Ground state solutions for asymptotically periodic modified Schrödinger-Poisson system involving critical exponent (Q2069881)

This paper deals with the existence of non-trivial solutions for a modified Schrödinger-Poisson system associated with the natural energy functional of the form \begin{align*} \mathcal{J}(u,\phi) &=\frac{1}{2} \int_{\mathbb{R}^3} |\nabla u|^2 + V(x) u^2 \; dx - \frac{1}{10} \int_{\mathbb{R}^3} K(x) \phi |u|^{10} \; dx \\ &+\int_{\mathbb{R}^3} u^2 |\nabla u|^2 \; dx - \int_{\mathbb{R}^3} \int_{0}^{u} g(x,s) ds \; dx + \frac{1}{20} \int_{\mathbb{R}^3} |\nabla \phi|^2 \; dx, \end{align*} where \(V\), \(K\) and \(g\) are asymptotically periodic functions of \(x\). The existence of ground state solution is ensured with the help of variational methods and the dual approach. Due to the presence of critical and superlinear growth terms and since the natural energy functional \(\mathcal{J}(u,\phi)\) is not well defined in \(H^1(\mathbb{R}^3) \times D^{1,2}(\mathbb{R}^3)\), where \(D^{1,2}(\mathbb{R}^3)\) is the completion of \(C_0^\infty(\mathbb{R}^3)\) with respect to the \(L^2\)-norm of \(\nabla u\), special treatment is required, including the use of Nehari manifold method, the Mountain Pass theorem and the concentration-compactness principle. The existing theory is therefore extended to handle negative non-local term in either the critical or subcritical case, which represents the main contribution of the paper.