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.