Abstract
We consider the Dirichlet problem for the inhomogeneous p-Laplace equation with p nonlinear source. New sufficient conditions are established for the existence of weak bounded radially symmetric solutions as well as a priori estimates of solution and of the gradient of solution. We obtain an explicit formula that shows the dependence of the existence of these solutions on the dimension of the problem, the size of the domain, the exponent p, the nonlinear source, and the exterior mass forces.
Similar content being viewed by others
References
Franca M., “Radial ground states and singular ground states for a spatial-dependent p-Laplace equation,” J. Differ. Equ., 218, 2629–2656 (2010).
Franchi B., Lanconelli E., and Serrin J., “Existence and uniqueness of nonnegative solutions of quasilinear equations in Rn,” Adv. Math., 118, 177–243 (1996).
Garcia-Huidobro M., and Duvan H., “On the uniqueness of positive solutions of a quasilinear equation containing a weighted p-Laplacian, the superlinear case,” Commun. Contemp. Math., 10, No. 3, 405–432 (2008).
Benedict J. and Drabek P., “Asymptotics for the principal eigenvalue of the p-Laplacian on the ball as p approaches 1,” Nonlinear Anal., 93, 23–29 (2013).
Cabre X., Capella A., and Sanchon M., “Regularity of radial minimizers of reaction equations involving the p-Laplacian,” Calc. Var. Partial Differ. Equ., 34, No. 4, 475–494 (2009).
Castro A. and Lazer A., “Infinitely many radially-symmetric solutions to a superlinear Dirichlet problem in a ball,” Proc. Amer. Math. Soc., 101, No. 1, 57–64 (1987).
El Hashimi A. and de Thelin F., “Infinitely many radially-symmetric solutions for a quasilinear elliptic problem in a ball,” J. Differ. Equ., 128, 78–102 (1996).
Garcia-Azorero J., Peral I., and Puel J. P., “Quasilinear problems with exponential growth in the reaction term,” Nonlinear Anal., 22, 481–498 (1994).
Garcia-Huidobro M., Manasevich R., and Schmitt K., “Positive radial solutions of quasilinear elliptic partial differential equations on a ball,” Nonlinear Anal., 35, 175–190 (1999).
Gazzola F., Serrin J., and Tang M., “Existence of ground states and free boundary problem for quasilinear elliptic operators,” Adv. Differ. Equ., 5, 1–30 (2000).
Pucci P. and Serrin J., “Uniqueness of ground states for quasilinear elliptic operators,” Indiana Univ. Math. J., 47, 501–528 (1998).
Zhang Zh. and Li Zh., “A universal bound for radial solutions of the quasilinear parabolic equation with p-Laplace operator,” J. Math. Anal. Appl., 385, 125–134 (2012).
Dai Q. and Peng L., “Necessary and sufficient conditions for the existence of nonnegative solutions of inhomogeneous p-Laplace equation,” Acta Math. Scientia, 27, No. 1, 34–56 (2007).
Fan X., “Positive solutions to p(x)-Laplacian–Dirichlet problems with sigh-changing nonlinearities,” Math. Nachr., 284, No. 11–12, 1435–1445 (2011).
Huang Y. X., “Existence of positive solutions for a class of the p-Laplace equations,” J. Aust. Math. Soc., Ser. B, 36, No. 2, 249–264 (1994).
Tersenov Ar. S., “On sufficient conditions for the existence of radially symmetric solutions of the p-Laplace equation,” Nonlinear Anal., 95, 362–373 (2014).
Tersenov Al. S., “Space dimension can prevent the blow-up of solutions for parabolic problems,” Electron. J. Differ. Equ., 2007, No. 165, 1–6 (2007).
Liang Z., “The role of the space dimension on the blow-up for a reaction-diffusion equation,” Appl. Math., 2, No. 5, 575–578 (2011).
Tersenov Ar. S., “New a priori estimates of solutions to anisotropic elliptic equations,” Sib. Math. J., 53, No. 3, 539–551 (2012).
Author information
Authors and Affiliations
Corresponding author
Additional information
The author was partially supported by the Russian Foundation for Basic Research (Grant 15–01–08275).
Novosibirsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 57, No. 5, pp. 1171–1183, September–October, 2016; DOI: 10.17377/smzh.2016.57.521.
Rights and permissions
About this article
Cite this article
Tersenov, A.S. Existence of radially symmetric solutions of the inhomogeneous p-Laplace equation. Sib Math J 57, 918–928 (2016). https://doi.org/10.1134/S0037446616050219
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446616050219