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

Volume 97 / 2010

Qingliu Yao Annales Polonici Mathematici 97 (2010), 35-50 MSC: 34B15, 34B16, 34B18. DOI: 10.4064/ap97-1-3

Abstract

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.

Authors

  • Qingliu YaoDepartment of Applied Mathematics
    Nanjing University of Finance and Economics
    Nanjing 210003, P.R. China
    e-mail

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