Uniqueness of the trivial solution of some inequalities with fractional Laplacian

We obtain sufficient conditions for the uniqueness of the trivial solution for some classes of nonlinear partial differential inequalities containing the fractional powers of the Laplace operator.


Introduction
In the present paper we obtain sufficient conditions for the uniqueness of the trivial solution for some new classes of nonlinear inequalities and systems with fractional powers of the Laplacian by using a modification of the test function method developed in [7,8].
However, this method cannot be used directly, since it was developed for other types of differential operators, in particular, for integer powers of the Laplacian.But it is known [1] that the solution sets for many problems containing operators of such types are relatively small.For instance, harmonic functions cannot approximate a function with interior maxima or minima, functions of a single variable with null second derivatives are necessarily affine linear, and so on, which facilitates choosing additional nonlinear terms that yield non-existence of solutions at all.In contrast, for fractional differential operators many new solutions can arise.Their set can even become locally dense in C(R n ), as in the case of s-harmonic functions (u such that (−∆) s u = 0), see [2], also in the case of higher order operators, see [1,6].Thus, in order to obtain non-existence results one has to exclude the existence of this larger solution set.Therefore non-existence results in the fractional setting are always a delicate matter, which requires a substantial modification of the known techniques, and were obtained up to now only in some special cases.Namely, this problem was considered in [2] for systems of equations with fractional powers of the Laplacian, and by the authors of the present paper in [5,9] for some respective inequalities and their systems.
The rest of the paper consists of four sections.In §2 we obtain some auxiliary estimates for the fractional Laplacian.Further, we prove uniqueness theorems: in §3, for some elliptic inequalities, in §4, for systems of such inequalities, and in §5, for respective parabolic problems.

Auxiliary estimates
We define the operator (−∆) s by the formula where for all functions such that the right-hand side of (2.1) makes sense at least in the distributional setting.
Remark 2.1.Note that this definition implies For u ∈ H 2s loc (R n ), this order can be reversed (see [3]).We will use definition (2.1) for the proof of the following Lemmas 2.2 and 2.4.Lemma 2.2.Let s ∈ IR + , q > p > 0 and α, β ∈ R. Consider a function ϕ 1 : IR n → IR defined as Remark 2.3.In the Mitidieri-Pohozaev approach such estimates were established by direct calculation of the iterated Laplacian of the test functions that were given explicitly.This does not work for the fractional Laplacian, so we need to establish some additional estimates.
Proof. of Lemma 2.2.It suffices to consider x ∈ R n such that 3 2 < |x| < 2, since otherwise the integrand is obviously regular and bounded.
We start with the case [s] = 0 using (2.1) with notation f (x, y) = ϕ 1 (x)−ϕ 1 (y) |x−y| n+2s , where s = {s}: where (here and below the singular integrals are understood in the sense of the Cauchy principal value).
On the other hand, the Lagrange Mean Value Theorem implies that for any ε > 0 and some constant c 4 > 0. Combining (2.5) and (2.7), we obtain which together with (2.3) implies with some constants c 5 , c 6 > 0 independent of x.Hence, in case [s] = 0 (2.4) follows by assumption λ > 2sq q−p − n, if ε > 0 is sufficiently small.For [s] > 0, we use the identity (2.2) and the representation of the radial Laplacian (2.10) It is easy to see that for 1 < |x| = r < 2 (2.2) and (2.10) imply (2.11) with some c > 0. This holds, both for 0 ≤ r ≤ 1 and for r > 2, since in these cases both parts of the inequality are zero.Differentiating the integral in the definition (2.1) up to order 2[s] and repeating the previous arguments for the respective derivatives (note that they can be exchanged with (−∆) s by Remark 2.1), we obtain which together with (2.10) implies (2.9) and hence (2.4) for arbitrary s ∈ R + .
Lemma 2.4.Let s ∈ R + , q > p > 0 and α, β ∈ R. For the family of functions ϕ R (x) = ϕ 1 x R , where R > 0, one has for every R > 0 and some c > 0 independent of R.
Sketch of the proof.By (2.1) and a change of variables ỹ = y R , we have Substituting (2.13) into the left-hand side of (2.12) and applying Lemma 2.2, we obtain the claim.

Single elliptic inequalities
Now consider the nonlinear elliptic inequality where s > 0, c > 0, q > p > 0 and α are real numbers.We define the class L α p,loc (IR n ) as that of all functions u such that for each compact set We will prove the following theorem.
Theorem 3.2.Inequality (3.1) has no nontrivial (i.e., distinct from zero a.e.) weak solutions for n + α − 2s > 0 and Proof.We make use of the test function ϕ R (x) = ϕ 1 x R defined in Lemma 2.4.Substituting ϕ(x) = ϕ R (x) into (3.1) and applying the Hölder inequality, we get Hence, From (3.5) by Lemma 2.4 we obtain Taking R → ∞, in case of strict inequality in (3.3) we come to a contradiction, which proves the claim.In case of equality, we have and by (3.4) IR n |u| q (1 + |x|) β dx = 0, which completes the proof in this case as well.
Remark 3.3.From the results of [7] it follows that at least for α = 0 and integer s the upper bound given for uniqueness of the trivial solution in (3.3) is optimal.Its optimality for α = 0 and/or non-integer s is an open problem.

Systems of elliptic inequalities
Here we consider a system of nonlinear elliptic inequalities where We will prove the following theorem.
Theorem 4.2.System (4.1) has no nontrivial (i.e., distinct from a pair of zero constants a.e.) weak solutions for Proof.Introduce a test function ϕ R (x) as in the proof of the previous theorems.Similarly to (3.4), we get Estimating the second factors in the right-hand sides of the obtained inequalities by Lemma 2.4 similarly to (2.4), we get and, substituting (4.5) into (4.4) and vice versa, Passing to the limit as R → ∞, we complete the proof of the theorem similarly to the previous ones, including the critical case.
Remark 4.3.From the results of [7] it follows that at least for α 1 = α 2 = 0 and integer s 1 , s 2 the upper bound given for uniqueness of the trivial solution in (4.3) is optimal.Its optimality for arbitrary α 1 , α 2 and/or non-integer s 1 , s 2 is an open problem.

Nonlinear parabolic inequalities
Now let u 0 ∈ L 1,loc (R n ), u 0 (x) ≥ 0 a.e. in R n .We consider the nonlinear parabolic inequality We prove the following theorem.Theorem 5.2.Inequality (5.1) with initial condition (5.2) has no nontrivial weak global solutions for α < 2s and where ϕ 1 is defined as in Lemma 2.2, and the parameter θ > 0 will be specified below.Substituting ϕ(x, t) = ϕ R,θ (x, t) into (3.1) and using the Young inequality, we get where d(c) > 0. Hence, where From (5.6) and (2.4) due to the definition of ϕ R,θ (x, t) we have with some C > 0. Choosing θ = 2s − α and taking R → ∞, in the case of a strict inequality in (5.4) we come to a contradiction, which proves the theorem.The case of equality is considered similarly to Theorem 3.2.
Remark 5.4.From the results of [7] it follows that at least for α = 0 and integer s the upper bound given for uniqueness of the trivial solution in (5.4) is optimal.Its optimality for α = 0 and/or non-integer s is an open problem.
Definition 4.1.A weak solution of system of inequalities (4.1) is a pair of functions