Positive distribution solutions of $-Δu + a²Δ²u = u^q$ in ℝ³ exist only for q>5, super-solutions only for q>3, and all such solutions are radial, differentiable, and decay like the Coulomb-type kernel $K_a(x)=(1-e^{-|x|/a})/|x|$.
Bopp,Eine lineare theorie des elektrons, Ann
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Distribution solutions of a static dispersion Schr\"{o}dinger equation
Positive distribution solutions of $-Δu + a²Δ²u = u^q$ in ℝ³ exist only for q>5, super-solutions only for q>3, and all such solutions are radial, differentiable, and decay like the Coulomb-type kernel $K_a(x)=(1-e^{-|x|/a})/|x|$.