Non-existence of positive solutions for a higher order fractional equation

In this paper, we consider a nonlinear equation involving fractional Laplacian of higher order on the whole space. We establish the equivalence between the pseudo-differential equation and an integral equation by applying the maximum principle and the Liouville theorem. For positive solutions to the equation, we obtained non-existence by applying the method of moving planes.

1. Introduction. In this paper, we investigate the following problem: where 0 < α < 2, s = α + 2 < n, a > −s, p > 0, the operator (−∆) α 2 +1 is defined by for u ∈ S, the Schwartz space of rapidly decreasing smooth functions on R n , and c n,α is a normalized positive constant depending on n and α . Denote |u(x)| 1 + |x| n+α+2 dx < ∞}, then the operator (−∆) α 2 +1 can be extended on a wider spaces L s ∩C s+ loc (R n ), > 0. For u ∈ S, the usual fractional Laplacian (−∆) α 2 as a non-local operator is defined by where c(n, α) is positive constant only depending on n and α. The operator (−∆) α 2 can also be equivalently described by the Fourier transform: satisfies the problem: The classical theories for local elliptic partial differential equations can be applied. We refer to [3,15] and references therein for broad applications of this method.
For u ∈ L s ∩ C s+ loc (R n ), it follows from [25] that We say that a function u : R n → R is a classic solution to problem (1), if u ∈ L s ∩ C s+ loc (R n ) satisfies problem (1) in the point-wise sense. In the following, we give our main results. (1) has no positive solutions. Theorem 1.2. For a = 0, p = n+s n−s , every positive solution u(x) of (1) is radially symmetric and decreasing about some point x 0 and therefore assumes the form with some positive constants c and t. Theorem 1.3. For a = 0, 0 < p < n+s n−s , problem (1) has no positive solutions. The fractional Laplacian operator comes from many phenomena, such as quantum mechanics, anomalous diffusion, turbulence, molecular dynamics, phase transitions and crystal dislocation [4,5,19,20,31,33]. In probability and finance, it also can be seen as the infinitesimal generator of Lévy stable diffusion processes. See [1,5,8,17,26,28,29,32,34] and the references therein.
In recent years, many authors investigated existence [3,4,5,19,20,30,31,35], regularity [3,4,5,7,31,33], Symmetry [9,16,19,20,21,35] and monotonicity [16,21,30] of the elliptic equations involving the fractional Laplacian operator. But there are few results (due to lack of maximum principle) for equations involving higher fractional Laplacian operator (−∆) α 2 +1 . An useful method to study the fractional Laplacian is the integral equations method, which turns a given fractional Laplacian equation into its equivalent integral equation, and then various properties of the original equation can be obtained by investigating the integral equation, see [11,13,14] and references therein. In [35], Zhuo,Chen,Cui and Yuan established the equivalence between a fractional Laplacian equation and an integral equation, and obtained radial symmetry and non-existence for positive solutions. In order to explore symmetry and non-existence of the solutions to (1), we first give an equivalent equation in integral form. Then we prove non-existence of positive solutions by applying the method of moving planes of integral form. This paper is organized as follows. Section 2 devotes to establish the equivalence between the equation and an equation in integral form. In Section 3, we prove non-existence of positive solutions to the equation.
2. The equivalent equation in integral form. The following Liouville theorem for α-harmonic function first appeared in [2]. An alternative proof was given in [35].
then u ≡ C.
The following maximum principle is also crucial for us.

Lemma 2.2. [31]
Let Ω ⊂ R n be a bounded open set, and let f be a lowersemicontinuous function inΩ such that (−∆) Theorem 2.3. Assume that u is a nonnegative solution of (1), then u also satisfies and vice versa.
Proof. According to Kulczycki[24], the Green function It is easy to verify A direct computation derives that Since Lemma 2.2 and (7) show that By the maximum principles of the Laplace, it follows that Hence v R (x) is well defined and the limit exists in the pointwise sense as R → ∞. Let |y| a u p (y) |x − y| n−s dy. Taking the limit for R → ∞ in (7) and (9), we get By the Liouville theorems of the fractional Laplacian and the fact (−∆) It is easy to see that b = 0 by contradiction. Applying Liouville theorems of the Laplace, there exists a constant c ≥ 0, such that Assume that c > 0, then which implies that c = 0. Hence (6) holds.

Remark 1.
In the proof of Theorem 2.3, the condition (−∆)u(x) ≥ 0 appearing in (8) is crucial to apply the maximum principle. It would be very interesting to know whether this assumption can be removed.
3. Proof of main results. For a = 0, p = n+s n−s , Chen, Li and Ou [13,14] obtained the following result for integral equation (6).
Theorem 3.1. [13,14] Every positive solution u(x) of (6) with a = 0, 0 < s < n, p = n+s n−s is radially symmetric and decreasing about some point x 0 and therefore assumes the form with some positive constants c and t.
Combining Theorem 2.3 with Theorem 3.1, we can obtain Theorem 1.2. For the integral equation (6) with a = 0, 0 < p < n+s n−s , Y.Y. Li [27] proved non-existence of positive solutions by using the method of moving sphere. For a = 0, 0 < p < n+s n−s , Theorem 1.3 follows directly from Theorem 3.2 and Theorem 2.3. Theorem 1.1 can be concluded by applying the method of moving planes, which is also valid for a = 0, 1 ≤ p < n+s n−s . We will present details in the following. Replacing x by x |x| 2 in (6), we get be the Kelvin transform of u. Then (6) can be rewritten as |x − y| n−s |y| n+s+a−p(n−s) dy.
Consider the x 1 direction. For x = (x 1 , x 2 , · · · , x n ) = (x 1 , x ) ∈ R n , λ < 0, denote be the reflection of the point x = (x 1 , x ) about plane T λ and Assume that u is a positive solution to (1), then

Proof. A direct computation yields
Hence the lemma is proved.
We need the following Hardy-Littlewood-Sobolev inequality, which is proved in [14]. Then T g L r ≤ C(n, s, r) g L nr n+sr (R n ) .
Proof of Theorem 1.1. We divide the proof into two steps.