Zero-mass gauged Schrödinger equations with supercritical exponential growth

We study the following gauged nonlinear Schr & ouml;dinger equation {-Delta u+(integral(infinity)(|x|)h(u)(s)/su(2)(s)ds+h(u)(2)(|x|)|x|(2))u = f(u)-a|u|(p-2)u, u(x) = u(|x|), where a > 0, p is an element of (1, 2), hu(s) = f(0)(s) r/2u(2)(r)dr and f possesses the supercritical exponential growth in the Trudinger-Moser sense at infinity. Via introducing a new type of Trudinger-Moser inequality in a suitable work space here, we shall exploit the general minimax principle and elliptic regular result to investigate the existence of mountain -pass type solutions for the equation using variational method. (c) 2024 Elsevier Inc. All rights reserved.