Existence results of positive solutions for Kirchhoff type equations via bifurcation methods

In this paper we address the following Kirchhoff type problem \begin{equation*} \left\{ \begin{array}{ll} -\Delta(g(|\nabla u|_2^2) u + u^r) = a u + b u^p& \mbox{in}~\Omega, u>0& \mbox{in}~\Omega, u= 0& \mbox{on}~\partial\Omega, \end{array} \right. \end{equation*} in a bounded and smooth domain $\Omega$ in ${\rm I}\hskip -0.85mm{\rm R}$. By using change of variables and bifurcation methods, we show, under suitable conditions on the parameters $a,b,p,r$ and the nonlinearity $g$, the existence of positive solutions.Comment: 18 pages, 1 figur

Life span assays in mutant <i>C. elegans</i> fed with SQDB and individual herbs.

<p>*P<0.005, Statistical significant; n.s. P>0.005 not significant.</p

Effect of <i>Cinnamomum cassia</i> (CIN) on life span in multiple pathway mutants of <i>C. elegans</i>.

<p>Age synchronized control (circles) or CIN fed (triangles) worms were seeded in liquid medium at the L1 stage. CIN treatment began 68 h after seeding (day 1 of adult life) and the fraction of animals alive was scored microscopically on the basis of movement. A. CIN significantly extends life span in wild type worms but not in B. <i>daf-16</i> C. <i>mev-1</i> or D. <i>ser-1</i> mutants. Survival rate was scored every day and is expressed as percentage of survival. The experiment was conducted at least three times with a minimum of 100 worms per trial. Statistical significant p<0.05, n.s. not significant.</p

Summary.

<p>Summary.</p

Representative epifluorescence image of transgenic <i>C. elegans</i> (CL2070, hsp-16- 2/GFP) fed with or without herbs followed by a heat-shock treatment for 2 hr.

<p>CL2070 (hsp-16-2/GFP) worms, grown at 20°C, were treated with vehicle control (Ctrl), SQDB (100 µg/ml), GS, CIN (10 µg/ml) for 48 h starting at 2 days of age. The worms were exposed to 35°C for 2 h and transferred to 20°C for 4 h to recover before fluorescence microscopy. <i>Integrated Density</i> means the sum of the values of the pixels in the image or selection. For quantifying a population of GFP reporter animals, each 40× image was analyzed using Image J software. Data are expressed as GFP integrated pixel density obtained from at least three independent experiments with at least 10 worms in each experimental group. **Statistically significant (independent t test, P<0.01); ***statistically significant, P<0.001.</p

Description of <i>C. elegans</i> strains.

<p>Description of <i>C. elegans</i> strains.</p

Paralysis assay in a transgenic <i>C. elegans</i> (CL4176) expressing Aβ.

<p>Age synchronized CL4176 worms were treated from egg until the L3 larval stage with CIN, GS (10 µg/ml) and SQDB (100 µg/ml). The animals were allowed to cultivate at 16°C for 38 h. At the 38 h time point, the temperature was up-shifted to 23°C. Worms began to paralyze 25 h after the temperature up-shift and were scored at 1 hr intervals. Data are expressed as percent (%) of worms not paralyzed. SQDB, GS and CIN alleviate β amyloid-induced toxicity in the transgenic <i>C. elegans</i> strain.</p

Life span assays in wild type <i>C. elegans</i> fed with SQDB and individual herbs.

<p>*P<0.005, Statistical significant; n.s. P>0.005 not significant.</p

Botanical names and origin of the formulas SQDB and HLXL.

<p>Botanical names and origin of the formulas SQDB and HLXL.</p

Life span assays in wild type <i>C. elegans</i> fed with HLXL and individual herbs.

<p>*P<0.005, Statistical significant; n.s. P>0.005 not significant.</p