Existence of S-shaped Type Bifurcation Curve with Dual Cusp Catastrophe Via Variational Methods

We discuss the existence of multiple positive solutions leading to the occurrence of an S-shaped bifurcation curve to the equations of the form −Δpu=f(u), with p>1. We deal with relatively unexplored cases when f is non-Lipschitz at 0, f(0)=0 and f(u)<0, u∈(0,r), for some r<+∞. Using the nonlinear generalized Rayleigh quotients method we find a range of parameters where the equation may have distinct branches of solutions. As a consequence, applying the variational methods, we prove that the equation has at least three positive solutions with two of them linearly unstable and one linearly stable. The results evidence that the bifurcation curve is S-shaped and exhibits the so-called dual cusp catastrophe. Our results are new even in the one-dimensional case and p=2.