Critical Schrödinger equations with Stein-Weiss convolution parts in [formula omitted

Abstract

We study the following planar Schrödinger equations with Stein-Weiss convolution parts−Δu+u=1|x|β(∫R2F(u)|x−y|μ|y|βdy)f(u),x∈R2, where β≥0, 0<μ<2 with 0<2β+μ<2 and F is the primitive of f that fulfills a critical exponential growth in the Trudinger-Moser sense. Via establishing a general version of Pohoz̆aev identity, we shall exploit the general minimax principle to investigate the existence of mountain-pass type solutions for the equation. Although there is no any monotonicity restriction on the nonlinearity f, we can show that the mountain-pass value equals to the least energy level. Our results extend the ones in, e.g. Yang-Rădulescu-Zhou (2022) [44] and Du-Gao-Yang (2022) [15], to the lower dimension.