Let
BR be a ball of radius
R in
35d9e" title="Click to view the MathML source">RN. We analyze the positive solutions to the problem
that branch out from the constant solution
u=1 as
35c56950a6c22388a0e06d00c4befa43" title="Click to view the MathML source">p grows from
2 to
3563e1caffbcef6244678" title="Click to view the MathML source">+∞. The nonzero constant positive solution is the unique positive solution for
35c56950a6c22388a0e06d00c4befa43" title="Click to view the MathML source">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
35c56950a6c22388a0e06d00c4befa43" title="Click to view the MathML source">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.