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.