Local analytic solutions to some nonhomogeneous problems with p-Laplacian

Applying the Briot-Bouquet theorem we show that there exists an unique analytic solution to the equation ( tΦp (y ′) ) ′ +(−1)tΦq(y) = 0, on (0, a), where Φr(y) := |y| r−1 y, 0 < r, p, q ∈ R, i = 0, 1, 1 ≤ n ∈ N, a is a small positive real number. The initial conditions to be added to the equation are y(0) = A 6= 0, y′(0) = 0, for any real number A. We present a method how the solution can be expanded in a power series for near zero. 1 Preliminaries We consider the quasilinear differential equation ∆pu+ (−1) i |u| q−1 u = 0, u = u(x), x ∈ R, where n ≥ 1, p and q are positive real numbers, i = 0, 1 and ∆p denotes the p−Laplacian ( ∆pu = div(|∇u| p−1 ∇u) ) . If n = 1, then the equation is reduced to (Φp (y )) ′ + (−1)Φq(y) = 0, where for r ∈ {p, q} Φr(y) := { |y| r−1 y, for y ∈ R\ {0} 0, for y = 0. We note that function Φr is an odd function. For n > 1 we restrict our attention to radially symmetric solutions. The problem under consideration is reduced to ( tΦp (y ) ′ + (−1)tΦq(y) = 0, on (0, a) (1) This paper is in final form and no version of it is submitted for publication elsewhere. EJQTDE, Proc. 8th Coll. QTDE, 2008 No. 4, p. 1 where a > 0. A solution of (1) means a function y ∈ C (0, a) for which tΦp (y ) ∈ C (0, a) and (1) is satisfied. We shall consider the initial values y(0) = A 6= 0, y(0) = 0, (2) for any A ∈ R. For the existence and uniqueness of radial solutions to (1) we refer to [9]. If n = 1 and i = 0, then it was showed that the initial value problem (1) − (2) has a unique solution defined on the whole R (see [3], and [4]), moreover, its solution can be given in closed form in terms of incomplete gamma functions [4]. If n = 1, i = 0, Lindqvist gives some properties of the solutions [8]. If n = 1 and p = q = 1, then (1) is a linear differential equation, and its solutions are well-known: if i = 0, the solution (1)− (2) with A = 1 is the cosine function, if i = 1, the solution (1)− (2) with A = 1 is the hyperbolic cosine function, and both the cosine and hyperbolic cosine functions can be expanded in power series. In the linear case, when n = 2, p = q = 1, i = 0, the solution of (1)−(2) with A = 1 is J0(t), the Bessel function of first kind with zero order, and for n = 3, p = q = 1, i = 0 then the solution of (1) − (2) with A = 1 is j0(t) = sin t/t, called the spherical Bessel function of first kind with zero order. In the cases above, for special values of parameteres n, p, q, i, we know the solution in the form of power series. The type of singularities of (1) − (2) was classified in [1] in the case when i = 0, and p = q. If n = 1, then a solution of (1) is not singular. Our purpose is to show the existence of the solution of problem (1)− (2) in power series form near the origin. We intend to examine the local existence of an analytic solution to problem (1)− (2) and we give a constructive procedure for calculating solution y in power series near zero. Moreover we present some numerical experiments. 2 Existence of an unique solution We will consider a system of certain differential equations, namely, the special Briot-Bouquet differential equations. For this type of differential equations we refer to the book of E. Hille [6] and E. L. Ince [7]. Theorem 1 (Briot-Bouquet Theorem) Let us assume that for the system of equations ξ dz1 dξ = u1(ξ, z1(ξ), z2(ξ)), ξ dz2 dξ = u2(ξ, z1(ξ), z2(ξ)), } (3) where functions u1 and u2 are holomorphic functions of ξ, z1(ξ), and z2(ξ) near the origin, moreover u1(0, 0, 0) = u2(0, 0, 0) = 0, then a holomorphic solution EJQTDE, Proc. 8th Coll. QTDE, 2008 No. 4, p. 2 of (3) satisfying the initial conditions z1(0) = 0, z2(0) = 0 exists if none of the eigenvalues of the matrix