Existence of Solutions of Some Nonlinear $φ$-Laplacian Equations with Neumann-Steklov Nonlinear Boundary Conditions
Charles Etienne Goli, Assohoun Adjé · Project Euclid (Cornell University) · 2017
We study the existence of solutions of the quasilinear equation $$(D(u(t))\\phi(u'(t)))'=f(t,u(t),u'(t)),\\qquad a.e. \\;\\;t\\in [0,T],$$ subject to nonlinear Neumann-Steklov boundary conditions on $[0,T]$, where $\\phi: (-a,a)\\rightarrow \\mathbb{R}$ (for $0 < a < \\infty$) is an increasing homeomorphism such that $\\phi(0)=0$, $f:[0,T]\\times\\mathbb{R}^{2} \\rightarrow \\mathbb{R}$ a $L^1$-Carathéodory function, $D$ : $\\mathbb{R}\\longrightarrow (0,\\infty)$ is a continuous function. Using topological methods, we obtain existence and multiplicity results.