Discrete Neumann boundary value problem for a nonlinear equation with singular ϕ-Laplacian

Abstract

Abstract Let I ⊂ R $I\subset\mathbb{R}$ be an open interval with 0 ∈ I $0\in I$ , and let g ∈ C 1 ( I , ( 0 , + ∞ ) ) $g\in C^{1}(I, (0,+\infty))$ . Let N ∈ N $N\in\mathbb{N}$ be an integer with N ≥ 4 $N\geq4$ , [ 2 , N − 1 ] Z : = { 2 , 3 , … , N − 1 } $[2, N-1]_{\mathbb{Z}}:=\{2, 3,\ldots,N-1\}$ . We are concerned with the existence of solutions for the discrete Neumann problem { ∇ ( k n − 1 △ v k 1 − ( △ v k ) 2 ) = n k n − 1 [ − g ′ ( ψ − 1 ( v k ) ) 1 − ( △ v k ) 2 + g ( ψ − 1 ( v k ) ) H ( ψ − 1 ( v k ) , k ) ] , k ∈ [ 2 , N − 1 ] Z , Δ v 1 = 0 = Δ v N − 1 $$\textstyle\begin{cases} \nabla(k^{n-1}\frac{\triangle v_{k}}{\sqrt{1-(\triangle v_{k})^{2}}} )=nk^{n-1}[-\frac{ g'(\psi^{-1}(v_{k}))}{\sqrt{1-(\triangle v_{k})^{2}}}+g(\psi^{-1}(v_{k}))H(\psi^{-1}(v_{k}),k)],\quad k\in[2, N-1]_{\mathbb{Z}},\\ \Delta v_{1}=0=\Delta v_{N-1} \end{cases} $$ which is a discrete analogue of the Neumann problem about the rotationally symmetric spacelike graphs with a prescribed mean curvature function in some Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes, where ψ ( s ) : = ∫ 0 s d t g ( t ) $\psi(s):=\int_{0}^{s}\frac{dt}{g(t)}$ , ψ − 1 $\psi ^{-1}$ is the inverse function of ψ, and H : R × [ 2 , N − 1 ] Z → R $H:\mathbb{R}\times[2, N-1]_{\mathbb{Z}}\to\mathbb{R}$ is continuous with respect to the first variable. The proofs of the main results are based upon the Brouwer degree theory.

Keywords

• prescribed mean curvature function
• singular ϕ-Laplacian
• existence
• Friedmann-Lemaître-Robertson-Walker spacetime
• Neumann problem
• Brouwer degree