On the existence and multiplicity of positive solutions of the $p$-Laplacian separated boundary value problem

Abstract

Using the lower and upper solutions method together with degree theory, we study the existence and multiplicity of positive solutions for the problem $$ (\varphi_{p}(u'))'+f(t,u)=0,\ \ a_{1}\varphi_{p}(u(a))-a_{2} \varphi_{p}(u'(a))=0,\ b_{1}\varphi_{p}(u(b))+b_{2} \varphi_{p}(u'(b))=0, $$ where $\varphi_{p} (s):=|s|^{p-2}s, \,p>1$, $a_1,b_1\in\Bbb R$, $a_2,b_2\in\Bbb R^+$, $a_1^2+a_2^2>0$, $b_1^2+b_2^2>0.$ The function $f$ satisfies assumptions related to the classically called sublinear, superlinear, subsuperlinear, or supersublinear cases. Our results improve the recent ones of L.H. Erbe-H. Wang ([21]) and L.H. Erbe-S. Hu-H. Wang ([20]).