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Existence and non-existence phenomena for nonlinear elliptic equations with $L^1$ data and singular reactions

Francescantonio Oliva, Francesco Petitta, Matheus F. Stapenhorst

TL;DR

This work analyzes the existence and nonexistence of positive solutions to a nonlinear elliptic equation with a p-Laplacian, an $L^1$ source, and a singular reaction $a(x)/u^{\gamma}$ under zero Dirichlet boundary conditions. The authors develop an iterative approximation scheme and derive barrier-type lower bounds using the first eigenfunction, enabling a priori estimates and compactness arguments that yield solutions for large forcing $μ$ and nonexistence for small $μ$. They extend the classical results of Diaz–Morel–Oswald to general $p$ (including the semilinear case $p=2$ with $\gamma=1$ under compatibility assumptions) and provide a refined analysis in the borderline $\gamma=1$ regime with compatibility conditions on the data. The paper also characterizes solution regularity in Marcinkiewicz spaces and discusses the precise thresholds and compatibility conditions that govern the existence-versus-nonexistence dichotomy. These results have implications for singular reaction-diffusion models with $L^1$ data and for materials and fluid contexts where non-Newtonian diffusion and boundary-singular effects arise.

Abstract

We study existence and non-existence of solutions for singular elliptic boundary value problems as \begin{equation}\label{eintro}\begin{cases}\tag{1} \displaystyle -Δ_p u+ \frac{a(x)}{u^γ}=μf(x) \ &\text{ in }Ω, \newline u>0&\text{ in }Ω, \newline u = 0 \ &\text{ on } \partialΩ, \end{cases} \end{equation} where $Ω$ is a smooth bounded open subset of $\mathbb{R}^N$ ($N\ge 2$), $Δ_p u$ is the $p$-Laplacian with $p>1$, $0<γ\leq 1$, and $a\geq0$ is bounded and non-trivial. For any positive $ f\in L^{1}(Ω)$ we show that problem \eqref{eintro} is solvable for any $μ>μ_0>0$, for some $μ_0$ large enough. As a reciprocal outcome we also show that no finite energy solution exists if $0<μ<μ_{0*}$, for some small $μ_{0*}$. This paper extends the celebrated one of J. I. Diaz, J. M. Morel and L. Oswald ([16]) to the case $p\neq2$. Our result is also new for $p=2$ provided the singular term has a critical growth near zero (i.e. $γ=1$).

Existence and non-existence phenomena for nonlinear elliptic equations with $L^1$ data and singular reactions

TL;DR

This work analyzes the existence and nonexistence of positive solutions to a nonlinear elliptic equation with a p-Laplacian, an source, and a singular reaction under zero Dirichlet boundary conditions. The authors develop an iterative approximation scheme and derive barrier-type lower bounds using the first eigenfunction, enabling a priori estimates and compactness arguments that yield solutions for large forcing and nonexistence for small . They extend the classical results of Diaz–Morel–Oswald to general (including the semilinear case with under compatibility assumptions) and provide a refined analysis in the borderline regime with compatibility conditions on the data. The paper also characterizes solution regularity in Marcinkiewicz spaces and discusses the precise thresholds and compatibility conditions that govern the existence-versus-nonexistence dichotomy. These results have implications for singular reaction-diffusion models with data and for materials and fluid contexts where non-Newtonian diffusion and boundary-singular effects arise.

Abstract

We study existence and non-existence of solutions for singular elliptic boundary value problems as \begin{equation}\label{eintro}\begin{cases}\tag{1} \displaystyle -Δ_p u+ \frac{a(x)}{u^γ}=μf(x) \ &\text{ in }Ω, \newline u>0&\text{ in }Ω, \newline u = 0 \ &\text{ on } \partialΩ, \end{cases} \end{equation} where is a smooth bounded open subset of (), is the -Laplacian with , , and is bounded and non-trivial. For any positive we show that problem \eqref{eintro} is solvable for any , for some large enough. As a reciprocal outcome we also show that no finite energy solution exists if , for some small . This paper extends the celebrated one of J. I. Diaz, J. M. Morel and L. Oswald ([16]) to the case . Our result is also new for provided the singular term has a critical growth near zero (i.e. ).
Paper Structure (9 sections, 11 theorems, 98 equations, 1 figure)

This paper contains 9 sections, 11 theorems, 98 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Main assumptions and results for $0<\gamma<1$
  4. Proof of existence and non-existence results
  5. A priori estimates and convergence results
  6. Existence for $\mu\geq \mu_{0}$
  7. Non-existence for $0<\mu<\mu_{0*}$
  8. Some compatibility conditions in the case $\gamma=1$.
  9. Non-existence result for $\gamma=1$

Key Result

Lemma 1.1

B1995 Let $u:\Omega\to\mathbb{R}$ be a measurable function such that $T_{k}(u)\in W^{1,1}_{loc}(\Omega)$ for every $k>0$. Then, there exists a measurable vectorial function $v:\Omega\to\mathbb{R}^{N}$ such that Furthermore, $u\in W^{1,1}_{loc}(\Omega)$ if and only if $v\in L^{1}_{loc}(\Omega)$ and then $v=\nabla u$ in the usual distributional sense.

Figures (1)

  • Figure 1: The $\varepsilon$-neighborhood of $\partial\Omega$

Theorems & Definitions (21)

  • Lemma 1.1
  • Lemma 1.2
  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Lemma 2.5
  • proof
  • Lemma 3.1
  • proof
  • ...and 11 more