Normalized solutions of Schrödinger equations involving Moser-Trudinger critical growth

Abstract

In this article, we are concerned with the nonlinear Schrödinger equation −Δu+λu=μ∣u∣p−2u+f(u),inR2,-\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 ∫R2∣u∣2dx=a2>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∈C1(R,R)f\in {C}^{1}\left({\mathbb{R}},{\mathbb{R}}) behaves like e4πu2{e}^{4\pi {u}^{2}} as ∣u∣→+∞| u| \to +\infty . For a L2{L}^{2}-critical or L2{L}^{2}-subcritical perturbation μ∣u∣p−2u\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→0a\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 RN{{\mathbb{R}}}^{N}, Calc. Var. Partial Differential Equations 61 (2022), no. 1, 1–24).

Keywords

• exponential critical growth
• moser-trudinger inequality
• schrödinger equation
• normalized solution
• variational method
• 35j20
• 35b33
• 35b45