SOLUTIONS TO NON-HOMOGENEOUS SCHRÖDINGER-POISSON SYSTEM INVOLVING A (P, Q)-LAPLACIAN OPERATOR

In this paper, we consider the following generalized Schrödinger-Poisson system $ \begin{equation*} \left\{ \begin{array}{ll} -\Delta_p u-\Delta_q u+|u|^{p-2}u+\lambda\phi |u|^{s-2}u=|u|^{l-2}u+f(x)\ \ &\text{in}\ \mathbb{R}^3,\\ -\Delta_r\phi=|u|^s &\hbox{in} \ \mathbb{R}^3,\\ \end{array}\right. \end{equation*} $ where $ p,q,r\in(1,3) $ with $ p<q $, $ \Delta_mu=\text{div}(|\nabla u|^{m-2}\nabla u) $ and $ m^*=\frac{3m}{3-m} $ with $ m\in\{p,q,r\} $, stand for the $ m $-Laplacian operator and Sobolev critical exponent respectively. $ \max\{1,\frac{p(r^*-1)}{r^*}, \frac{q(r-1)}{r}\}<s<\frac{q^*(r^*-1)}{r^*}, $ $ q<l<q^* $, $ \lambda $ is a positive parameter, $ f $ satisfies certain integrability conditions. First, one solution with negative energy is obtained by Ekeland variational principle for any $ \lambda>0 $ and $ l\in(q,q^*) $. Second, according to the range of $ l $, we use two methods to obtain one solution with positive energy. Precisely, for the case $ \frac{qr(s+1)}{r+q(r-1)}<l<q^* $, we employ a scaling technique to demonstrate the boundedness of Palais-Smale sequence, furthermore, one solution with positive energy is got by mountain pass theorem for any $ \lambda>0 $; for the case $ q<l<\frac{qr(s+1)}{r+q(r-1)} $, the cut-off technique is used to obtain a bounded Palais-Smale sequence for $ \lambda>0 $ small enough, then one solution with positive energy is also obtained for $ \lambda>0 $ small enough.