A third-order p-laplacian boundary value problem on an unbounded domain

Abstract

In this work, we apply Leray-Schauder continuation principle to establish the existence of at least one solution to the third order p-Laplacian boundary value problem on an unbounded domain of the form \begin{equation*} (w(t) \varphi_{p}( u^{\prime\prime}(t)))^{\prime} = K ( t, u(t) , u^{\prime}(t), u^{\prime\prime}(t) ) , t \in ( 0, \infty) \end{equation*} \begin{equation*} u(0)= 0, \, u^{\prime} (0) = \sum^{m}_{i=1} \alpha _{i} \int_{0}^{\xi_{i}} u(t) dt, \, \lim_{t \rightarrow\infty} ( w(t)\varphi_{p} ( u^{\prime \prime} (t)) = 0 \end{equation*} under the nonresonant condition $ \sum_{i=1}^{m} \alpha_{i} \xi^{2} \neq 2. $