Exact multiplicity and bifurcation diagrams of positive solutions of a one-dimensional multiparameter prescribed mean curvature problem

We study the exact multiplicity and bifurcation diagrams of positive solutions u is an element of C-2(-L, L) boolean AND C[-L, L] of the one-dimensional multiparameter prescribed mean curvature problem {-(u'(x)/root 1+(u'(x))(2))' = lambda(up + uq), -L < x <L, u(-L) = u(L) = 0, where lambda > 0 is a bifurcation parameter, L > 0, the radius of the one-dimensional ball (-L, L), is an evolution parameter, and 0 <= p < q < infinity are two constants. We prove that the problem has at most two positive solutions for any 0 <= p< q < infinity and lambda, L > 0. In addition, if 0 <= p < q <= <(q)over tilde>(p) = p + 1 + 2 root p+1, we give a classification of totally three qualitatively different bifurcation diagrams on the (lambda, parallel to u parallel to(infinity))-plane for any L > 0. For any fixed p >= 0 and q >= (q) over bar (p) = p + 2 + 2.root 2p+3, we prove that there exist positive L* < L* such that the bifurcation diagiams on the (lambda, parallel to u parallel to(infinity))-plane are individually qualitatively different for the cases (i) 0 < L < L-* or L > L-*, (ii) L= L*, (iii) L-* <= L < L*. (C) 2014 Elsevier Inc. All rights reserved.