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On mixed local-nonlocal problems with Hardy potential
Analysis of PDEs · Submitted 9 Jul 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In particular, we will discuss the existence, non-existence and uniqueness of solutions in terms of the summability of $f$ and of the value of the parameter $\gamma$.
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The abstract, sentence by sentence
In this paper we study the effect of the Hardy potential on existence, uniqueness and optimal summability of solutions of the mixed local-nonlocal elliptic problem $$-\Delta u + (-\Delta)^s u - \gamma \frac{u}{|x|^2}=f \text{ in } \Omega, \ u=0 \text{ in } \mathbb{R}^n \setminus \Omega,$$ where $\Omega$ is a bounded domain in $\mathbb{R}^n$ containing the origin and $\gamma> 0$. In particular, we will discuss the existence, non-existence and uniqueness of solutions in terms of the summability of $f$ and of the value of the parameter $\gamma$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this paper we study the effect of the Hardy potential on existence, uniqueness and optimal summability of solutions of the mixed local-nonlocal elliptic problem $$-\Delta u + (-\Delta)^s u - \gamma \frac{u}{|x|^2}=f \text{ in } \Omega, \ u=0 \text{ in } \mathbb{R}^n \setminus \Omega,$$ where $\Omega$ is a bounded domain in $\mathbb{R}^n$ containing the origin and $\gamma> 0$. | contribution 0.94 | 0.70 | 0.08 | asserted 0.92 |
| s1 | In particular, we will discuss the existence, non-existence and uniqueness of solutions in terms of the summability of $f$ and of the value of the parameter $\gamma$. | contribution 0.91 | 0.84 | 0.23 | asserted 0.72 |
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Original abstract
In this paper we study the effect of the Hardy potential on existence, uniqueness and optimal summability of solutions of the mixed local-nonlocal elliptic problem $$-\Delta u + (-\Delta)^s u - \gamma \frac{u}{|x|^2}=f \text{ in } \Omega, \ u=0 \text{ in } \mathbb{R}^n \setminus \Omega,$$ where $\Omega$ is a bounded domain in $\mathbb{R}^n$ containing the origin and $\gamma> 0$. In particular, we will discuss the existence, non-existence and uniqueness of solutions in terms of the summability of $f$ and of the value of the parameter $\gamma$.
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We just mention that other singular PDEs driven by $ℒ$ have been considered so far, see e.g. [ref] .
Multiplicity of positive solutions for mixed local-nonlocal singular critical problems · Introduction · 0.92
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