Two positive solutions for one-dimensional \(p\)-Laplacian with a singular weight. (English) Zbl 1302.34035

Kajikiya, Ryuji; Lee, Yong-Hoon; Sim, Inbo

Topol. Methods Nonlinear Anal. 42, No. 2, 427-448 (2013).

From the introduction: We study one-dimensional \(p\)-Laplacian with a singular weight \[ \begin{aligned} \varphi_p(u'(t))''+\lambda h(t) f(u(t)) &= 0\qquad \text{a.e. in }(0,1),\\ u(0)= u(1) &= 0,\end{aligned} \] where \(\varphi_p(s)= |s|^{p-2} s\), \(p>1\), \(\lambda\) is a nonnegative parameter, \(h\) is a nonnegative measurable function on \((0,1)\), \(h\not\equiv 0\) on any open subinterval in \((0,1)\) which may be singular at \(t=0,1\) and \(f\in C(\mathbb{R},\mathbb{R})\).
Using super-subsolution method and mountain pass lemma, we prove the existence of at least two positive solutions, at least one positive solution and no positive solution according to the range of a bifurcation parameter.

MSC:

34B09	Boundary eigenvalue problems for ordinary differential equations
34B16	Singular nonlinear boundary value problems for ordinary differential equations
34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations
34C23	Bifurcation theory for ordinary differential equations

Keywords:

\(p\)-Laplace equation; bifurcation; super-subsolution; mountain pass lemma