Critical Schrödinger equations with Stein-Weiss convolution parts in \(\mathbb{R}^2\)
In this paper the authors study the equation \[ -\Delta u + u =\frac{1}{|x|^\beta}\left( \int_{\mathbb{R}^2} \frac{F(u)}{|x-y|^\mu|y|^\beta} \ dy\right)f(u), \ \ x\in \mathbb{R}^2, \] where \(f\) is a function with critical exponential growth, and \(F\) is a primitive. Using a general version of the Pohožaev identity, the authors employ a general minimax principle to investigate the existence of mountain-pass type solutions for the equation. Even without a monotonicity restriction on the nonlinearity \(f\), it is shown that the mountain-pass value equals the least energy level.
scientific article; zbMATH DE number 7623988
scientific article