Combined effects of logarithmic and superlinear nonlinearities in fractional Laplacian systems

Abstract

In this paper, we consider the existence and multiplicity of solutions for the following fractional Laplacian system with logarithmic nonlinearity

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^{s}u=\lambda h_1(x)u\ln |u|+\frac{p}{p+q}b(x)|v|^{q}|u|^{p-2}u\ \ &{}x\in \Omega , \\ (-\Delta )^t v=\mu h_2(x)v\ln |v|+\frac{q}{p+q}b(x)|u|^p|v|^{q-2}v\ \ &{}x\in \Omega , \\ u=v=0\ \ &{}x\in \mathbb {R}^N{\setminus }\Omega , \end{array}\right. } \end{aligned}$$

where $s,t\in (0,1),\ N>\max \{2s,2t\}$, $\lambda ,\mu >0$, $2<p+q<\min \{\frac{2N}{N-2s},\frac{2N}{N-2t}\}$, $\Omega \subset \mathbb {R}^N$ is a bounded domain with Lipschitz boundary, $h_1,h_2,b\in C(\overline{\Omega })$ and $(-\Delta )^{s}$ is the fractional Laplacian. When $h_1,h_2,b$ are positive functions, the existence of ground state solutions for the problem is obtained. When $h_1,h_2$ are sign-changing functions and b is a positive function, two nontrivial and nonnegative solutions are obtained. Our results are new even in the case of a single equation.