Normalized ground states for a nonlinear Schrödinger equation with double critical nonlinearities

We consider the constrained nonlinear Schrödinger problem in $\mathbb{R}^{N}$ ($N\ge 3$) where $2< p,m<2+\tfrac{4}{N}$, $2^{*}=\tfrac{2N}{N-2}$, $a,\mu ,\kappa >0$, and $\gamma >0$ is a parameter. We prove that there exists an explicit threshold $\gamma _{0}=\gamma _{0}(a,\mu ,\kappa ,N,p,m)>0$, determined by the variational inequalities and Sobolev embedding constants, such that for every $0<\gamma <\gamma _{0}$, problem $(P_{a})$ admits a positive normalized ground state $(\lambda ,u)\in \mathbb{R}\times H^{1}(\mathbb{R}^{N})$ with $\lambda <0$. The proof is variational and relies on a careful analysis of the energy functional on the $L^{2}$-sphere, the construction of a Pohozaev manifold as a natural constraint, and concentration–compactness arguments adapted to handle the Sobolev-critical term. The double-subcritical structure ($|u|^{p-2}u$ and $|u|^{m-2}u$) competes with the critical term, and the smallness of γ ensures compactness of minimizing sequences.