Existence and Multiplicity of Solutions for a Mixed Local–Nonlocal System with Logarithmic Nonlinearities

Abstract

In this paper, we consider the following mixed local-nonlocal system with logarithmic nonlinearities

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u_j+(-\Delta )^su_j=\lambda _ja_j(x)u_j\ln |u_j|+\sum \limits _{i\ne j}\beta _{ij}|u_i|^q|u_j|^{q-2}u_j \quad &{} \text{ in } \Omega ,\\ u_j=0\quad &{} \text{ on } \partial \Omega , \end{array}\right. } \end{aligned}$$

where $s\in (0,1)$, $\Omega \subset \mathbb {R}^N$ is a bounded domain with Lipschitz boundary, $N>2$, $a_j\in C({\overline{\Omega }})$, $\lambda _j>0$ is k parameters, $\beta _{ij}>0$ for all $1\le i<j\le k$, $\beta _{ij}=\beta _{ji}$ for $i\ne j$, $j=1,2,\cdots ,k$. When $1<2q<2<2^*=\frac{2N}{N-2}$ and $a_j$ are sign-changing functions, two distinct solutions are obtained by using Nehari manifold and fibering map method. When $2<q<2q<2^*=\frac{2N}{N-2}$ and $a_j$ are positive constant functions, the existence of ground state solution is obtained by using minimization method.