Radial solutions of p-Laplace equations with nonlinear gradient terms on exterior domains

Abstract

Abstract This paper studies the existence of radial solutions of the boundary value problem of p-Laplace equation with gradient term { − Δ p u = K ( | x | ) f ( | x | , u , | ∇ u | ) , x ∈ Ω , ∂ u ∂ n = 0 , x ∈ ∂ Ω , lim | x | → ∞ u ( x ) = 0 , $$ \textstyle\begin{cases} -\Delta_{p} u= K( \vert x \vert ) f( \vert x \vert , u, \vert \nabla u \vert ) ,\quad x\in\Omega , \\ \frac{\partial u}{\partial n}=0 ,\quad x\in\partial\Omega, \\ \lim_{ \vert x \vert \to\infty}u(x)=0 , \end{cases} $$ where Ω = { x ∈ R N : | x | > r 0 } $\Omega=\{x\in\mathbb{R}^{N}: |x|>r_{0}\}$ , N ≥ 3 $N\ge3$ , 1 < p ≤ 2 $1< p\le2$ , K : [ r 0 , ∞ ) → R + $K: [r_{0}, \infty)\to\mathbb{R}^{+}$ , and f : [ r 0 , ∞ ) × R × R + → R $f: [r_{0}, \infty)\times\mathbb{R}\times\mathbb{R}^{+}\to \mathbb{R}$ are continuous. Under certain inequality conditions of f, the existence results of radial solutions are obtained.

Keywords

• p-Laplace equation
• Radial solution
• Exterior domain
• Gradient term