Multiple radial positive solutions of semilinear elliptic problems with Neumann boundary conditions

Let BR be a ball of radius R in RN. We analyze the positive solutions to the problem {−Δu+u=|u|p−2u,in  BR,∂νu=0,on  ∂BR, that branch out from the constant solution u=1 as p grows from 2 to +∞. The nonzero constant positive solution is the unique positive solution for p close to 2. We show that there exist arbitrarily many positive solutions as p→∞ (in particular, for supercritical exponents) or as R→∞ for any fixed value of p>2, partially answering a conjecture in Bonheure et al. (2012). We give explicit lower bounds for p and R so that a given number of solutions exist. The geometrical properties of those solutions are studied and illustrated numerically. Our simulations motivate additional conjectures. The structure of the least energy solutions (among all or only among radial solutions) and other related problems are also discussed.