Positive radial solutions for the Minkowski-curvature equation with Neumann boundary conditions

Figure 1.  Number of half-turns performed by the solutions of the Cauchy problem $ (u(R_1), v(R_1)) = (d, 0) $ associated with (5) varying with the initial condition $ u(R_1) = d $. The existence of a bound from above $ d^* $ for the initial data $ d $ in correspondence to which the solutions of the Cauchy problem perform at least one half-turn in the phase plane is a consequence of the fact that $ |u'|\le 1 $, see (5)

Figure 2.  (a) Partial bifurcation diagram in dimension $ N = 1 $, with $ R_1 = 0 $, $ R_2 = 1 $, and $ s_0 = 1 $. (b) Graphs of eight solutions belonging to the four branches represented in (a). The colour of each solution is the same as the branch it belongs to. For each branch we have selected two solutions, one with $ u(0)>1 $ and the other with $ u(0)<1 $. The solutions displayed correspond to different values of $ q $

Figure 3.  (a) Partial bifurcation diagram in a unit disk (i.e., $ R_1 = 0 $, $ R_2 = 1 $, and $ N = 2 $), with $ s_0 = 1 $. (b) Solutions corresponding to $ q = 70 $

Figure 4.  A solution $ (u_d, v_d) $, with $ 0<d<s_0 $, in the phase plane $ (u, v) $. The solution is also given in polar coordinates $ (\theta_d, \rho_d) $ with $ \alpha = 1 $. It can be noted from the picture that $ u_d(\bar r) = s_0 $ if and only if $ \cos\theta_d(\bar r) = 0 $ and that $ v_d(r) = 0 $ for some $ r\in [R_1, R_2] $ if and only if $ \sin\theta_d(r) = 0 $