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Siberian Mathematical Journal, Vol. 57, No. 5, pp. 918–928, 2016Original Russian Text Copyright c ? 2016 Tersenov Ar.S.EXISTENCE OF RADIALLY SYMMETRIC SOLUTIONSOF THE INHOMOGENEOUS p-LAPLACE EQUATIONAr. S. Tersenov UDC 517.9Abstract: We consider the Dirichlet problem for the inhomogeneous p-Laplace equation with p non-linear source. New suf f i cient conditions are established for the existence of weak bounded radiallysymmetric solutions as well as a priori estimates of solution and of the gradient of solution. We obtainan explicit formula that shows the dependence of the existence of these solutions on the dimension ofthe problem, the size of the domain, the exponent p, the nonlinear source, and the exterior mass forces.DOI: 10.1134/S0037446616050219Keywords: inhomogeneous p-Laplace equation with nonlinear source, radially symmetric solutions,a priori estimates§1. Introduction and the Main ResultsIn this paper we consider the Dirichlet boundary value problem for the inhomogeneous p-Laplaceequation with a nonlinear source and exterior mass forces−div(|∇u|p−2 ∇u) = c(|x|)g(u) + f(|x|)in B R ⊂R n , (1.1)u = 0 on ∂B R , (1.2)where B R is a ball of radius R, p > 1. Throughout the paper we suppose that g(u) : R ? → R is a continuousfunction, c and f belong to L ∞ , where f is not identically zero. Without loss of generality we assumethat g(0) = 0.We are interested in the existence of bounded radially symmetric solutions of (1.1), (1.2). It is wellknown that the radially symmetric solution satisf ies the equation−(|u ? |p−2 u ?) ? −n − 1r|u ? |p−2 u ? = c(r)g(u) + f(r)in r ∈(0,R) (1.3)and the boundary conditionsu ? (0) = 0, u(R) = 0, (1.4)where as usual r = |x|. There are by now many papers about the qualitative behavior of radial solutionsto (1.3), (1.4). The case that f ≡0 was investigated in [1,2], where many interesting results are obtainedon existence or nonexistence of the weak radially symmetric solutions to (1.3), (1.4) as well as the groundand singular ground state solutions to (1.3). In [3] there was studied the uniqueness of positive groundand singular ground state solutions to (1.3) as well as the uniqueness of positive radially symmetricsolutions of the Dirichlet problem in the ball. The case that f is identically zero was studied in [4–12]too. The opposite case was considered in [13–15]. Unlike the above-mentioned papers, here we obtainan explicit suf f i cient condition for the existence of a weak radial solution to (1.3), (1.4). This conditionshows the dependence of the existence of a weak solution on p, g, n, R, c 0 = ||c||L ∞ , and f 0 = ||f|| L ∞ .We also study the inf luence of the term f(x), which models the presence of the exterior mass forces, onthe existence of weak solutions to (1.3), (1.4).The author was partially supported by the Russian Foundation for Basic Research (Grant 15–01–08275).Novosibirsk. Translated from Sibirski? ? Matematicheski? ? Zhurnal, Vol. 57, No. 5, pp. 1171–1183, September–October,2016; DOI:10.17377/smzh.2016.57.521. Original article submitted November 11, 2015.918 0037-4466/16/5705–0918c ?