Nontrivial solutions for the Laplace equation with a nonlinear Goldstein-Wentzell boundary condition
<p>The paper deals with the existence and multiplicity of nontrivial solutions for the doubly elliptic problem</p> <p class="disp_formula">$ \begin{cases} \Delta u = 0 \qquad &\text{in}~~ \Omega , \\ u = 0 &\text{on}~~ \Gamma_0 , \\ -\Delta_\Gamma u...
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-11-01
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| Series: | Communications in Analysis and Mechanics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/cam.2023039?viewType=HTML |
| _version_ | 1797361225777020928 |
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| author | Enzo Vitillaro |
| author_facet | Enzo Vitillaro |
| author_sort | Enzo Vitillaro |
| collection | DOAJ |
| description | <p>The paper deals with the existence and multiplicity of nontrivial solutions for the doubly elliptic problem</p>
<p class="disp_formula">$ \begin{cases} \Delta u = 0 \qquad &\text{in}~~ \Omega , \\ u = 0 &\text{on}~~ \Gamma_0 , \\ -\Delta_\Gamma u +\partial_\nu u = |u|^{p-2}u\qquad &\text{on}~~ \Gamma_1 , \end{cases} $</p>
<p>where $ \Omega $ is a bounded open subset of $ \mathbb R^N $ ($ N\ge 2 $) with $ C^1 $ boundary $ \partial\Omega = \Gamma_0\cup\Gamma_1 $, $ \Gamma_0\cap\Gamma_1 = \emptyset $, $ \Gamma_1 $ being nonempty and relatively open on $ \Gamma $, $ \mathcal{H}^{N-1}(\Gamma_0) > 0 $ and $ p > 2 $ being subcritical with respect to Sobolev embedding on $ \partial\Omega $.</p>
<p>We prove that the problem admits nontrivial solutions at the potential-well depth energy level, which is the minimal energy level for nontrivial solutions. We also prove that the problem has infinitely many solutions at higher energy levels.</p> |
| first_indexed | 2024-03-08T15:50:51Z |
| format | Article |
| id | doaj.art-85b94668d47645778c053e288ff2bc0e |
| institution | Directory Open Access Journal |
| issn | 2836-3310 |
| language | English |
| last_indexed | 2024-03-08T15:50:51Z |
| publishDate | 2023-11-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Communications in Analysis and Mechanics |
| spelling | doaj.art-85b94668d47645778c053e288ff2bc0e2024-01-09T06:09:28ZengAIMS PressCommunications in Analysis and Mechanics2836-33102023-11-0115481183010.3934/cam.2023039Nontrivial solutions for the Laplace equation with a nonlinear Goldstein-Wentzell boundary conditionEnzo Vitillaro0Dipartimento di Matematica e Informatica, Università di Perugia, Via Vanvitelli, 1 06123 Perugia, Italy<p>The paper deals with the existence and multiplicity of nontrivial solutions for the doubly elliptic problem</p> <p class="disp_formula">$ \begin{cases} \Delta u = 0 \qquad &\text{in}~~ \Omega , \\ u = 0 &\text{on}~~ \Gamma_0 , \\ -\Delta_\Gamma u +\partial_\nu u = |u|^{p-2}u\qquad &\text{on}~~ \Gamma_1 , \end{cases} $</p> <p>where $ \Omega $ is a bounded open subset of $ \mathbb R^N $ ($ N\ge 2 $) with $ C^1 $ boundary $ \partial\Omega = \Gamma_0\cup\Gamma_1 $, $ \Gamma_0\cap\Gamma_1 = \emptyset $, $ \Gamma_1 $ being nonempty and relatively open on $ \Gamma $, $ \mathcal{H}^{N-1}(\Gamma_0) > 0 $ and $ p > 2 $ being subcritical with respect to Sobolev embedding on $ \partial\Omega $.</p> <p>We prove that the problem admits nontrivial solutions at the potential-well depth energy level, which is the minimal energy level for nontrivial solutions. We also prove that the problem has infinitely many solutions at higher energy levels.</p>https://www.aimspress.com/article/doi/10.3934/cam.2023039?viewType=HTMLlaplace equationlaplace-beltrami operatorexistence and multiplicity for nontrivial solutionswentzell boundary conditionsventcel boundary conditionsmountain pass theorem |
| spellingShingle | Enzo Vitillaro Nontrivial solutions for the Laplace equation with a nonlinear Goldstein-Wentzell boundary condition Communications in Analysis and Mechanics laplace equation laplace-beltrami operator existence and multiplicity for nontrivial solutions wentzell boundary conditions ventcel boundary conditions mountain pass theorem |
| title | Nontrivial solutions for the Laplace equation with a nonlinear Goldstein-Wentzell boundary condition |
| title_full | Nontrivial solutions for the Laplace equation with a nonlinear Goldstein-Wentzell boundary condition |
| title_fullStr | Nontrivial solutions for the Laplace equation with a nonlinear Goldstein-Wentzell boundary condition |
| title_full_unstemmed | Nontrivial solutions for the Laplace equation with a nonlinear Goldstein-Wentzell boundary condition |
| title_short | Nontrivial solutions for the Laplace equation with a nonlinear Goldstein-Wentzell boundary condition |
| title_sort | nontrivial solutions for the laplace equation with a nonlinear goldstein wentzell boundary condition |
| topic | laplace equation laplace-beltrami operator existence and multiplicity for nontrivial solutions wentzell boundary conditions ventcel boundary conditions mountain pass theorem |
| url | https://www.aimspress.com/article/doi/10.3934/cam.2023039?viewType=HTML |
| work_keys_str_mv | AT enzovitillaro nontrivialsolutionsforthelaplaceequationwithanonlineargoldsteinwentzellboundarycondition |