Every nonnegative solution of the static Schrödinger-Hartree-Maxwell equations with higher-order or fractional Laplacians is either zero, or in the critical case an explicit rescaled bubble.
A Liouville Theorem for the Fractional Laplacian
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We extend the classical Liouville Theorem from Laplacian to the fractional Laplacian, that is, we prove Every $\alpha$-harmonic function bounded either above or below in all of $R^n$ must be constant.
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Classification of nonnegative solutions to static Schr\"{o}dinger-Hartree-Maxwell type equations
Every nonnegative solution of the static Schrödinger-Hartree-Maxwell equations with higher-order or fractional Laplacians is either zero, or in the critical case an explicit rescaled bubble.