Normalized Solutions to $(p,q)$-Laplacian Equations: Existence and Multiplicity

In this paper we investigate the existence and multiplicity of normalized solutions for the following $(p,q)$-Laplacian problem \[ \begin{cases} -\Delta_{p}u - \Delta_{q}u = \lambda |u|^{p-2}u + |u|^{p-2}u \log |u|^{p} + V(\varepsilon x) f(u) &\textrm{in $\mathbb{R}^{N}$}, \\ \int_{\mathbb{R}^{N}} |u|^{p} \, dx = a^{p}, \end{cases} \] where $1 \lt p \lt q \lt p+p^{2}/N \lt N$, $a, \varepsilon \gt 0$. $\Delta_{i} = \operatorname{div}(|\nabla u|^{i-2} \nabla u)$ with $i \in \{p,q\}$ is the $i$-Laplacian operator. $\lambda$ emerges as an unknown Lagrange multiplier, while $V$ denotes a continuous function subject to appropriate assumptions. The function $f$ is continuous with subcritical growth of $L^{p}$-mass type. Employing variational methods, we establish the existence of multiple normalized solutions for all $\varepsilon$ that is sufficiently small. Moreover, the number of normalized solutions is at least twice the number of global maximum points of the function $V$.