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Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity

  • * Corresponding author: Junping Shi

    * Corresponding author: Junping Shi 

This work is partially supported by the National Natural Science Foundation of China (No: 11571370)

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  • It is shown that the planar Schrödinger-Poisson system with a general nonlinear interaction function has a nontrivial solution of mountain-pass type and a ground state solution of Nehari-Pohozaev type. The conditions on the nonlinear functions are much weaker and flexible than previous ones, and new variational and analytic techniques are used in the proof.

    Mathematics Subject Classification: Primary: 35J20; Secondary: 35Q55.


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  • Table 1.  Examples of nonlinear functions $ f(u) $ satisfying conditions in Theorems 1.1 and 1.2. Here $ b_0 = \frac{q(p-2)}{(q-1)(3-q)}\left[\frac{(p-1)(p-3)}{p(q-2)}\right]^{\frac{q-2}{p-2}}[2(p-q)]^{\frac{q-p}{p-2}}. $

    $ f(u) $ (F4) (F5) (F6)
    $ f_1(u)=|u|^{p-2}u $ $ 3\le p $ $ 2 <p\le 3 $ $ 3\le p $
    $ f_2(u)=\left(|u|^{p-2}+b|u|^{q-2}\right)u $ $ 2<q<3< p $ $ 2<q<p\le 3 $ $ 2<q<3\le p $
    $ 0\le b\le b_0 $
    $ f_3(u)=u\left[1-\frac{1}{\ln(e+u^2)}\right] $ YES YES NO
    $ f_4(u)=u\ln (1+u^2) $ NO YES NO
    $ f_5(u)=|u|u\ln (1+u^2) $ YES NO YES
    $ f_6(u)=3|u|u\ln\left(1+u^2\right)+\frac{2|u|^3u}{1+u^2} $ YES NO YES
    $ f_7(u)=4u^3\int_{0}^{u}|s|^{1+\sin s}s\mathrm{d}s+|u|^{5+\sin u}u $ YES NO NO
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