Normalized solutions for the $(2,q)$-Laplacian operator between mass-critical exponents

This paper concerns the existence of normalized solutions to a class of $(2,q)$-Laplacian equations with a power type nonlinearity in the intermediate regime between the two mass critical exponents $2(1+2/N)$, $q(1+2/N)$. More precisely, we prove the existence of solutions with negative energy obtained through a global minimization procedure, and of solutions with positive energy established via a local minimization technique and a mountain-pass argument. Furthermore, we derive both existence and nonexistence results for the zero-mass case $\lambda=0$, highlighting the role of the mixed diffusion in determining the qualitative behavior of solutions. Specifically, this paper’s novelty lies in providing a comprehensive understanding of the intermediate cases that arise when the non-homogeneous $(2,q)$-Laplacian operator appears. Our analysis combines variational methods, compactness arguments, and delicate energy estimates adapted to the nonhomogeneous nature of the $(2,q)$-Laplacian operator.