Nodal solutions for Logarithmic weighted $N$-Laplacian problem with exponential nonlinearities

In this article, we study the following problem $$-\mathrm{d}\mathrm{i}\mathrm{v}(\omega (x)\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{N-2}\mathrm{\nabla }u)=\lambda f(x,u)inB,u=0on\mathrm{\partial }B,$$ where $B$ is the unit ball of ${\mathbb{R}}^{\mathbb{N}}$, $N\ge 2$ and $w(x)$ a singular weight of logarithm type. The reaction source $f(x,u)$ is a radial function with respect to $x$ and is subcritical or critical with respect to a maximal growth of exponential type. By using the constrained minimization in Nehari set coupled with the quantitative deformation lemma and degree theory, we prove the existence of nodal solutions.