Uniqueness of entire solutions to quasilinear equations of p-Laplace type
<p>We prove the uniqueness property for a class of entire solutions to the equation</p> <p class="disp_formula"> $ \begin{equation*} \left\{ \begin{array}{ll} -{\rm div}\, \mathcal{A}(x,\nabla u) = \sigma, \quad u\geq 0 \quad {\text{in }} \mathbb{R}^n, \\ {\liminf\...
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-09-01
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| Series: | Mathematics in Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mine.2023068?viewType=HTML |
| _version_ | 1797659016986361856 |
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| author | Nguyen Cong Phuc Igor E. Verbitsky |
| author_facet | Nguyen Cong Phuc Igor E. Verbitsky |
| author_sort | Nguyen Cong Phuc |
| collection | DOAJ |
| description | <p>We prove the uniqueness property for a class of entire solutions to the equation</p>
<p class="disp_formula"> $ \begin{equation*} \left\{ \begin{array}{ll} -{\rm div}\, \mathcal{A}(x,\nabla u) = \sigma, \quad u\geq 0 \quad {\text{in }} \mathbb{R}^n, \\ {\liminf\limits_{|x|\rightarrow \infty}}\, u = 0, \end{array} \right. \end{equation*} $ </p>
<p>where $ \sigma $ is a nonnegative locally finite measure in $ \mathbb{R}^n $, absolutely continuous with respect to the $ p $-capacity, and $ {\rm div}\, \mathcal{A}(x, \nabla u) $ is the $ \mathcal{A} $-Laplace operator, under standard growth and monotonicity assumptions of order $ p $ ($ 1 < p < \infty $) on $ \mathcal{A}(x, \xi) $ ($ x, \xi \in \mathbb{R}^n $); the model case $ \mathcal{A}(x, \xi) = \xi | \xi |^{p-2} $ corresponds to the $ p $-Laplace operator $ \Delta_p $ on $ \mathbb{R}^n $. Our main results establish uniqueness of solutions to a similar problem,</p>
<p class="disp_formula">$ \begin{equation*} \left\{ \begin{array}{ll} -{\rm div}\, \mathcal{A}(x,\nabla u) = \sigma u^q +\mu, \quad u\geq 0 \quad {\text{in }} \mathbb{R}^n, \\ {\liminf\limits_{|x|\rightarrow \infty}}\, u = 0, \end{array} \right. \end{equation*} $</p>
<p>in the sub-natural growth case $ 0 < q < p-1 $, where $ \mu, \sigma $ are nonnegative locally finite measures in $ \mathbb{R}^n $, absolutely continuous with respect to the $ p $-capacity, and $ \mathcal{A}(x, \xi) $ satisfies an additional homogeneity condition, which holds in particular for the $ p $-Laplace operator.</p> |
| first_indexed | 2024-03-11T18:07:53Z |
| format | Article |
| id | doaj.art-0c95eb586adb4177b446869e910c7c29 |
| institution | Directory Open Access Journal |
| issn | 2640-3501 |
| language | English |
| last_indexed | 2024-03-11T18:07:53Z |
| publishDate | 2023-09-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematics in Engineering |
| spelling | doaj.art-0c95eb586adb4177b446869e910c7c292023-10-17T01:15:50ZengAIMS PressMathematics in Engineering2640-35012023-09-015313310.3934/mine.2023068Uniqueness of entire solutions to quasilinear equations of p-Laplace typeNguyen Cong Phuc0Igor E. Verbitsky11. Department of Mathematics, Louisiana State University, 303 Lockett Hall, Baton Rouge, LA 70803, USA2. Department of Mathematics, University of Missouri, Columbia, MO 65211, USA<p>We prove the uniqueness property for a class of entire solutions to the equation</p> <p class="disp_formula"> $ \begin{equation*} \left\{ \begin{array}{ll} -{\rm div}\, \mathcal{A}(x,\nabla u) = \sigma, \quad u\geq 0 \quad {\text{in }} \mathbb{R}^n, \\ {\liminf\limits_{|x|\rightarrow \infty}}\, u = 0, \end{array} \right. \end{equation*} $ </p> <p>where $ \sigma $ is a nonnegative locally finite measure in $ \mathbb{R}^n $, absolutely continuous with respect to the $ p $-capacity, and $ {\rm div}\, \mathcal{A}(x, \nabla u) $ is the $ \mathcal{A} $-Laplace operator, under standard growth and monotonicity assumptions of order $ p $ ($ 1 < p < \infty $) on $ \mathcal{A}(x, \xi) $ ($ x, \xi \in \mathbb{R}^n $); the model case $ \mathcal{A}(x, \xi) = \xi | \xi |^{p-2} $ corresponds to the $ p $-Laplace operator $ \Delta_p $ on $ \mathbb{R}^n $. Our main results establish uniqueness of solutions to a similar problem,</p> <p class="disp_formula">$ \begin{equation*} \left\{ \begin{array}{ll} -{\rm div}\, \mathcal{A}(x,\nabla u) = \sigma u^q +\mu, \quad u\geq 0 \quad {\text{in }} \mathbb{R}^n, \\ {\liminf\limits_{|x|\rightarrow \infty}}\, u = 0, \end{array} \right. \end{equation*} $</p> <p>in the sub-natural growth case $ 0 < q < p-1 $, where $ \mu, \sigma $ are nonnegative locally finite measures in $ \mathbb{R}^n $, absolutely continuous with respect to the $ p $-capacity, and $ \mathcal{A}(x, \xi) $ satisfies an additional homogeneity condition, which holds in particular for the $ p $-Laplace operator.</p>https://www.aimspress.com/article/doi/10.3934/mine.2023068?viewType=HTMLquasilinear equationsp-laplaceentire solutionsuniqueness of solutionssub-natural growth |
| spellingShingle | Nguyen Cong Phuc Igor E. Verbitsky Uniqueness of entire solutions to quasilinear equations of p-Laplace type Mathematics in Engineering quasilinear equations p-laplace entire solutions uniqueness of solutions sub-natural growth |
| title | Uniqueness of entire solutions to quasilinear equations of p-Laplace type |
| title_full | Uniqueness of entire solutions to quasilinear equations of p-Laplace type |
| title_fullStr | Uniqueness of entire solutions to quasilinear equations of p-Laplace type |
| title_full_unstemmed | Uniqueness of entire solutions to quasilinear equations of p-Laplace type |
| title_short | Uniqueness of entire solutions to quasilinear equations of p-Laplace type |
| title_sort | uniqueness of entire solutions to quasilinear equations of p laplace type |
| topic | quasilinear equations p-laplace entire solutions uniqueness of solutions sub-natural growth |
| url | https://www.aimspress.com/article/doi/10.3934/mine.2023068?viewType=HTML |
| work_keys_str_mv | AT nguyencongphuc uniquenessofentiresolutionstoquasilinearequationsofplaplacetype AT igoreverbitsky uniquenessofentiresolutionstoquasilinearequationsofplaplacetype |