Optimal existence theory for single and multiple positive solutions to second order Neumann boundary value problems
The paper deals with Neumann boundary value problems of the form \[ -u''+\rho_1^2 u =f(t,u),\quad u'(0)=u'(1)=0, \tag{1} \] or \[ u''+\rho_2^2 u =f(t,u),\quad u'(0)=u'(1)=0, \tag{2} \] where \(f:[0,1]\times [0,\infty)\) is nonnegative continuous, and \(\rho_1>0\), \(\rho_2\in (0,\pi/2)\). The authors provide sufficient conditions for the existence of at least one positive solution of (1) or (2) and for the existence of at least two positive solutions of (1) or (2). These conditions are formulated in terms of an asymptotic behaviour of \({f(t,x)\over x}\) near \(0\) and \(\infty\). The proofs are based on Krasnoselskii's fixed-point theorem in cones.
scientific article; zbMATH DE number 2141228