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Infinitely many solutions for nonlinear Schrödinger equations with slow decaying of potential

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  • In the paper we prove the multiplicity existence of both nonlinear Schrödinger equation and Schrödinger system with slow decaying rate of electric potential at infinity. Namely, for any $\mathit{\boldsymbol{m}},\mathit{\boldsymbol{n > }}{\bf{0}}$ , the potentials $P, Q$ have the asymptotic behavior

    $\left\{ \begin{array}{l}P(r) = 1 + \frac{a}{{{r^m}}} + O\left( {\frac{1}{{{r^{m + \theta }}}}} \right),{\rm{ }}\;\;\;\;\theta > 0\\Q(r) = 1 + \frac{b}{{{r^n}}} + O\left( {\frac{1}{{{r^{n + \widetilde \theta }}}}} \right),\;\;\;\;\;\widetilde \theta > 0\end{array} \right.$

    then Schrödinger equation and Schrödinger system have infinitely many solutions with arbitrarily large energy, which extends the results of [37] for single Schrödinger equation and [30] for Schrödinger system.

    Mathematics Subject Classification: Primary:35B40, 35B45;Secondary:35J40.


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