Higher-order p-Laplacian boundary value problem at resonance on an unbounded domain

Abstract

In this work, we employ the extension of Mawhin's coincidence degree by Ge and Ren to investigate the solvability of the p-Laplacian higher-order boundary value problems of the form $\begin{array}{c}{\left(w\left(t\right){\varphi }_p\left(x^{(n-1)}\left(t\right)\right)\right)}^{\text{'}}=h\left(t,x\left(t\right),\cdots ,x^{(n-1)}\left(t\right)\right),0<t<\infty ,\\ x^{(n-2)}\left(0\right)=\frac{(n-2)!}{{\eta }^{n-2}}x\left(\eta \right),x^{(n-1)}\left(0\right)=x^{\left(i\right)}\left(0\right)=0,i=\mathrm{1,2},\cdots ,n-3,n\ge 3,\end{array}$ $x^{(n-2)}\left(\infty \right)=\underset{0}{\overset{\eta }{\int }}{x}^{(n-2)}\left(s\right)dA\left(s\right),$ Where $\eta >0,h:\left[0,\infty \right)\times {ℝ}^n\to ℝ$ is a Caratheodory's function with $A\left(0\right)=0,A\left(\eta \right)=1,w\in C\left[0,\infty \right),w\left(t\right)>0$ for all $t\text{≥}0,{\text{ϕ}}_p\left(s\right)={\left|s\right|}^{p-2}s$ .

Keywords: Coincidence degree; Higher-order; Mathematics; Resonance; Unbounded domain; p-Laplacian.