Symmetric positive solutions of nonlinear boundary value problems (Q865343)

The authors study the following nonlinear boundary value problem \[ u^{2m}=f(t,u,u',\cdots,u^{(2m-2)}), \;t\in(0,1), \] \[ u^{(2i)}(0)=u^{(2i)}(1)=0, \;i=0,\cdots,m-1. \] The existence of symmetric positive solutions of the above problem is discussed. Sufficient conditions are obtained for the problem to have one, any finite number, and a countably infinite number of such solutions. Two examples are given to illustrate the main results.