arXiv subjects

Ricardo Ziegele

Publications and source records attributed to Ricardo Ziegele.

5 recordsLinked to original sources

arXiv2026

Multiplicity via symmetry for the Minkowski mean curvature equation: Dirichlet and Neumann problems

We consider the Minkowski mean curvature equation $$ -\operatorname{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right) =λu+μh(x,u) \quad \text{in } Ω, \quad \mathcal B u = 0 \quad \text{on } \partial Ω, $$ where $\mathcal B \in \{ \mathcal D, \mathcal N \}$ is a Dirichlet or Neumann boundary condition and $Ω$ is a smooth bounded domain, required to be convex in the Neumann case. Moreover, $λ, μ\geq0$ are parameters, and $h(x,u)$ is odd with respect to $u$ and positive for $u > 0$. Using minmax methods in nonsmooth critical point theory, we prove that $λ>λ^\mathcal B_k$ yields at least $k$ pairs of nontrivial solutions with negative energy for every $μ\geq0$, where $λ_k^\mathcal B$ are the corresponding Laplacian eigenvalues, indexed by $λ^\mathcal B_0=0$. Moreover, if $H(x,u):=\int_0^u h(x,t)\,dt=o(u^2)$ as $u\to0$, we prove that for $λ\in[λ^\mathcal B_k,λ^\mathcal B_{k+1})$ and every $j\geq\max\{1,k\}$, sufficiently large $μ$ yields at least $2j-k$ distinct pairs of solutions: $j$ with negative energy and $j-k$ with positive energy.

math.AP↗

arXiv2026

Positive solutions for an indefinite-weight Minkowski mean curvature Neumann problem: multiplicity and asymptotic behaviour

We study the Neumann boundary value problem $$ -\operatorname{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right) = λa(x)g(u) \quad \text{in } Ω, \qquad \frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\cdot \mathbf n = 0 \quad \text{on } \partialΩ, $$ where $Ω\subset \mathbb R^N$ is a bounded convex domain, $a\in L^\infty(Ω)$ is an indefinite weight with $\int_Ωa(x) \, dx< 0$, and $g$ is a nonlinearity. Under suitable assumptions on $g$, we prove the existence of two positive solutions for sufficiently large $λ>0$, using Szulkin's theory for nonsmooth functionals: a global minimizer $u_λ^{(l)}$ with negative energy, and a mountain-pass critical point $u_λ^{(s)}$ with positive energy. Furthermore, we study the asymptotic behaviour as $λ\to+\infty$ of both families of solutions. For the ground states we prove that, up to a subsequence, $u_λ^{(l)}\to u_\infty$ uniformly, where $u_\infty$ satisfies $\|\nabla u_\infty\|_{L^\infty(Ω)}=1$ and is characterized by being a maximizer of $\int_Ωa(x) G(u) \, dx$ and a minimizer of the kinetic energy among all such maximizers. In the model case $g(u)=|u|^{p-2}u$, $p>2$, we also show that the mountain-pass energy level satisfies the upper bound $c_λ=O(λ^{-2/(p-2)})$, and that $u_λ^{(s)}\to0$ in $C(\overlineΩ)$.

math.AP↗

arXiv2026

Collapse and saturation for radial nodal solutions of a Lorentz--Minkowski mean curvature equation

We consider the Dirichlet problem for the Lorentz--Minkowski mean curvature operator in a radial setting, $$ -\operatorname{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right) = λu + μg(u) \quad \text{in } \mathcal B_R, \qquad u = 0 \quad \text{on } \partial\mathcal B_R, $$ on a ball of radius $R$ centered at the origin $\mathcal B_R \subset \mathbb{R}^N$, where $λ, μ\geq 0$ are parameters, and the nonlinearity $g$ satisfies suitable growth assumptions at zero. We first prove an a priori dichotomy for the asymptotic behaviour, as $μ\to +\infty$, of solutions with a prescribed number of zeros: either they converge uniformly to $0$, or they converge to a piecewise affine limit function $u_\infty$, with $|u_\infty'|=1$ a.e., having the same nodal properties. We then show that, when the number of zeros is appropriately prescribed, both alternatives are actually realized. More precisely, building on the multiplicity result in [A. Boscaggin, F. Colasuonno, and R. Ziegele, \emph{Positive and nodal solutions for the Minkowski mean curvature equation: multiplicity and asymptotics}, arXiv:2607.15956 (2026)], we refine the shooting construction therein to obtain a family of small solutions converging to zero and, through a backward shooting argument, a family of large solutions converging to a saturated piecewise affine profile.

math.AP↗

arXiv2026

Positive and nodal solutions for the Minkowski mean curvature equation: multiplicity and asymptotics

We consider the Dirichlet problem for the mean curvature operator in Minkowski space, \[ -\operatorname{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right) = λu + μh(x,u) \quad \text{in } Ω, \qquad u = 0 \quad \text{on } \partialΩ, \] in a bounded domain $Ω\subset \mathbb{R}^N$, where $λ, μ$ are real parameters, and the nonlinearity $h$ is superlinear at $u = 0$. In particular, we study the combined effect of the parameters $λ,\,μ$ on the multiplicity of solutions. In the general setting, following Szulkin's approach for nonsmooth functionals, we prove the existence, for $λ$ not belonging to the spectrum of the Dirichlet Laplacian and $μ$ sufficiently large, of a global minimizing solution (with negative action level) and of a min-max solution (with positive action level). Moreover, we characterize the limiting profiles of these solutions as $μ\to +\infty$. More precisely, when the global minimizer is positive, its limit profile is $\mathrm{dist}(\cdot,\partialΩ)$, thus saturating, in the limit, the geometric constraint $|\nabla u|\le1$, while min-max solutions collapse uniformly to zero as $μ\to+\infty$. A nonexistence criterion is also given for suitable values of $λ$ and $μ$. Finally, when the domain $Ω$ is a ball, using a shooting approach, we establish the existence of arbitrarily many nodal radial solutions for every $λ\ge 0$ and for $μ$ sufficiently large.

math.AP↗

arXiv2025

Existence and a priori bounds for fully nonlinear PDEs with a harmonic map-like structure

In this paper, we study a new class of fully nonlinear uniformly elliptic equations with a so-called harmonic map-like structure, whose model case is given by \begin{equation*} \mathcal{M}^{\pm}_{λ,Λ}(D^2u) \pm b(x) |Du| \pm β(u)\langle M(x) Du,Du \rangle \pm c(x) u = f(x)\; \textrm{ in } Ω, \end{equation*} where $Ω\subset \mathbb{R}^n$ is a bounded $C^{1,1}$ domain, $\mathcal{M}^{\pm}$ are the Pucci extremal operators, $β(s) = s^k$ for some $k \in \mathbb{N} $ odd, $b \in L^{q}_{+}(Ω)$, $c,f \in L^p(Ω)$, and $n \leq p \leq q$, $q>n$. We obtain existence results under a smallness regime on the coefficients, along with some classical results such as the Aleksandrov--Bakelman--Pucci estimate and the comparison principle, as well as a priori bounds for the respective Dirichlet problem in the noncoercive case. We also establish multiplicity results and qualitative behavior, which seem to be new in the case of the Laplacian operator.

math.AP↗