Nodal solutions for a supercritical problem with variable exponent and logarithmic nonlinearity

We study the existence of sign-changing solutions for following supercritical problem with variable exponent and logarithmic nonlinearity$ \begin{equation*} \left\{ \begin{aligned} - \Delta u& = |u|^{2^{*}-2} u\big( \ln( \tau +|u|) \big)^{ |x|^{\beta} } & \ \mbox{in} B_{1}, \\ 0 \mbox{on} \partial \end{aligned} \right. \end{equation*} $where $ B_{1} is unit ball in \mathbb{R}^{N} $, N\geq 3 2^* 2N/(N-2) critical Sobolev exponent, \tau\geq 1 \beta>0 are constants. For any k\in \mathbb{N} if <\beta<(N-2)/2 we show that there exist one pair which change sign exactly k times by variational methods.