On the existence and multiplicity of positive solutions of the \(p\)-Laplacian separated boundary value problem
The paper deals with positive solutions of the problem \[ \begin{gathered} (\varphi _p(u'))' + f(t,u) = 0 \qquad t\in (a,b) ,\\ a_1\varphi _p(u(a))-a_2\varphi _p(u'(a))=0 , \qquad b_1\varphi _p(u(b))+b_2\varphi _p(u'(b))=0 , \end{gathered} \] where \(\varphi _p(s):=|s|^{p-2}s\) \ (\(p>1\), \(a_2,b_2\geq 0\)) and \(f(t,u)\) is an \(L^1\)-Carathéodory function. According to limits \(u\to 0\) and \(u\to \infty \) of \(f(t,u)/u\) taking their values \(0,\infty \) four different cases are distinguished: sublinear, superlinear, sub-superlinear and super-sublinear. In these cases using lower and upper solution techniques based on the degree theory existence and multiplicity of positive solutions are studied. The results generalize the results of \textit{L. H. Erbe} and \textit{H. Wang} [Proc. Am. Math. Soc. 120, No. 3, 743-748 (1994; Zbl 0802.34018)] and \textit{L. H. Erbe, S. Hu} and \textit{H. Wang} [J. Math. Anal. Appl. 184, No. 3, 640-648 (1994; Zbl 0805.34021)].
scientific article; zbMATH DE number 1450170