Non-Degeneracy and Infinitely Many Solutions for Critical SchrÖDinger-Maxwell Type Problem

In this paper, we consider the following Schrödinger-Maxwell type equation with critical exponent \(-\Delta u=K(y)\Big (\frac{1}{|x|^{n-2}}*K(x)|u|^{\frac{n+2}{n-2}}\Big )u^{\frac{4}{n-2}},\quad {in}\,\, \mathbb {R}^n, \qquad \text {(0.1)}\) where the function K satisfies the assumption \(\mathcal {F}\), and \(*\) stands for the standard convolution. We first derived the non-degeneracy result for the critical Schrödinger-Maxwell equation. Then, as an application, we proved that problem Eq. (0.1) admits infinitely many non-radial positive solutions with arbitrary large energy. We believe that the various new ideas and technique computations that we used in this paper would be useful to deal with other related elliptic problems involving convolution nonlinear terms.