Classification of nonnegative classical solutions to third-order equations

In this paper, we are concerned with the third-order equations (1.1) and (1.11) with critical or subcritical nonlinearities. By applying the method of moving planes to the third-order PDEs (1.1) and (1.11), we prove that nonnegative classical solutions u to (1.1) and (1.11) are radially symmetric about some point x₀∈Rⁿ and derive the explicit forms for u in the critical case. We also prove the non-existence of nontrivial nonnegative classical solutions in the subcritical cases (see Theorem 1.1 and Theorem 1.3).