INFINITE RESONANT SOLUTIONS AND TURNING POINTS IN A PROBLEM WITH UNBOUNDED BIFURCATION

We consider an elliptic equation -Delta u + u = 0 with nonlinear boundary conditions partial derivative u/partial derivative n = lambda u + g(lambda, x, u), where (g(lambda, x, s))/s -> 0, as vertical bar s vertical bar -> infinity. In [Arrieta et al., 2007, 2009] the authors proved the existence of unbounded branches of solutions near a Steklov eigenvalue of odd multiplicity and, among other things, provided tools to decide whether the branch is subcritical or supercritical. In this work, we give conditions on the nonlinearity, guaranteeing the existence of a bifurcating branch which is neither subcritical nor supercritical, having an infinite number of turning points and an infinite number of resonant solutions.