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

• * Corresponding author: Francesca Colasuonno

Dedicated to Professor Patrizia Pucci on the occasion of her 65th birthday, with great esteem

This work was partially supported by the INdAM - GNAMPA Project 2019 "Il modello di Born-Infeld per l'elettromagnetismo nonlineare: esistenza, regolarità e molteplicità di soluzioni". A. Boscaggin and B. Noris acknowledge also the support of the project ERC Advanced Grant 2013 n. 339958: "Complex Patterns for Strongly Interacting Dynamical Systems – COMPAT"

• 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.

Mathematics Subject Classification: Primary: 35J62, 35B05, 35A24, 34B18; Secondary: 35B09.

 Citation: • • 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 $\sin\theta_d(r) = 0$

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