Positive radial solutions of p-Laplace equations on exterior domains

Abstract: This paper deals with the existence of positive radial solutions of the p-Laplace equation  −∆p u = K(|x|) f (u) , x ∈ Ω , ∂u ∂n = 0 , x ∈ ∂Ω, lim|x|→∞ u(x) = 0 , where Ω = {x ∈ RN : |x| > r0}, N ≥ 2, 1 < p < N, K : [r0, ∞) → R is continuous and 0 < ∫ ∞ r0 rN−1K(r) dr < ∞, f ∈ C(R, R). Under the inequality conditions related to the asymptotic behaviour of f (u)/up−1 at 0 and infinity, the existence results of positive radial solutions are obtained. The discussion is based on the fixed point index theory in cones.


Introduction
Boundary value problems with p-Laplace operator ∆ p u = div(|∇u| p−2 ∇u) arise in many different areas of applied mathematics and physics, such as non-Newtonian fluids, reaction-diffusion problems, non-linear elasticity, etc. But little is known about the p-Laplace operator cases (p 2) compared to the vast amount of knowledge for the Laplace operator (p = 2). In this paper, we discuss the existence of positive radial solution for the p-Laplace boundary value problem (BVP) x ∈ ∂Ω, in the exterior domain Ω = {x ∈ R N : |x| > r 0 }, where N ≥ 2, r 0 > 0, 1 < p < N, ∂u ∂n is the outward normal derivative of u on ∂Ω, K : [r 0 , ∞) → R + is a coefficient function, f : R + → R is a nonlinear function. Throughout this paper, we assume that the following conditions hold: (A1) K ∈ C([r 0 , ∞), R + ) and 0 < ∞ r 0 r N−1 K(r) dr < ∞; (A2) f ∈ C(R + , R + ); For the special case of p = 2, namely the Laplace boundary value problem x ∈ ∂Ω, the existence of positive radial solutions has been discussed by many authors, see [1][2][3][4][5][6][7]. The authors of references [1][2][3][4][5][6] obtained some existence results by using upper and lower solutions method, priori estimates technique and fixed point index theory. In [7], the present author built an eigenvalue criteria of existing positive radial solutions. The eigenvalue criterion is related to the principle eigenvalue λ 1 of the corresponding radially symmetric Laplace eigenvalue problem (EVP) Specifically, if f satisfies one of the following eigenvalue conditions: (H1) f 0 < λ 1 , f ∞ > λ 1 ; (H2) f ∞ < λ 1 , f 0 > λ 1 , the BVP(1.2) has a classical positive radial solution, where See [7,Theorem 1.1]. This criterion first appeared in a boundary value problem of second-order ordinary differential equations, and built by Zhaoli Liu and Fuyi Li in [8]. Then it was extended to general boundary value problems of ordinary differential equations, See [9,10]. In [11,12], the radially symmetric solutions of the more general Hessian equations are discussed. The purpose of this paper is to establish a similar existence result of positive radial solution of BVP (1.1). Our results are related to the principle eigenvalue λ p,1 of the radially symmetric p-Laplce eigenvalue problem (EVP) (1.5) But now we can only prove a weaker version of it: In second inequality of (H1) and (H2), λ p,1 needs to be replaced by the larger number where a ∈ C + (0, 1] is given by (2.4) and Ψ ∈ C(R) is given by (2.7). Our result is as follows: Theorem 1.1. Suppose that Assumptions (A1) and (A2) hold. If the nonlinear function f satisfies one of the the following conditions: then BVP (1.1) has at least one classical positive radial solution.
As an example of the application of Theorem 1.1, we consider the following p-Laplace boundary value problem has a positive radial solution.
The proof of Theorem 1.1 is based on the fixed point index theory in cones, which will be given in Section 3. Some preliminaries to discuss BVP (1.1) are presented in Section 2.

Preliminaries
For the radially symmetric solution u = u(|x|) of BVP (1.1), setting r = |x|, since Let q > 1 be the constant satisfying 1 p + 1 q = 1. To solve BVP (2.1), make the variable transformations Then BVP (2.1) is converted to the ordinary differential equation BVP in (0, 1] 3) is a quasilinear ordinary differential equation boundary value problem with singularity at To discuss BVP (2.3), we first consider the corresponding simple boundary value problem where h ∈ C + (I) is a given function. Let then w = Φ(v) is a strictly monotone increasing continuous function on R and its inverse function is also a strictly monotone increasing continuous function.
Hence a ∈ L(I).
For every h ∈ C(I), we verify that is a unique solution of BVP (2.5). Since the function G(s) := 1 s a(τ)h(τ)dτ ∈ C(I), from (2.10) it follows that v ∈ C 1 (I) and Proof. Let h ∈ C + (I) and v = S h. By (2.10) and (2.11), for every t ∈ I v(t) ≥ 0 and v (t) ≥ 0. Hence, v(t) is a nonnegative monotone increasing function and v C = max t∈I v(t) = v(1). From (2.11) and the monotonicity of Ψ, we notice that v (t) is a monotone decreasing function on I. For every t ∈ (0, 1), by Lagrange's mean value theorem, there exist ξ 1 ∈ (0, t) and ξ 2 ∈ (t, 1), such that Hence Obviously, when t = 0 or 1, this inequality also holds. The proof is completed.
Proof. For the radially symmetric eigenvalue problem EVP (1.4), writing r = |x| and making the variable transformations of (2.2), it is converted to the one-dimensional weighted p-Laplace eigenvalue problem (EVP) where v(t) = u(r(t)). Clearly, λ ∈ R is an eigenvalue of EVP (1.4) if and only if it is an eigenvalue of EVP (2.12). By (2.4) and (2.9), a ∈ C + (0, 1] ∩ L(I) and 1 0 a(s)ds > 0. This guarantees that EVP (2.12) has a minimum positive real eigenvalue λ p,1 , which given by (2.14) By Lemma 2.2, S (C + (I)) ⊂ K. Let f ∈ C(R + , R + ), and define a mapping F : K → C + (I) by One important fact is that if i (A, K ∩ D, K) 0, then A has a fixed point in K ∩ D. In next section, we will use the following two lemmas in [16,17] to find the nonzero fixed point of the mapping A defined by (2.16).

Proof of the main result
Proof of Theorem 1.1. We only consider the case that (H1)* holds, and the case that (H2)* holds can be proved by a similar way.
Let K ⊂ C(I) be the closed convex cone defined by (2.14) and A : K → K be the completely continuous mapping defined by (2.16). If v ∈ K is a nontrivial fixed point of A, then by the definitions of S and A, v(t) is a positive solution of BVP (2.3) and u = v(r 0 N−2 /|x| N−2 ) is a classical positive radial solution of BVP (1.1). Let 0 < R 1 < R 2 < +∞ and set We prove that A has a fixed point in K ∩ (D 2 \ D 1 ) when R 1 is small enough and R 2 large enough. Since f p 0 < λ p,1 , by the definition of f p 0 , there exist ε ∈ (0, λ p,1 ) and δ > 0, such that Choosing R 1 ∈ (0, δ), we prove that A satisfies the condition of Lemma 2.4 in K ∩ ∂D 1 , namely Hence, v 0 ∈ C 1 (I) satisfies the differential equation Since v 0 ∈ K ∩ ∂D 1 , by the definitions of K and D 1 , Hence by (3.2), By this inequality and Eq (3.4), we have Multiplying this inequality by v 0 (t) and integrating on (0, 1], then using integration by parts for the left side, we have (3.5) Since v 0 ∈ K ∩ ∂D, by the definition of K, Hence, by (2.13) and (3.5) we obtain that which is a contradiction. This means that ( Choosing R 2 > max{δ, H/σ 0 }, we show that (3.10) For ∀ v ∈ K ∩ ∂D 2 and t ∈ [σ 0 , 1], by the definitions of K and D 2 v(t) ≥ t v C ≥ σ 0 R 2 > H.
The proof of Theorem 1.1 is complete.