On radial positive normalized solutions of the Nonlinear Schrödinger equation in an annulus

We are interested in the following semilinear elliptic problem: { -Delta u+lambda u=u(p-1),x is an element of T, u>0,u=0on partial derivative T, integral(T)u(2)dx= cwhere T={x is an element of R-N:1<|x|<2}is an annulus inRN,N >= 2,p>1isSobolev-subcritical, searching for conditions (aboutc,Nandp) for the existence of positive radial solutions. We analyze the asymptotic behaviorofcas lambda ->+infinity and lambda ->-lambda 1to get the existence, non-existence and multiplicity of normalized solutions. Additionally, based on the properties of these solutions, we extend the results obtained in Pierotti et al. in CalcVar Partial Differ Equ 56:1-27, 2017. In contrast of the earlier results, a positive radial solution with arbitrarily large mass can be obtained whenN >= 3orifN= 2 and p<6. Our paper also includes the demonstrationof orbital stability/instability results.