Positive solutions to a class of elastic beam equations with semipositone nonlinearity

Qingliu Yao¹

¹ Department of Applied Mathematics Nanjing University of Finance and Economics Nanjing 210003, P.R. China

Annales Polonici Mathematici, Tome 97 (2010) no. 1, pp. 35-50

Résumé

Let $h\in L^{1}[0,1]\cap C(0,1)$ be nonnegative and $f(t,u,v)+h(t)\geq 0$. We study the existence and multiplicity of positive solutions for the nonlinear fourth-order two-point boundary value problem $$ u^{(4)}(t)=f(t,u(t),u^{\prime }(t)),\quad\ 0 t 1,\quad\ u(0)=u^{\prime }(0)=u^{\prime }(1)=u^{\prime \prime \prime }(1)=0, $$ where the nonlinear term $f(t,u,v)$ may be singular at $t=0$ and $t=1$. By constructing a suitable cone and integrating certain height functions of $f(t,u,v)$ on some bounded sets, several new results are obtained. In mechanics, the problem models the deflection of an elastic beam fixed at the left end and clamped at the right end by sliding clamps.

Détail

DOI : 10.4064/ap97-1-3

Keywords: cap nonnegative v geq study existence multiplicity positive solutions nonlinear fourth order two point boundary value problem t prime quad quad prime prime prime prime prime where nonlinear term v may singular constructing suitable cone integrating certain height functions v bounded sets several results obtained mechanics problem models deflection elastic beam fixed end clamped right end sliding clamps