## 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 $$s\neq0$$.

### MSC:

 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
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### References:

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