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

作者：Xu, Man; Ma, Ruyun^*

来源：Advances in Difference Equations, 2018, 2018(1): 9.

DOI：10.1186/s13662-017-1462-1

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

Let I subset of R be an open interval with 0 is an element of I, and let g is an element of C-1(I, (0,+infinity)). Let N is an element of N be an integer with N >= 4, [2, N - 1](Z) := {2, 3,..., N - 1}. We are concerned with the existence of solutions for the discrete Neumann problem @@@ {del(k(n-1) Delta v(k)/root 1-(Delta v(k))(2)) = nk(n-1)[- g'(psi(-1)(v(k)))/root 1-(Delta v(k))(2) + g(psi(-1)(v(k)))H(psi(-1)((v)k), k)], k is an element of [2,N - 1](Z), @@@ Delta v(1) = 0 = Delta v(N-1) @@@ which is a discrete analogue of the Neumann problem about the rotationally symmetric spacelike graphs with a prescribed mean curvature function in some Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetimes, where psi (s) := integral(s)(0) dt/g(t), psi(-1) is the inverse function of psi, and H : R x [2, N - 1](Z) -> R is continuous with respect to the first variable. The proofs of the main results are based upon the Brouwer degree theory.

• 出版日期2018-1-10
• 单位西北师范大学