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Existence of positive ground state solutions for Choquard equation with variable exponent growth

Supported by National Natural Science Foundation of China (No.11471267).
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  • In this paper, we investigate the following Choquard equation

    $ \begin{equation*} -\Delta u = (I_\alpha*|u|^{f(x)})|u|^{f(x)-2}u \ \ \ \ {\rm in} \ \mathbb{R}^N, \end{equation*} $

    where $ N\geq 3 $, $ \alpha\in (0,N) $ and $ I_\alpha $ is the Riesz potential. If

    $ \begin{equation*} f(x) = \begin{cases} p, &x\in\Omega,\\ ( N+\alpha)/(N-2), &x\in\mathbb{R}^N \backslash\Omega, \end{cases} \end{equation*} $

    where $ 1< p <\frac{N+\alpha}{N-2} $ and $ \Omega\subset\mathbb{R}^N $ is a bounded set with nonempty, we obtain the existence of positive ground state solutions by using the Nehari manifold.

    Mathematics Subject Classification: Primary: 35A15, 35B50; Secondary: 35J61.

    Citation:

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