Nonexistence of Nontrivial Stationary Solutions with Decay Order for Some Nonlinear Evolution Equations

We show that there are no nontrivial stationary solutions of certain decay order for some applied nonlinear evolution equations which include the thin epitaxial film model with slope selection and the square phase field crystal (SPFC) equation. The method is to use the Morawetz multiplier and the Gagliardo-Nirenberg inequality. 2010 Mathematics Subject Classifications Primary 35J60, 35J61, 35J62.

In this paper, we will study the equation that is more general than the above equations (1.3) and (1.4), where a and b are constants and f is a continuous function.We will show that under certain conditions of a, b, and ( , ) f x u , there is no nontrivial solution with certain decay order for (1.5).The method is to use the Morawetz multiplier (Levandosky & Strauss, 2000;Morawetz 1968) and the Gagliardo-Nirenberg inequality (Bellazzini, Frank, & Visciglia, 2014).
Nonlinear fourth-order elliptic equations related to (1.5) have attracted a lot of attentions lately.For example, for the equation 2 p uu   , it is known (Lin 1998;Wei and Xu 1999;Fazly, Wei, & Xu 2015)   As usual, 12 ( , ,..., )  denotes the gradient of u, u  denotes the divergence of u, and rx  .Also the subscript denotes the partial derivative, thus . We also use the notation CR is the space of functions whose partial derivatives of order up to and including k are continuously differentiable.
Define the following four sets of functions D h, k (R n ), E m (R n ), F n (R n ), and NF(R n ), which we will use in this article:  , for all multi-indices α and β ϵ 0 N n such that |α| ≤ h and |β| ≤ k, where α where F(x, u) is the antiderivative of f(x, u) with respect to u such that F(x, 0) = 0.
Remark 1.A function u is said to be of decay order (h, k) if and only if u ϵ D h , k (R n ).All the functions are assumed to be real-valued.

Morawetz Multiplier
Multiplying both sides of Equation ( 1) 0 where Y depends on ζ and u as well as their partial derivatives up to and including the third order and F(x, u), and To show the assertion (a), we recall the Gagliardo-Nirenberg inequality (Bellazzini, Frank, & Visciglia, 2014) where C n , depending only on n, is the least positive constant such that the above inequality (3.2) holds for all u in H 2 (R n ).
Therefore, since a > 0, classified.Thus it is natural to consider the case of equation (1.5) which combines the effect of nonlinearity, fourth order, and the p-Laplacian operator (Drá bek 2007).