We study the existence of positive solutions for the non-autonomous Schrödinger-Poisson system: $\left\{ {\begin{array}{*{20}{l}} { - \Delta u + u + \lambda K\left( x \right)\phi u = a\left( x \right){{\left| u \right|}^{p - 2}}u}&{{\text{in }}{\mathbb{R}^3},} \\ { - \Delta \phi = K\left( x \right){u^2}}&{{\text{in }}{\mathbb{R}^3},} \end{array}} \right.$ where $\lambda >0$, $2 < p \le 4$ and both $K\left( x\right) $ and $a\left( x\right) $ are nonnegative functions in $\mathbb{R}^{3}$, which satisfy the given conditions, but not require any symmetry property. Assuming that $% \lim_{\left\vert x\right\vert \rightarrow \infty }K\left( x\right) = K_{\infty }\geq 0$ and $\lim_{\left\vert x\right\vert \rightarrow \infty }a\left( x\right) = a_{\infty }>0$, we explore the existence of positive solutions, depending on the parameters $\lambda$ and $p$. More importantly, we establish the existence of ground state solutions in the case of $3.18 \approx \frac{{1 + \sqrt {73} }}{3} < P \le 4$.

In this paper, under certain parametric conditions we are concerned with the first integrals of the Duffing-van der Pol-type oscillator system, which include the van der Pol oscillator and the damped Duffing oscillator etc as particular cases. After applying the method of differentiable dynamics to analyze the bifurcation set and bifurcations of equilibrium points, we use the Lie symmetry reduction method to find two nontrivial infinitesimal generators and use them to construct canonical variables. Through the inverse transformations we obtain the first integrals of the original oscillator system under the given parametric conditions, and some particular cases such as the damped Duffing equation and the van der Pol oscillator system are included accordingly.

We study the following non-autonomous Schrödinger–Poisson system where 2 < p < 4, and both and are non-negative functions in . By proposing a new analytical method along with the Lusternik–Schnirelmann category, we prove the existence of two bounded positive solutions in , whose energy levels are positive and lie in a definite range. The main feature which distinguishes this paper from other related works lies in the fact that it is the first attempt to study two positive solutions of the above Schrödinger–Poisson system with 2 < p < 4.