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Existence of Solutions of Some Nonlinear φ-Laplacian Equations with Neumann-Steklov Nonlinear Boundary Conditions

Volume 20, Number 2 (2017), 16 - 38

Existence of Solutions of Some Nonlinear φ-Laplacian Equations with Neumann-Steklov Nonlinear Boundary Conditions

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Abstract

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 $L^1$-Carath\'eodory function, $D$ : $\mathbb{R}\longrightarrow (0,\infty)$ is a continuous function. Using topological methods, we obtain existence and multiplicity results.