Ground state solutions for some Schrödinger-Poisson systems with periodic potentials
Let \(V:\mathbb{R}^3\rightarrow \mathbb{R}\) and \(f:\mathbb{R}^3\times \mathbb{R}\rightarrow \mathbb{R}\) be two continuous functions with \(\inf_{\mathbb{R}^3}V>0\). The authors investigate the existence of ground state solutions to the following Schrödinger-Poisson system \[ \begin{cases} -\Delta u+V(x)u +\phi u = f(x,u) &\text{ in} \;\mathbb{R}^3,\\ -\Delta \phi =u^2 &\text{in} \;\mathbb{R}^3.\end{cases} \] The existence of at least a ground state solution is proved under the following conditions: {\parindent=6mm \begin{itemize} \item[(i)] \ \(f(\cdot,t)\) and \(V\) are \(1\)-periodic in each of the variables \(x_1,x_2,x_3\); \item [(ii)] \ \(f(x,\cdot)\) is subcritical at infinity and superlinear at \(0\), uniformly in \(x\in \mathbb{R}^3\); \item [(iii)] \ \(\lim_{|t|\rightarrow \infty}\displaystyle{\frac{\int_0^tf(x,s)ds}{t^4}}=+\infty\), uniformly in \(x\in \mathbb{R}^3\); \item [(iv)] \ for each \(x\in \mathbb{R}^3\), the function \(t\rightarrow t^{-3} f(x,t)\) is positive in \(\mathbb{R}\setminus\{0\}\), strictly decreasing in \((-\infty,0)\) and strictly increasing in \((0,+\infty)\). \end{itemize}} The authors also prove that the set of the ground state solutions is compact (up to translation) and, if \(f(x,\cdot)\) is odd, contains infinitely many geometrically distinct solutions (i.e. with distinct orbits under the action of \(\mathbb{Z}^3\)) as well. The ground state solutions are found by minimizing the energy functional on the Nehari manifold. Existence results of ground state solutions are also established for the critical growth case with a nonlinearity of the form \(f(x,t)=K(x)|t|^4t+\lambda g(x,t)\), where \(\lambda\) is a positive parameter, \(g\) is a subcritical perturbation and \(K\) is a positive continuous function. Finally, the authors consider the autonomous case with different assumptions on the nonlinearity \(f\) and prove the existence of critical points of the energy functional using the Jeanjean's monotonicity trick. Then, a least energy solution is found by minimizing the energy functional over the set of all critical points.
scientific article; zbMATH DE number 6517066
scientific article