Multiple Solutions for Problems Involving p(x)-Laplacian and p(x)-Biharmonic Operators
In this paper, we consider the following $p(x)$-biharmonic problem with Hardy nonlinearity:\begin{equation*}\left\{\begin{aligned}&\Delta_{p(x)}^{2}u-\Delta_{p(x)}u =\lambda \frac{|u|^{p(x)-2}u}{\delta(x)^{2p(x)}}+f(x,u) && \mbox{in }\Omega,\\&u=0 && \...
Збережено в:
| Дата: | 2024 |
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| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна Національної академії наук України
2024
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| Теми: | |
| Онлайн доступ: | https://jmag.ilt.kharkiv.ua/index.php/jmag/article/view/1070 |
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| Назва журналу: | Journal of Mathematical Physics, Analysis, Geometry |
Репозитарії
Journal of Mathematical Physics, Analysis, Geometry| Резюме: | In this paper, we consider the following $p(x)$-biharmonic problem with Hardy nonlinearity:\begin{equation*}\left\{\begin{aligned}&\Delta_{p(x)}^{2}u-\Delta_{p(x)}u =\lambda \frac{|u|^{p(x)-2}u}{\delta(x)^{2p(x)}}+f(x,u) && \mbox{in }\Omega,\\&u=0 && \mbox{on }\partial\Omega,\\&|\nabla u|^{p(x)-2}\frac{\partial u}{\partial n}=g(x,u) && \mbox{on }\partial\Omega,\end{aligned}\right.\end{equation*}where $ \Omega\subset R^{N}$ $( N\geq 3 )$, $\Delta_{p(x)}$ is the $p(x)$-Laplacian and $\Delta_{p(x)}^{2}$ is the $p(x)$-biharmonic operator. More precisely, under some appropriate conditions on the nonlinearities $f$ and $g$, we combine the variational methods with the theory of the generalized Lebesgue and Sobolev spaces to prove the existence and the multiplicity of solutions.
Mathematical Subject Classification 2020: 35J20, 35J60, 35G30, 35J35 |
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