DCDS
On S-shaped bifurcation curves for a two-point boundary value problem arising in a theory of thermal explosion
Shao-Yuan Huang Shin-Hwa Wang
We study the bifurcation curve and exact multiplicity of positive solutions of a two-point boundary value problem arising in a theory of thermal explosion \begin{equation*} \left\{ \begin{array}{l} u^{\prime\prime}(x) + \lambda \exp ( \frac{au}{a+u}) =0,     -1 < x < 1, \\ u(-1)=u(1)=0, \end{array} \right. \end{equation*} where $\lambda >0$ is the Frank--Kamenetskii parameter and $a>0$ is the activation energy parameter. By developing some new time-map techniques and applying Sturm's theorem, we prove that, if $a\geq a^{\ast \ast }\approx 4.107$, the bifurcation curve is S-shaped on the $(\lambda ,\Vert u \Vert _{\infty })$-plane. Our result improves one of the main results in Hung and Wang (J. Differential Equations 251 (2011) 223--237).
keywords: turning point Positive solution S-shaped bifurcation curve exact multiplicity Sturm's theorem.
CPAA
Exact multiplicity and bifurcation curves of positive solutions of a one-dimensional Minkowski-curvature problem and its application
Shao-Yuan Huang
In this paper, we discuss exact multiplicity and bifurcation curves of positive solutions of the one-dimensional Minkowski-curvature problem
$\begin{equation*}\left\{\begin{array}{l}\left[ -u^{\prime }/\sqrt{1-u^{\prime 2}}\right] ^{\prime }=\lambda f(u),\,\,\,\,\,\,-L < x < L, \\u(-L)=u(L)=0,\end{array}\right.\end{equation*} $
where $λ >0$ is a bifurcation parameter, $L>0$ is an evolution parameter,
$f∈ C[0, ∞)\cap C^{2}(0, ∞), $
$f(u)>0$
for
$u>0$
, and
$f^{\prime \prime }(u)$
is not sign-changing on
$\left( 0,\infty \right)$
.We find that if
$f^{\prime \prime }(u)≤q 0$
for
$u>0$
, the shapes of bifurcation curves are monotone increasing for $L>0$, and if
$f^{\prime \prime }(u)>0$
for
$u>0$
and
$f(u)$
satisfies some suitable hypotheses, the shapes of bifurcation curves has three possibilities. Furthermore, we study, in the
$(\lambda ,L,{\left\| u \right\|_\infty })$
-space, the shapes and structures of the bifurcation surfaces. Finally, we give an application for this problem with a nonlinear term
$f(u) = u^{p}+u^{q}$
where
$q≥p>0$
satisfy some conditions.
keywords: Minkowski-curvature bifurcation curve exact multiplicity $\subset $-shaped positive solution
DCDS
A global bifurcation theorem for a positone multiparameter problem and its application
Kuo-Chih Hung Shao-Yuan Huang Shin-Hwa Wang
We study the global bifurcation and exact multiplicity of positive solutions for the positone multiparameter problem
$\left\{ \begin{align} &{{u}^{\prime \prime }}(x)+\lambda {{f}_{\varepsilon }}(u)=0\text{,}\ \ -1<x<1\text{,} \\ &u(-1)=u(1)=0\text{,} \\ \end{align} \right.$
where λ > 0 is a bifurcation parameter and
$\varepsilon >0$
is an evolution parameter. Under some suitable hypotheses on
$f_{\varepsilon }(u)$
, we prove that there exists
$\tilde{\varepsilon}>0$
such that, on the
$(λ ,||u||_{∞ })$
-plane, the bifurcation curve is S-shaped for
$0<\varepsilon <\tilde{\varepsilon}$
and is monotone increasing for
$\varepsilon ≥q \tilde{\varepsilon}$
. We give an application for this problem with a class of polynomial nonlinearities
$f_{\varepsilon}(u)=-\varepsilon u^{p}+bu^{2}+cu+d\ $
of degree pq 3 and coefficients
$\varepsilon ,b,d>0,$
cq 0. Our results generalize those in Hung and Wang (Trans. Amer. Math. Soc. 365 (2013) 1933-1956.)
keywords: Global bifurcation multiparameter problem S-shaped bifurcation curve exact multiplicity positive solution

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