Volume 26, Issue 3
On an Anisotropic Equation with Critical Exponent and Non-standard Growth Condition

Nguyen Thanh Chung

J. Part. Diff. Eq., 26 (2013), pp. 217-225.

Published online: 2013-09

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  • Abstract

Using variational methods, we prove the existence of a nontrivial weak solution for the problem \begin{align*} \left\{  \begin{array}{ll} -\sum_{i=1}^N\partial_{x_i}\Big(|\partial_{x_i}u|^{p_i-2}\partial_{x_i}u\Big) =\lambda a(x)|u|^{q(x)-2}u+|u|^{p^\ast-2}u, & \text{ in } \Omega,  \\ u = 0, & \text{ in } \partial\Omega, \end{array} \right. \end{align*} where $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) is a bounded domain with smooth boundary $\partial\Omega$, $2 \leq p_i < N$, $i = \overline{1, N}$, $q: \overline \Omega \to (1, p^\ast)$ is a continuous function, $p^\ast=\frac{N}{\sum_{i=1}^N \frac{1}{p_i}-1}$ is the critical exponent for this class of problem, and $\lambda$ is a parameter.

  • Keywords

Anisotropic equation critical exponent variational methods existence

  • AMS Subject Headings

35D05, 35J60, 35J70, 58E05, 68T40

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

ntchung82@yahoo.com (Nguyen Thanh Chung)

  • BibTex
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@Article{JPDE-26-217, author = {Chung , Nguyen Thanh}, title = {On an Anisotropic Equation with Critical Exponent and Non-standard Growth Condition}, journal = {Journal of Partial Differential Equations}, year = {2013}, volume = {26}, number = {3}, pages = {217--225}, abstract = {

Using variational methods, we prove the existence of a nontrivial weak solution for the problem \begin{align*} \left\{  \begin{array}{ll} -\sum_{i=1}^N\partial_{x_i}\Big(|\partial_{x_i}u|^{p_i-2}\partial_{x_i}u\Big) =\lambda a(x)|u|^{q(x)-2}u+|u|^{p^\ast-2}u, & \text{ in } \Omega,  \\ u = 0, & \text{ in } \partial\Omega, \end{array} \right. \end{align*} where $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) is a bounded domain with smooth boundary $\partial\Omega$, $2 \leq p_i < N$, $i = \overline{1, N}$, $q: \overline \Omega \to (1, p^\ast)$ is a continuous function, $p^\ast=\frac{N}{\sum_{i=1}^N \frac{1}{p_i}-1}$ is the critical exponent for this class of problem, and $\lambda$ is a parameter.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v26.n3.2}, url = {http://global-sci.org/intro/article_detail/jpde/5162.html} }
TY - JOUR T1 - On an Anisotropic Equation with Critical Exponent and Non-standard Growth Condition AU - Chung , Nguyen Thanh JO - Journal of Partial Differential Equations VL - 3 SP - 217 EP - 225 PY - 2013 DA - 2013/09 SN - 26 DO - http://doi.org/10.4208/jpde.v26.n3.2 UR - https://global-sci.org/intro/article_detail/jpde/5162.html KW - Anisotropic equation KW - critical exponent KW - variational methods KW - existence AB -

Using variational methods, we prove the existence of a nontrivial weak solution for the problem \begin{align*} \left\{  \begin{array}{ll} -\sum_{i=1}^N\partial_{x_i}\Big(|\partial_{x_i}u|^{p_i-2}\partial_{x_i}u\Big) =\lambda a(x)|u|^{q(x)-2}u+|u|^{p^\ast-2}u, & \text{ in } \Omega,  \\ u = 0, & \text{ in } \partial\Omega, \end{array} \right. \end{align*} where $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) is a bounded domain with smooth boundary $\partial\Omega$, $2 \leq p_i < N$, $i = \overline{1, N}$, $q: \overline \Omega \to (1, p^\ast)$ is a continuous function, $p^\ast=\frac{N}{\sum_{i=1}^N \frac{1}{p_i}-1}$ is the critical exponent for this class of problem, and $\lambda$ is a parameter.

Nguyen Thanh Chung. (2019). On an Anisotropic Equation with Critical Exponent and Non-standard Growth Condition. Journal of Partial Differential Equations. 26 (3). 217-225. doi:10.4208/jpde.v26.n3.2
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