Normalized Solutions to N-Laplacian Equations in $${\mathbb {R}}^N$$ with Exponential Critical Growth

Abstract

In this paper, we are concerned with normalized solutions \((u,\lambda )\in W^{1,N}(\mathbb {R}^N)\times \mathbb {R}^+\) to the following N-Laplacian problem

$$\begin{aligned} -{\text {div}}(|\nabla u|^{N-2} \nabla u)+\lambda |u|^{N-2} u=f(u) \text{ in } \mathbb {R}^N,~N \ge 2, \end{aligned}$$

satisfying the normalization constraint \(\int _{\mathbb {R}^N}|u|^N\textrm{d}x=c^N\). The nonlinearity f(s) is an exponential critical growth function, i.e., behaves like \(\exp (\alpha |s|^{N /(N-1)})\) for some \(\alpha >0\) as \(|s| \rightarrow \infty \). Under some mild conditions, we show the existence of normalized mountain pass type solution via the variational method. We also emphasize the normalized ground state solution has a mountain pass characterization under some further assumption. Our existence results in present paper also solve a Soave’s type open problem (J Funct Anal 279(6):108610, 2020) on the nonlinearities having an exponential critical growth.