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The effect of Q-condition in elliptic equations involving Hardy potential and singular convection term

Fessel Achhoud, Abdelkader Bouajaja, Hicham Redwane

TL;DR

This work analyzes quasi-linear elliptic equations with a Hardy potential and singular convection on bounded domains, proving existence and uniqueness of weak solutions under a size condition on coefficients. The authors introduce a regularizing lower-order term $a(x)h(u)$ and develop a truncation-based approximation framework that yields a priori estimates, enabling existence and regularity results for both regular data and $f\in L^1$. They also establish weak and strong maximum principles under sign conditions on the data, and provide an example showing the sharpness of the regularity threshold. The results extend Hardy-type elliptic problems to singular data and nonlinear lower-order terms, with implications for the solvability and qualitative behavior of solutions in critical Sobolev settings.

Abstract

Using an approach by contradiction we prove the existence and uniqueness of a weak solution to a quasi-linear elliptic boundary value problem with singular convection term and Hardy Potential. Whose simplest model is \begin{equation*} \Scale[0.8]{\ds u \in W_0^{1,2}(\mathcal{O})\cap L^\infty(\mathcal{O}) : -Δu=-\mathcal{A}\text{div}\left(\frac{x}{\vert x\vert^2}u\right)+λ\frac{u}{\vert x\vert^2}+f(x),} \end{equation*} where \(\mathcal{O}\) is a bounded open set in \(\mathbb{R}^N\), $\left(\mathcal{A},λ\right) \in \left(0, \infty\right)^2$ and \(f\in W^{-1,2}(\mathcal{O})\). Additionally, by taking advantage of the regularizing effect of the interaction between the coefficient of the zero order term and the datum, we establish the existence, uniqueness and regularity of a weak solution to a quasi-linear boundary value problem whose simplest example is \begin{equation*} \Scale[0.8]{\ds u \in W_0^{1,2}(\mathcal{O})\cap L^\infty(\mathcal{O}) : -Δu +a(x)\vert u\vert^{p-2}u=-\mathcal{A}\text{div}\left(\frac{x}{\vert x\vert^2}u\right)+λ\frac{u}{\vert x\vert^2}+f(x),} \end{equation*} under suitable assumptions on $a$ and $f$.

The effect of Q-condition in elliptic equations involving Hardy potential and singular convection term

TL;DR

This work analyzes quasi-linear elliptic equations with a Hardy potential and singular convection on bounded domains, proving existence and uniqueness of weak solutions under a size condition on coefficients. The authors introduce a regularizing lower-order term and develop a truncation-based approximation framework that yields a priori estimates, enabling existence and regularity results for both regular data and . They also establish weak and strong maximum principles under sign conditions on the data, and provide an example showing the sharpness of the regularity threshold. The results extend Hardy-type elliptic problems to singular data and nonlinear lower-order terms, with implications for the solvability and qualitative behavior of solutions in critical Sobolev settings.

Abstract

Using an approach by contradiction we prove the existence and uniqueness of a weak solution to a quasi-linear elliptic boundary value problem with singular convection term and Hardy Potential. Whose simplest model is \begin{equation*} \Scale[0.8]{\ds u \in W_0^{1,2}(\mathcal{O})\cap L^\infty(\mathcal{O}) : -Δu=-\mathcal{A}\text{div}\left(\frac{x}{\vert x\vert^2}u\right)+λ\frac{u}{\vert x\vert^2}+f(x),} \end{equation*} where is a bounded open set in , and \(f\in W^{-1,2}(\mathcal{O})\). Additionally, by taking advantage of the regularizing effect of the interaction between the coefficient of the zero order term and the datum, we establish the existence, uniqueness and regularity of a weak solution to a quasi-linear boundary value problem whose simplest example is \begin{equation*} \Scale[0.8]{\ds u \in W_0^{1,2}(\mathcal{O})\cap L^\infty(\mathcal{O}) : -Δu +a(x)\vert u\vert^{p-2}u=-\mathcal{A}\text{div}\left(\frac{x}{\vert x\vert^2}u\right)+λ\frac{u}{\vert x\vert^2}+f(x),} \end{equation*} under suitable assumptions on and .

Paper Structure

This paper contains 10 sections, 6 theorems, 128 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Regular data: $f\in W^{-1,p^\prime}(\mathcal{O})$
  3. A priori estimates
  4. Proof of Theorem \ref{['th1']}.
  5. No regular data: $f\in L^1(\mathcal{O})$
  6. A priori estimates
  7. Proof of Theorem \ref{['th2']}
  8. Example
  9. Proof of the Theorem \ref{['wmp']}
  10. Proof of the Theorem \ref{['smp']}

Key Result

Theorem 1

Assume that c1, c2 and c3 hold. Let $\mathcal{A}$ and $\lambda$ be positive constants that satisfy then there exists a unique weak solution for the problem prb1, that is a function $u\in W^{1,2}_0(\mathcal{O})$ such that for every $\varphi$ in $W_{0}^{1,2}(\mathcal{O}) \cap L^{\infty}(\mathcal{O})$.

Figures (1)

  • Figure 1: Graph of the function $\mathcal{F}(m)=\frac{21}{m}(1-\frac{3}{m})+\frac{1}{2}(\frac{6}{m}-1)$.

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Theorem 3
  • Theorem 4
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof