Classification of nonnegative solutions to Schrödinger equation with logarithmic nonlinearity

In this paper, we study the physically interesting static Schrödinger equation with logarithmic nonlinearity: (-Δ)su(x)=c1(1|·|σ∗log(1+up1))uq1(x)+c2log(1+up2(x))uq2(x),inRn,involving higher-order or higher-order fractional Laplacians, where n≥ 1 , 0<s:=m+α2<n2, m≥ 0 is an integer, 0 < α≤ 2 , 0 < σ< n, 0<p1≤2n-σn-2s, 0<q1≤n+2s-σn-2s, q₂≥ 0 , p₂> 1 and 1<p2+q2<n+2sn-2s if α= 2 , p₂> 0 and 0<p2+q2<n+2sn-2s if 0 < α< 2. We first prove the super poly-harmonic properties of nonnegative classical solutions to the above PDEs and then derive the equivalence between the PDEs and the corresponding integral equation. Finally, we derive the explicit forms for positive solution u in the critical case and the non-existence of non-trivial nonnegative solutions in the subcritical cases via the method of moving spheres in integral form. In other words, we obtain the classification results of nonnegative classical solutions for the above PDEs equation.