Radial Solutions for p -Laplacian Neumann Problems Involving Gradient Term Without Growth Restrictions
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Radial Solutions for p-Laplacian Neumann Problems Involving Gradient Term Without Growth Restrictions Minghe Pei1 · Libo Wang1
· Xuezhe Lv1
Received: 3 September 2020 / Revised: 21 October 2020 / Accepted: 30 October 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract We study the existence of radial solutions for the p-Laplacian Neumann problem with gradient term of the type
− p u = f (|x|, u, x · ∇u) in Ω, ∂u = 0 on ∂Ω, ∂n
where p u = div(|∇u| p−2 ∇u) is the p-Laplace operator with p > 1, Ω ⊂ R N (N ≥ 2) is a ball. We do not impose any growth restrictions on the nonlinearity. By using the topological transversality method together with the barrier strip technique, the existence of radial solutions to the above problem is obtained. Keywords p-Laplacian Neumann problem · Existence of radial solutions · Barrier strip technique · Topological transversality method Mathematics Subject Classification 35J92 · 35J62 · 34B15
Communicated by Maria Alessandra Ragusa. The project was sponsored by the Education Department of JiLin Province of P. R. China (JJKH20200029KJ).
B 1
Libo Wang [email protected] School of Mathematics and Statistics, Beihua University, JiLin City 132013, People’s Republic of China
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M. Pei et al.
1 Introduction In this paper, we study the existence of radial solutions to the p-Laplacian Neumann problem with gradient term of the form
− p u = f (|x|, u, x · ∇u) in Ω, ∂u = 0 on ∂Ω, ∂n
(1.1)
where p u = div(|∇u| p−2 ∇u) is the p-Laplace operator with p > 1, Ω = {x ∈ R N : |x| < R} with N ≥ 2, the function f : [0, R] × R2 → R is continuous, | · | indicates the Euclidean norm, and n is the outward unit normal vector of the boundary ∂Ω. The typical model equation is, for suitable a, b, g, − p u + b(|x|)x · ∇u + |u| p−2 u = a(|x|)g(u) in Ω,
(1.2)
where Ω = {x ∈ R N : |x| < R} and p > 1. This kind of equations with Neumann boundary conditions and p = 2 has been studied extensively via various methods in the literature. Particularly, for the case of p = 2 and b(·) ≡ 0, see Serra and Tilli [13], Bonheure et al. [4] and the references therein, for the case of p = 2 and b(·) ≡ 0, see Bonheure et al. [3], Ma et al. [8] and the references therein. However, Eq. (1.2) with Neumann boundary conditions and p = 2 does not seem to have been deeply investigated. The only result that we are aware of is that of Secchi [12] in case of p = 2 and b(·) ≡ 0. Up to now, we have not seen the solvability results of the radial solution of Eq. (1.2) with Neumann boundary conditions when p = 2 and b(·) ≡ 0. For other works concerned with Eq. (1.2) or more general equations on infinite domains, we refer the readers to Yin [16] or Zhang [17], and for the works concerned with Neumann problems involving gradient term, we refer the readers to Cianciaruso [5] and references therein. In addition, see [2,9– 11,14,15] and references therein for works concerned with more general equations driven by the ( p, q)-Laplace operator or fractional integral operator. Inspir
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