We analyze existence, multiplicity and oscillatory behavior of positive radial solutions to a class of quasilinear equations governed by the Lorentz-Minkowski mean curvature operator. The equation is set in a ball or an annulus of $ \mathbb R^N $, is subject to homogeneous Neumann boundary conditions, and involves a nonlinear term on which we do not impose any growth condition at infinity. The main tool that we use is the shooting method for ODEs.
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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 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 $
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Number of half-turns performed by the solutions of the Cauchy problem
(a) Partial bifurcation diagram in dimension
(a) Partial bifurcation diagram in a unit disk (i.e.,
A solution