафаровР. Ч.SafarovR. Ch.

Сафаров Расул Чориёр Угли 

ул. Пирогова, 1, Новосибирск 630090 

ул. Кучабаг, 17, Карши 180119, Узбекистан 

Rasul Ch. Safarov 

1 Pirogov Street, Novosibirsk 630090 

17 Kuchabag Street, Karshi 180119 

r.safarov1@g.nsu.ruИнститут математики им. С. Л. Соболева СО РАНРоссияSobolev Institute of MathematicsRussian FederationНовосибирский государственный университет ; Каршинский государственный университетРоссияNovosibirsk State University ; Karshi State UniversityRussian Federation2024301220243146481Copyright © Терсенов А.С., Сафаров Р.Ч., 20242024Терсенов А.С., Сафаров Р.Ч.Tersenov A.S., Safarov R.C.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://matmess.elpub.ru/jour/article/view/138

Рассматривается третья краевая задача для уравнения с p-лапласианом с младшим членом, не удовлетворяющим условию Бернштейна — Нагумо. Исследуется разрешимость задачи в классе радиально-симметричных решений. Определен класс градиентных нелинейностей, для которого доказано существование слабого соболевского радиально-симметричного решения с производной, непрерывной по Гёльдеру с показателем 1 p−1 . Показано, что нелинейность по градиенту может быть произвольной при условии, что младший член, содержащий градиент, непрерывен по Липшицу по пространственной переменной и строго монотонен по переменной u. Решение исходной задачи аппроксимируется классическими решениями соответствующей регуляризованной задачи. Полученные для регуляризованной задачи априорные оценки не зависят от параметра регуляризации, что позволяет предельным переходом получить решение исходной задачи указанной гладкости.

We consider the third boundary value problem for a p-Laplace equation with a low-order term that does not satisfy the Bernstein–Nagumo condition. The solvability of the problem in the class of radially symmetric solutions is investigated. A class of gradient nonlinearities is defined, for which the existence of a weak Sobolev radially symmetric solution with a Hölder continuous derivative with exponent 1 p−1 is proven. It is shown that nonlinearity in the gradient can be arbitrary, provided that the low order term containing the gradient is Lipschitz continuous in the spatial variable and strictly monotone in the variable u. The solution to the original problem is approximated by classical solutions to the corresponding regularized problem. The a priori estimates obtained for the regularized problem do not depend on the regularization parameter, which allows us to obtain a solution to the original problem of the specified smoothness by passing to the limit.

уравнение с p-лапласианомусловие Бернштейна — Нагуморадиально-симметричные решенияаприорные оценкиp-Laplace equationBernstein–Nagumo conditionradially symmetric solutionsa priori estimatesReferencesDall’Aglio A., Giachetti D., Segura de Leon S. Global existence for parabolic problems involving the p-Laplacian and a critical gradient term // Indiana Univ. Math. J. 2009. V. 58, N 1. P. 1–48.Dall’Aglio A., Giachetti D., and Segura de Leon S., “Global existence for parabolic problems involving the p-Laplacian and a critical gradient term,” Indiana Univ. Math. J., 58, No. 1, 1–48 (2009).Dall’Aglio A., De Cicco V., Giachetti D., Puel J.-P. Existence of bounded solutions for nonlinear elliptic equations in unbounded domains // Nonlinear Differ. Equ. Appl. 2004. V. 11, N 4. P. 431–450.Dall’Aglio A., De Cicco V., Giachetti D., and Puel J.-P., “Existence of bounded solutions for nonlinear elliptic equations in unbounded domains,” Nonlinear Differ. Equ. Appl., 11, No. 4, 431–450 (2004).Nakao M., Chen C. Global existence and gradient estimates for the quasilinear parabolic equations of m-Laplacian type with a nonlinear convection term // J. Differ. Equ. 2000. V. 162, N 1. P. 224–250.Nakao M. and Chen C., “Global existence and gradient estimates for the quasilinear parabolic equations of m-Laplacian type with a nonlinear convection term,” J. Differ. Equ., 162, No. 1, 224–250 (2000).Sbai A., El Hadfi Y., Zeng S. Nonlinear singular elliptic equations of p-Laplace type with superlinear growth in the gradient // Mediterr. J. Math. 2023. V. 20. Article N 32. P. 1–20.Sbai A., El Hadfi Y., and Zeng S., “Nonlinear singular elliptic equations of p-Laplace type with superlinear growth in the gradient,” Mediterr. J. Math., 20, article No. 32, 1–20 (2023).Bueno H., Ercole G., Ferreira W. M., Zumpano A. Positive solutions for the p-Laplacian with dependence on the gradient // Nonlinearity. 2012. V. 25, N 4. P. 1211–1234.Bueno H., Ercole G., Ferreira W. M., and Zumpano A., “Positive solutions for the p-Laplacian with dependence on the gradient,” Nonlinearity, 25, No. 4, 1211–1234 (2012).Figueiredo D. G., Sanchez J., Ubilla P. Quasilinear equations with dependence on the gradient // Nonlinear Anal. 2009. V. 71. P. 4862–4868.Figueiredo D. G., Sanchez J., and Ubilla P., “Quasilinear equations with dependence on the gradient,” Nonlinear Anal., 71, 4862–4868 (2009).Iturriaga L., Lorca S., Sanchez J. Existence and multiplicity results for the p-Laplacian with a p-gradient term // Nonlinear Differ. Equ. Appl. 2008. V. 15. P. 729–743.Iturriaga L., Lorca S., and Sanchez J., “Existence and multiplicity results for the p-Laplacian with a p-gradient term,” Nonlinear Differ. Equ. Appl., 15, 729–743 (2008).Li J., Yin J.„ Ke Yu. Existence of positive solutions for the p-Laplacian with p-gradient term // J. Math. Anal. Appl. 2011. V. 383. P. 147–158.Li J., Yin J., and Ke Yu., “Existence of positive solutions for the p-Laplacian with p-gradient term,” J. Math. Anal. Appl., 383, 147–158 (2011).Ruiz D. A priori estimates and existence of positive solutions for strongly nonlinear problems // J. Differ. Equ. 2004. V. 199. P. 96–114.Ruiz D., “A priori estimates and existence of positive solutions for strongly nonlinear problems,” J. Differ. Equ., 199, 96–114 (2004).Zou H. H. A priori estimates and existence for quasilinear elliptic equations // Calc. Var. Partial Differ. Equ. 2008. V. 33. P. 417–437.Zou H. H., “A priori estimates and existence for quasilinear elliptic equations,” Calc. Var. Partial Differ. Equ., 33, 417–437 (2008).Dwivedi G., Gupta S. An existence result for p-Laplace equation with gradient nonlinearity in RN // Commun. Math. 2022. V. 30, N 1. P. 149–159.Dwivedi G. and Gupta S., “An existence result for p-Laplace equation with gradient nonlinearity in RN,” Commun. Math., 30, No. 1, 149–159 (2022).Medina M., Ochoa P. On viscosity and weak solutions for non-homogeneous p-Laplace equations // Adv. Nonlinear Anal. 2019. V. 8, N 1. P. 468–481.Medina M. and Ochoa P., “On viscosity and weak solutions for non-homogeneous p-Laplace equations,” Adv. Nonlinear Anal., 8, No. 1, 468–481 (2019).Leonori T., Porretta A., Riey G. Comparison principles for p-Laplace equations with lower order terms // Ann. Mat. Pura Appl. V. 2017. V. 196. P. 877–903.Leonori T., Porretta A., and Riey G., “Comparison principles for p-Laplace equations with lower order terms,” Ann. Mat. Pura Appl. V, 196, 877–903 (2017).Razani A. Game-theoretic p-Laplace operator involving the gradient // Miskolc Math. Notes. 2022. V. 23, N 2. P. 867–879.Razani A., “Game-theoretic p-Laplace operator involving the gradient,” Miskolc Math. Notes, 23, No. 2, 867–879 (2022).Bueno H., Ercole G. A quasilinear problem with fast growing gradient // Appl. Math. Let. 2013. V. 26, N 4. P. 520-523.Bueno H. and Ercole G., “A quasilinear problem with fast growing gradient,” Appl. Math. Lett., 26, No. 4, 520–523 (2013).Tersenov Ar. S. Influence of gradient terms on the existence of solutins to the Dirichlet problem for p-Laplacian // J. Math. Sci. 2018. V. 228, N 4. P. 463–474.Tersenov Ar. S., “Influence of gradient terms on the existence of solutins to the Dirichlet problem for p-Laplacian,” J. Math. Sci., 228, No. 4, 463–474 (2018).Tersenov Ar. S. Radially symmetric solutions of the p-Laplace equation with gradient terms // J. Appl. Ind. Math. 2018. V. 12, N 4. P. 770–784.Tersenov Ar. S., “Radially symmetric solutions of the p-Laplace equation with gradient terms,” J. Appl. Ind. Math., 12, No. 4, 770–784 (2018).Tersenov Ar. S. On the existence of radially symmetric solutios for the p-Laplace equation with strong gradient nonlinearities // Sib. Math. J. 2023. V. 64, N 6. P. 1443–1454.Tersenov Ar. S., “On the existence of radially symmetric solutios for the p-Laplace equation with strong gradient nonlinearities,” Sib. Math. J., 64, No. 6, 1443–1454 (2023).Alotaibi T., Hai D. D. Existence and nonexistence of positive radial solutions for a class of p-Laplacian superlinear problem with nonlinear boundary conditions // Commun. Pure Appl. Anal. 2020. V. 19, N 9. P. 4655–4666.Alotaibi T. and Hai D. D., “Existence and nonexistence of positive radial solutions for a class of p-Laplacian superlinear problems with nonlinear boundary conditions,” Commun. Pure Appl. Anal., 19, No. 9, 4655–4666 (2020).Franchi B., Lanconelli E., Serrin J. Existence and uniqueness of nonnegative solutions of quasilinear equations in Rn // Adv. Math. 1996. V. 118. P. 177–243.Franchi B., Lanconelli E., and Serrin J., “Existence and uniqueness of nonnegative solutions of quasilinear equations in Rn,” Adv. Math., 118, 177–243 (1996).Gilbarg D., Trudinger N. S. Elliptic partial differential equations of second order. 2nd ed. Berlin; Heidelberg: Springer, 1983.Gilbarg D. and Trudinger N. S., Elliptic Partial Differential Equations of Second Order, Springer, Berlin; Heidelberg (1983).

The authors declare that there are no conflicts of interest present.