On the existence and multiplicity of positive solutions of the \(p\)-Laplacian separated boundary value problem. (English) Zbl 0940.35086

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 L. H. Erbe and H. Wang [Proc. Am. Math. Soc. 120, No. 3, 743-748 (1994; Zbl 0802.34018)] and L. H. Erbe, S. Hu and H. Wang [J. Math. Anal. Appl. 184, No. 3, 640-648 (1994; Zbl 0805.34021)].
Reviewer: J.Franců (Brno)


35J65 Nonlinear boundary value problems for linear elliptic equations
35J15 Second-order elliptic equations
35J60 Nonlinear elliptic equations