Unexpected phenomena in a one dimensional elliptic equation with a singular first order divergence term

2024/07/05 by Giachetti, Daniela, Martínez-Aparicio, Pedro J., Murat, François +1
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2407.04611

Abstract

We study existence of a weak solution for one-dimensional problems as \begincases -(d)/(dx)(a(x) (d u)/(dx)) = - (d ϕ(u) )/(dx)- (d g(x) )/(dx)amp; in (0,L), u(0)=u(L)=0 , amp; \endcases where a is a positive bounded function, g∈ L²(0,L), and ϕ:ℝ↦ ℝ∪ \+∞\ is continuous as a function with values in ℝ∪ \+∞\. Some relevant qualitative and quantitative facts concerning such problems and their solutions are described. In particular a precise characterization of the behaviour of suitable approximating solution is provided. Of particular (and independent) interest is the study of an associated ODE for which, we prove existence, uniqueness and comparison results. As a consequence of our arguments, a delicate stability result as well a quite unexpected multiplicity result is shown for problems as in \eqrefintro.

Related

• New definitions of a solution of a one-dimensional elliptic equation with a singular first order divergence term
• Elliptic problems involving the 1--Laplacian and a singular lower order term
• Singular solutions for divergence-form elliptic equations involving regular variation theory: Existence and classification
• A blow-up approach for singular elliptic problems with natural growth in the gradient
• Existence and non-existence phenomena for nonlinear elliptic equations with L¹ data and singular reactions