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A positive bound state for an asymptotically linear or superlinear Schrödinger equation in exterior domains

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    * Corresponding author
Research supported by FAPDF 193.000.939/2015 and 0193.001300/2016, CNPq/PQ 308173/2014-7 and PROEX/CAPES (Brazil).
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  • We establish the existence of a positive solution for semilinear elliptic equation in exterior domains

    $\begin{array}{lc}-Δ u + V(x) u = f(u),\ \ {\rm{in}} \ \ Ω \subseteq \mathbb{R}^N &&&(P_V)\end{array}$

    where $N≥2$, $Ω$ is an open subset of $\mathbb{R}^N$ and $ \mathbb{R}^N \setminus Ω $ is bounded and not empty but there is no restriction on its size, nor any symmetry assumption. The nonlinear term $f$ is a non homogeneous, asymptotically linear or superlinear function at infinity. Moreover, the potential Ⅴ is a positive function, not necessarily symmetric. The existence of a solution is established in situations where this problem does not have a ground state.

    Mathematics Subject Classification: Primary: 35Q55, 35J61; Secondary: 35J20.


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