Existence and concentration behavior of solutions to 1-Laplace equations on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msup></mml:math>

Li, Zhouxin; Liu, Ruishu

doi:10.1016/j.jde.2020.09.041

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The Main Results Of Weak Solutions1

Discussion1

Introduction1

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Cited by 7 publications

(3 citation statements)

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“…In this section, we first recall some useful definitions and known results, which can be found in [13][14][15][16][17][18]. Denote…”

Section: The Main Results Of Weak Solutionsmentioning

confidence: 99%

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Study of Weak Solutions for Degenerate Parabolic Inequalities with Nonlocal Nonlinearities

Dong

2022

Symmetry

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“…In this section, we first recall some useful definitions and known results, which can be found in [13][14][15][16][17][18]. Denote…”

Section: The Main Results Of Weak Solutionsmentioning

confidence: 99%

“…Refs. [12][13][14][15][16] study the existence, uniqueness, solvability, and stability of solutions of parabolic equations. Since these literature works are not concerned with variational inequalities, it is not necessary to give the comparison principle of parabolic operator and construct penalty functions when analyzing the existence.…”

Section: Discussionmentioning

confidence: 99%

Study of Weak Solutions for Degenerate Parabolic Inequalities with Nonlocal Nonlinearities

Dong

2022

Symmetry

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“…From the mathematical point of view, there is huge literature concerning existence, uniqueness and regularity of solutions to problems involving the -Laplace operator under Dirichlet boundary conditions; even the case has been dealt with but, unsurprisingly, the literature is more limited. The study of this type of problems is a very active branch as shown by recent works such as [1, 10, 17, 24, 25, 30].…”

Section: Introductionmentioning

confidence: 99%

Behaviour of solutions to p-Laplacian with Robin boundary conditions as p goes to 1

Pietra

Oliva²,

León³

2023

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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We study the asymptotic behaviour, as $p\to 1^+$ , of the solutions of the following inhomogeneous Robin boundary value problem: P \begin{equation*} \begin{cases} \displaystyle -\Delta_p u_p = f & \text{ in }\Omega,\\ \displaystyle |\nabla u_p|^{p-2}\nabla u_p\cdot \nu +\lambda |u_p|^{p-2}u_p = g & \text{ on } \partial\Omega, \end{cases} \end{equation*} where $\Omega$ is a bounded domain in $\mathbb {R}^{N}$ with sufficiently smooth boundary, $\nu$ is its unit outward normal vector and $\Delta _p v$ is the $p$ -Laplacian operator with $p>1$ . The data $f\in L^{N,\infty }(\Omega )$ (which denotes the Marcinkiewicz space) and $\lambda,\,g$ are bounded functions defined on $\partial \Omega$ with $\lambda \ge 0$ . We find the threshold below which the family of $p$ –solutions goes to 0 and above which this family blows up. As a second interest we deal with the $1$ -Laplacian problem formally arising by taking $p\to 1^+$ in (P).

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Minimization to the Zhang's energy on BV(Ω) and sharp affine Poincaré-Sobolev inequalities

Leite

Montenegro

2022

Journal of Functional Analysis

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Existence and concentration behavior of solutions to 1-Laplace equations on RN

Cited by 7 publications

References 21 publications

Study of Weak Solutions for Degenerate Parabolic Inequalities with Nonlocal Nonlinearities

Study of Weak Solutions for Degenerate Parabolic Inequalities with Nonlocal Nonlinearities

Behaviour of solutions to p-Laplacian with Robin boundary conditions as p goes to 1

Minimization to the Zhang's energy on BV(Ω) and sharp affine Poincaré-Sobolev inequalities

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