Old mathematical challenges : precedents to the millennium problems

The millennium problems set out by the Clay Mathematics Institute became a stimulus for mathematical research. The aim of this article is to highlight some previous challenges that were also a stimulus to finding proof for some interesting results. With this pretext, we present three moments in the history of mathematics that were important for the development of new lines of research. We briefly analyse the Tartaglia challenge, which brought about the discovery of a formula for third degree equations; Johan Bernoulli?s problem of the curve of fastest descent, which originated the calculus of variations; and the incidence of the problems posed by David Hilbert in 1900, focusing on the first problem in the list: the continuum hypothesis

Gelfand-type problems involving the 1-Laplacian operator

In this paper, the theory of Gelfand problems is adapted to the 1-Laplacian setting. Concretely, we deal with the following problem: −∆1u = λf(u) in Ω,u = 0 on ∂Ω, where Ω ⊂ RN (N ≥ 1) is a domain, λ ≥ 0, and f : [0, +∞[ → ]0, +∞[ is any continuous increasing and unbounded function with f(0) > 0. We prove the existence of a threshold λ∗ = h(Ω) f(0) (h(Ω) being the Cheeger constant of Ω) such that there exists no solution when λ > λ∗ and the trivial function is always a solution when λ ≤ λ∗. The radial case is analyzed in more detail, showing the existence of multiple (even singular) solutions as well as the behavior of solutions to problems involving the p-Laplacian as p tends to 1, which allows us to identify proper solutions through an extra condition

Elliptic equations having a singular quadratic gradient term and a changing sign datum

In this paper we study a singular elliptic problem whose model is \begin{eqnarray*} - \Delta u= \frac{|\nabla u|^2}{|u|^\theta}+f(x), in \Omega\\ u = 0, on \partial \Omega; \end{eqnarray*} where $\theta\in (0,1)$ and $f \in L^m (\Omega)$, with $m\geq \frac{N}{2}$. We do not assume any sign condition on the lower order term, nor assume the datum $f$ has a constant sign. We carefully define the meaning of solution to this problem giving sense to the gradient term where $u=0$, and prove the existence of such a solution. We also discuss related questions as the existence of solutions when the datum $f$ is less regular or the boundedness of the solutions when the datum $f \in L^m (\Omega)$ with $m> \frac{N}{2}$

The Dirichlet problem for the 11-Laplacian with a general singular term and L1L^1-data

We study the Dirichlet problem for an elliptic equation involving the $1$-Laplace operator and a reaction term, namely: $$\left\{\begin{array}{ll} \displaystyle -\Delta_1 u =h(u)f(x)&\hbox{in }\Omega\,,\\ u=0&\hbox{on }\partial\Omega\,, \end{array}\right.$$ where $\Omega \subset \mathbb{R}^N$ is an open bounded set having Lipschitz boundary, $f\in L^1(\Omega)$ is nonnegative, and $h$ is a continuous real function that may possibly blow up at zero. We investigate optimal ranges for the data in order to obtain existence, nonexistence and (whenever expected) uniqueness of nonnegative solutions

Elliptic equations involving the 1-Laplacian and a subcritical source term

In this paper we deal with a Dirichlet problem for an elliptic equation involving the 1-Laplacian operator and a source term. We prove that, when the growth of the source is subcritical, there exist two bounded nontrivial solutions to our problem. Moreover, a Pohoz̆aev type identity is proved, which holds even when the growth is supercritical. We also show explicit examples of our results

Gelfand-type problems involving the 1-Laplacian operator

This paper is concerned with the Gelfand problem for the p-Laplacian with p=1 and a strictly positive and increasing nonlinearity f. It is shown that there exists a sharp threshold λ∗, depending on the Cheeger constant of the domain, such that for λ0, while no solutions exist for λ>λ∗. The analysis of the many solutions found in the radial case allows the identification of bounded solutions, as well as the understanding of the limit as p→1 of solutions of the p-Laplace-Gelfand problem