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Ground states for Kirchhoff-type equations with critical growth

This work is supported in part by the National Natural Science Foundation of China (11501403; 11461023; 11701322; 11561072) and the Shanxi Province Science Foundation for Youths under grant 2013021001-3 and the Honghe University Doctoral Research Programs (XJ17B11, XJ17B12) and the Yunnan Province Local University (Part) Basic Research Joint Project (2017FH001-013).
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  • In this paper, we study the following Kirchhoff-type equation with critical growth

    $-(a+b\int {_{\mathbb{R}^3}} |\nabla u|^2dx)\triangle u+V(x)u = λ f(x,u)+|u|^4u, \; x \; ∈\mathbb{R}^3,$

    where a>0, b>0, λ>0 and f is a continuous superlinear but subcritical nonlinearity. When V and f are asymptotically periodic in x, we prove that the equation has a ground state solution for large λ by Nehari method. Moreover, we regard b as a parameter and obtain a convergence property of the ground state solution as $b\searrow 0$.

    Mathematics Subject Classification: 35J20, 35J70, 35P05, 35P30, 34B15, 58E05, 47H04.


    \begin{equation} \\ \end{equation}
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