Normalized solutions of Schrödinger equations involving Moser-Trudinger critical growth

Abstract

Abstract In this article, we are concerned with the nonlinear Schrödinger equation − Δ u + λ u = μ ∣ u ∣ p − 2 u + f ( u ) , in R 2 , -\Delta u+\lambda u=\mu {| u| }^{p-2}u+f\left(u),\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{2}, having prescribed mass ∫ R 2 ∣ u ∣ 2 d x = a 2 > 0 , \mathop{\int }\limits_{{{\mathbb{R}}}^{2}}{| u| }^{2}{\rm{d}}x={a}^{2}\gt 0, where λ \lambda arises as a Lagrange multiplier, μ > 0 \mu \gt 0 , p ∈ ( 2 , 4 ] p\in \left(2,4] , and the nonlinearity f ∈ C 1 ( R , R ) f\in {C}^{1}\left({\mathbb{R}},{\mathbb{R}}) behaves like e 4 π u 2 {e}^{4\pi {u}^{2}} as ∣ u ∣ → + ∞ | u| \to +\infty . For a L 2 {L}^{2} -critical or L 2 {L}^{2} -subcritical perturbation μ ∣ u ∣ p − 2 u \mu {| u| }^{p-2}u , we investigate the existence of normalized solutions to the aforementioned problem. Moreover, the limiting profiles of solutions have been considered as μ → 0 \mu \to 0 or a → 0 a\to 0 . This result can be considered as a supplement to the work of Soave (Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case, J. Funct. Anal. 279 (2020), no. 6, 1–43) and Alves et al. (Normalized solutions for a Schrödinger equation with critical growth in R N {{\mathbb{R}}}^{N} , Calc. Var. Partial Differential Equations 61 (2022), no. 1, 1–24).