Normalized Solutions to N-Laplacian Equations in RN with Exponential Critical Growth

In this paper, we are concerned with normalized solutions (u, lambda) is an element of W-1,W-N(R-N) x R+ to the following N-Laplacian problem -div(|del u|(N-2)del u) + lambda|u|(N-2)u = f(u) in R-N, N >= 2, satisfying the normalization constraint integral(N)(R) |u|(N) dx = 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| -> infinity. 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.