Bifurcation from infinity for the special class of nonlinear differential equations (Q1080013)

The article was inspired by the works of P. H. Rabinowitz and H. Berestycki. We consider the problem \(Lu=\lambda \rho u+H(x,u,u',\lambda)\) with separated boundary conditions on [0,\(\pi\) ], where \(Lu=\lambda \rho u\) is a classical linear Sturm-Liouville second order ordinary differential equation. We assume that the nonlinear function H is of the form \(H=f_ 1+h\), \(f_ 1\), h being continuous, with h satisfying an \(o(| u| +| u'|)\) condition at \(\infty\), and \(| f_ 1(x,u,u',\lambda)| \leq M| u|\) for \(| u|,| u'| \geq 1\), uniformly in x and in \(\lambda\). For such an equation, we show the existence of two families of continua of solutions, \({\mathcal D}^+_ k\) and \({\mathcal D}^-_ k\) corresponding to the usual nodal properties and bifurcating from an interval \(I_ k\times \{\infty \}\). An analogous theorem is given for nonlinear elliptic partial differential equations (for \(k=1)\).