Abstract
<abstract><p>In this paper, we study the following non-autonomous Schrödinger-Poisson equation with a critical nonlocal term and a critical nonlinearity:</p>
<p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{\begin{aligned} & -\Delta u +V(x) u + \lambda \phi |u|^3 u = f(u) + (u^+)^5,\ \ {\rm in } \ \ \ \ \mathbb{R}^3,\\ & -\Delta \phi = |u|^5, \ \ {\rm in } \ \ \ \ \mathbb{R}^3. \end{aligned}\right. \end{equation*} $\end{document} </tex-math></disp-formula></p>
<p>First, we consider the case that the nonlinearity satisfies the Berestycki-Lions type condition with critical growth. Second, we consider the case that $ \mathrm{int}V^{-1}(0) $ is contained in a spherical shell. By using variational methods, we obtain the existence and asymptotic behavior of positive solutions.</p></abstract>
Publisher
American Institute of Mathematical Sciences (AIMS)
Reference31 articles.
1. A. Ambrosetti, P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349–381. https://doi.org/10.1016/0022-1236(73)90051-7
2. A. Azzollini, A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345 (2008), 90–108. http://doi.org/10.1016/j.jmaa.2008.03.057
3. V. Benci, D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Method Nonl. An., 11 (1998), 283–293. http://doi.org/10.12775/TMNA.1998.019
4. H. Berestycki, T. Gallouët, O. Kavian, Equations de champs scalaires euclidiens non linéaire dans le plan, C. R. Acad. Sci. Paris Ser. I Math., 297 (1983), 307–310.
5. H. Berestycki, P.-L. Lions, Nonlinear scalar field equations I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313–345. https://doi.org/10.1007/BF00250555