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Positive Radial Solutions for Elliptic Equations with Nonlinear Gradient Terms on the Unit Ball Yongxiang Li Abstract. This paper deals with the existence of positive radial solutions of the elliptic equation with nonlinear gradient term  −Δu = f (|x|, u, |∇u|) , x ∈ Ω, u|∂Ω = 0 , where Ω = {x ∈ RN : |x| < 1}, N ≥ 2, f : [0, 1] × R+ × R+ → R are continuous, R+ = [0, ∞). Under some inequality conditions, the existence results of positive radial solution are obtained. The proofs of the main results are based on the method of lower and upper solutions and truncating function technique. Mathematic Subject Classification. 35J25, 35J60, 47H11, 47N20. Keywords. Elliptic equation, nonlinear gradient term, positive radial solution, lower and upper solutions.

1. Introduction In this paper, we discuss the existence of positive radial solution for the elliptic boundary value problem (BVP) with nonlinear gradient term  −Δu = f (|x|, u, |∇u|) , x ∈ Ω, (1.1) u|∂Ω = 0 , on the unit ball Ω = {x ∈ RN : |x| < 1} in RN , where N ≥ 2, f : I × R+ × R+ → R is a nonlinear function, I = [0, 1]. For the special case of BVP (1.1) that f does not contain gradient term, namely the simply elliptic boundary problem  −Δu = f (|x|, u) , x ∈ Ω, (1.2) u|∂Ω = 0 , Research was supported by NNSFs of China (11661071, 11761063).

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the existence of radial solutions has been considered by many authors with different methods and techniques; see [1–4] and references therein. When Ω is an annulus or exterior domain, the existence of radial solutions has also been widely discussed; for the annulus see [5–7], and the exterior domain see [8–11] and references therein. For the annulus and exterior domain, one well-known result is when f (r, ξ) is nonnegative, and superlinear growth on ξ at origin and infinity BVP (1.2) has a positive radial solution [8,11]. But this result is not true for the ball. Grossi [3] has pointed out, when Ω is the unit ball and N +2 p≥ N −2 (supercritical growth case), the elliptic boundary value problem  x ∈ Ω, −Δu = |u|p , (1.3) u|∂Ω = 0 has no positive solution. Corresponding to BVP (1.2), f (r, ξ) = |ξ|p is superlinear growth on ξ at origin and infinity. By [4, Theorem 1], when Ω is an annulus, BVP (1.3) has at least one positive radial solution. In general, the existence of positive radial solutions to elliptic boundary value problems on the ball is more complicated than one on the annulus and exterior domain. The elliptic boundary value problems with general gradient term arise in many different areas of applied mathematics, and the existence of the solution is considered by several authors [12–15]. The purpose of this paper is to obtain existence results of positive radial solutions for BVP (1.1). Because of the influence of the gradient term, BVP (1.1) is more difficult than BVP (1.2). For the case that Ω is an annulus or exterior domain, the authors of references [12,13] recently extended the existence results of positive radial solution for BVP (1.2) in [8,11] to BVP (1.1). T