Normalized groundstates for mixed $(p,2)$-Laplacian equations in ${\mathbb{R}}^2$ with exponential critical growth

We investigate normalized groundstates for mixed $(p,2)$-Laplacian equations \begin{align*} \begin{cases} -Δ_p u-Δu+λu=f(u) & $\text{in }$ $\mathbb{R}$^2, \displaystyle \int_{$\mathbb{R}$^2}|u|^2,$\mathrm{d}$x=m, u\in H^1($\mathbb{R}$^2)\cap D^{1,p}($\mathbb{R}$^2), \end{cases} \end{align*} where ${\mathrm{\Delta }}_p$ denotes the $p$-Laplacian with $1<p<2$, $\lambda \in \mathbb{R}$ represents a Lagrange multiplier and the nonlinerity $f$ exhibits exponential critical growth. Compared to the single-Laplacian case, the lack of regularity here precludes the Pohozaev identity, and the exponential critical growth severely compromises the restoration of compactness. To address these issues, we introduce a refined Moser iteration technique adapted to exponential critical growth, which establishes the Pohozaev identity for weak solutions under the mere assumption of $C_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,\alpha }$-regularity. By combining constrained minimization on the Pohozaev manifold within a closed $L^2$-ball with a minimax characterization, we prove the existence of normalized groundstates for any prescribed mass $m>0$. Notably, our approach works independently of the sign of the Lagrange multiplier $\lambda$, thereby surmounting the fundamental barrier in recovering compactness for mixed Laplacian problems.