Existence results of sign-changing solutions for singular one-dimensional \(p\)-Laplacian problems. (English) Zbl 1138.34010

Applying the global bifurcation theorem and deriving the shape of the unbounded subcontinua of solutions, the authors establish existence and multiplicity of sign-changing solutions for the boundary value problem
\[ (\varphi_p(u'(t)))'+h(t)f(u(t))=0,\;t\in(0,1), \]
\[ u(0)=0=u(1), \]
where \(\varphi_p(x)=| x| ^{p-2}x, p>1,\) \(h\) may by singular at \(t=0\) and/or \(t=1\) and \(f\in C(\mathbb R,\mathbb R)\). The cases
\[ f_0:=\lim_{| u| \to0}{f(u)\over \varphi_p(u)}=0\text{ and }f_0=\infty \]
are considered. In the first case the authors suppose in addition that
\[ f_\infty:=\lim_{| u| \to\infty}{f(u)\over \varphi_p(u)}=\infty,\quad h\in L^1(0,1),\;h\geq0\text{ for a.e. }t\in(0,1),\;\int_Ih(s)\,ds>0, \]
where \(I\) is any compact subinterval of \((0,1)\), and \(sf(s)>0\) for \(s\neq0\). Under these assumptions, they prove that for each \(k\in \mathbb N\), the considered problem has two solutions \(u^+_k\) and \(u^-_k\) such that \(u^+_k\) has exactly \(k-1\) zeros and is positive near \(t=0\), and \(u^-_k\) has exactly \(k-1\) zeros and is negative near \(t=0\). A similar result is obtained in the second case under the following assumptions:
\[ f_\infty=0,\quad h\in C^1(0,1)\cap L^1(0,1),\;h\geq0\text{ for a.e. }t\in(0,1), \]
\(\lim_{t\to0^{+}}th(t)\) and \(\lim_{t\to1^{-}}(1-t)h(t)\) exist and \(sf(s)>0\) for \(s\neq0\).


34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
Full Text: DOI


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