环形区域上含非线性梯度项的p-Laplace方程的径向对称解

Radial Symmetric Solutions of p-Laplace Equations with Nonlinear Gradient Terms on Annular Domains

摘要

用Leray-Schauder不动点定理,讨论ℝ^N中环形区域Ω={x∈ℝ^N: r₁<|x|<r₂}上含非线性梯度项的p-Laplace方程边值问题

$\left\{\begin{array}{l}-\Delta_{p} u=f(|x|, u,|\nabla u|), \quad x \in \Omega, \\\left.u\right|_{\partial \Omega}=0\end{array}\right. $

径向对称解的存在性,其中p>1,f:[r1,r2]×ℝ×ℝ+→ℝ连续.当1＜p≤2时,在非线性项f满足p-Laplace算子-Δpu在边界条件u|∂Ω=0的第一特征值λp,1的最优不等式条件下,获得了其径向对称解的存在性结果.

Abstract

By using the Leray-Schauder fixed point theorem, we discuss the existence of radial symmetric solutions for the boundary value problem of the p-Laplace equation with nonlinear gradient term

$\left\{\begin{array}{l}-\Delta_{p} u=f(|x|, u,|\nabla u|), \quad x \in \Omega, \\\left.u\right|_{\partial \Omega}=0\end{array}\right. $

where Ω={x∈ℝ^N: r₁<|x|<r₂} is an annular domain in ℝ^N, p>1, f:[r₁,r₂]×ℝ×ℝ⁺→ℝ is continuous. When 1<p≤2, under the optimal inequality condition of the nonlinear term f which satisfy the first eigenvalue λ_p,1 of the p-Laplace operator -Δ_pu under the boundary condition u|_∂Ω=0, an existence result of radial symmetric solutions is obtained.

关键词

p-Laplace方程 / 环形区域 / 径向对称解 / 非线性梯度项 / Leray-Schauder不动点定理

Key words

p-Laplace equation / annular domain / radial symmetric solution / nonlinear gradient term / Leray-Schauder fixed point theorem

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基金资助

国家自然科学基金(12061062)