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Bound state solutions of Schrödinger-Poisson system with critical exponent


Research supported by the National Natural Science Foundation of China (No. 11571187)

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  • In this paper, we consider the following Schrödinger-Poisson problem

    $\tag{P}\label{0.1} \begin{cases}- Δ u+V(x)u+K(x)φ u=|u|^{2^*-2}u, &x∈ \mathbb{R}^3,\\-Δ φ=K(x)u^2,&x∈ \mathbb{R}^3,\end{cases}$

    where $2^*=6 $ is the critical exponent in $\mathbb R^3$, $ K∈ L^{\frac{1}{2}}(\mathbb{R}^3)$ and $V∈ L^{\frac{3}{2}}(\mathbb{R}^3)$ are given nonnegative functions. When $|V|_{\frac{3}{2}}+|K|_{\frac{1}{2}}$ is suitable small, we prove that problem (P) has at least one bound state solution via a linking theorem.

    Mathematics Subject Classification: Primary:35J20;Secondary:35J60.


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