Concentration phenomena for magnetic Kirchhoff equations with critical growth

Abstract

<p style='text-indent:20px;'>In this paper, we study the following nonlinear magnetic Kirchhoff equation with critical growth</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} \left\{ \begin{aligned} &amp;-\Big(a\epsilon^{2}+b\epsilon\, [u]_{A/\epsilon}^{2}\Big)\Delta_{A/\epsilon} u+V(x)u = f(|u|^{2})u+\vert u\vert^{4}u \quad \hbox{in }\mathbb{R}^3, \\ &amp;u\in H^{1}(\mathbb{R}^{3}, \mathbb{C}), \end{aligned} \right. \end{align*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \epsilon&gt;0 $\end{document}</tex-math></inline-formula> is a parameter, <inline-formula><tex-math id="M2">\begin{document}$ a, b&gt;0 $\end{document}</tex-math></inline-formula> are constants, <inline-formula><tex-math id="M3">\begin{document}$ V:\mathbb{R}^{3}\rightarrow \mathbb{R} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ A: \mathbb{R}^{3}\rightarrow \mathbb{R}^{3} $\end{document}</tex-math></inline-formula> are continuous potentials, and <inline-formula><tex-math id="M5">\begin{document}$ f: \mathbb{R}\rightarrow \mathbb{R} $\end{document}</tex-math></inline-formula> is a nonlinear term with subcritical growth. Under a local assumption on the potential <inline-formula><tex-math id="M6">\begin{document}$ V $\end{document}</tex-math></inline-formula>, combining variational methods, penalization techniques and the Ljusternik-Schnirelmann theory, we establish multiplicity and concentration properties of solutions to the above problem for <inline-formula><tex-math id="M7">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula> small. A feature of this paper is that the function <inline-formula><tex-math id="M8">\begin{document}$ f $\end{document}</tex-math></inline-formula> is assumed to be only continuous, which allows to consider larger classes of nonlinearities in the reaction.</p>