Multi-bump solutions of the magnetic p-Laplacian Schrödinger equations with critical and logarithmic nonlinearities

This paper focuses on the following magnetic p-Laplacian Schrödinger equations with critical and logarithmic nonlinearities in $R^N$:$-{\Delta }_{p,A}u+\left(\lambda Z\right(x)+V(x\left)\right)|u{|}^{p-2}u=\beta |u{|}^{p-2}u\log |u{|}^p+|u{|}^{p^{⁎}-2}u,x\in R^N,$ where $N\ge 3$, $p\in [2,N)$, ${\Delta }_{p,A}=\text{div}\left(\right|\nabla u+iA\left(x\right)u{|}^{p-2}(\nabla u+iA(x\left)u\right))$ denotes the magnetic p-Laplacian, the magnetic potential $A\in L_{loc}^p(R^N,R^N)$ and the parameters $\lambda ,\beta \ge 1$, $Z\left(x\right),V\left(x\right):R^N\to R$ are the nonnegative continuous functions, $p^{⁎}=\frac{Np}{N-p}$ is the Sobolev critical exponent. Applying variational methods, multiple multi-bump solutions for the above equation have been obtained. More precisely, our findings demonstrate that if the zero set of Z possesses several isolated connected components ${\Omega }_1,\cdots ,{\Omega }_k$ such that the interior of ${\Omega }_i$ is not empty and ∂Ω is smooth, then for $\lambda \ge 1$ sufficiently large, a bump solution is confined in a neighborhood of ${\bigcup }_{j\in \Gamma }{\Omega }_j$ for every non-empty subset $\Gamma \subset \{1,\cdots ,k\}$. Furthermore, let $\lambda \ge 1$ be large enough, we also show that the above equation has at least $2^k-1$ multi-bump solutions. The novelty and characteristic of this paper lie in the simultaneous appearance of critical and logarithmic nonlinearities in this equation, and the results in this paper extend the subcritical case [39] to the critical case.