On Nonexistence of Solutions to Some Nonlinear Inequalities with Transformed Argument

We obtain results on nonexistence of nontrivial solutions for several classes of nonlinear partial differential inequalities and systems of such inequalities with transformed argument.


Introduction
In recent years conditions for nonexistence of solutions to nonlinear partial differential equations and inequalities attract the attention of many mathematicians.This problem is not only of interest of its own, but also has important mathematical and physical applications.In particular, Liouville type theorems of nonexistence of nontrivial positive solutions to nonlinear equations in the whole space or half-space can be used for obtaining a priori estimates of solutions to respective problems in bounded domains [1,4].
In [5][6][7] (see also references therein) sufficient conditions for nonexistence of solutions were obtained for different classes of nonlinear elliptic and parabolic inequalities using the nonlinear capacity method developed by S. Pohozaev [8].On the other hand, there exists an elaborated theory of partial differential equations with transformed argument due to A. Skubachevskii [9].But the problem of sufficient conditions for nonexistence of solutions to respective inequalities with transformed argument remained open.Some special cases of such problems were treated in [2,3].
In this paper we obtain sufficient conditions for nonexistence of solutions to several classes of elliptic and parabolic inequalities with transformed argument and for systems of elliptic inequalities of this type.
The structure of the paper is as follows.In §2, we prove nonexistence theorems for semilinear elliptic inequalities of higher order; in §3, for quasilinear elliptic inequalities; in §4, for Email: olga.a.salieva@gmail.comO. Salieva systems of quasilinear elliptic inequalities; and in §5, for nonlinear parabolic inequalities with a shifted time argument.
The letter c with different subscripts or without them denotes positive constants that may depend on the parameters of the inequalities and systems under consideration.
In some situations assumption (g2) can be replaced by a weaker one: and Here and below q = q q−1 .
Proof.Assume for contradiction that a nontrivial solutions of (2.1) does exist.Let 0 < R < ∞ (in particular, the case R = 1 is possible).The function where ϕ(s) is from Lemma 2.6, will be used as a test function for inequality (2.1).Multiplying both sides of (2.1) by the test function ϕ R and integrating by parts 2k times, we get Using (g1), (g2), and the monotonicity of ϕ R , one can estimate the right-hand side of (2.5) from below as On the other hand, applying the parametric Young inequality to the left-hand side of (2.5), we get with some constants c 1 , c 2 > 0. Combining (2.5)-(2.7),we have Restricting the integration domain in the left-hand side of the inequality, we obtain Taking R → ∞, we get a contradiction for n − 2kq < 0, which proves the theorem in all cases except the critical one (where n − 2kq = 0).In the critical case we get But (2.5), (2.6) and the Hölder inequality imply O. Salieva and therefore since the second factor on the right-hand side of (2.8) can be estimated from above by c 2 R n−2kq as before, where n − 2kq = 0. Thus for a nontrivial u we obtain a contradiction in this case as well.This completes the proof.
Theorem 2.8.Let either n ≤ 2k and q > 1, or n > 2k and 1 < q ≤ n n−2k .Suppose that g satisfies assumptions (g1) and (g'2).Then inequality (2.1) has no nontrivial solutions u ∈ L q,loc (R n ) such that Proof.Similarly to estimate (2.6), for R > ρ we get (2.10) Then (2.5) and (2.7)-(2.10)imply where c 1 , c 2 > 0, and the constant c 1 can be chosen arbitrarily small.Hence by assumption (2.9) for c 1 < 1 2l ρ +1 and sufficiently large R we have i.e., the conclusion of Theorem 2.7 for any subcritical q remains valid in this case as well.The critical case can be treated similarly to the previous theorem.
Proof.Multiplying both sides of (2.11) by the test function ϕ R and integrating by parts 2k − 1 times, we get (2.12) Using (g1) and (g2), we can estimate the right-hand side of (2.12) from below as (2.13) On the other hand, applying the parametric Young inequality to the left-hand side of (2.12), we get with some constants c 1 , c 2 > 0. Combining (2.12)-(2.14),we have Restricting the integration domain in the left-hand side of the inequality, we obtain Taking R → ∞, we get a contradiction for n − (2k − 1)q < 0. The critical case can be treated similarly to the previous theorems.
On the other hand, applying the parametric Young inequality with exponents p p−1 and p to the integrand at the left-hand side of (3.2) represented as with 0 < ε < |λ|, and then applying it again with exponents q+λ λ+p−1 and q+λ q−p+1 (note that these exponents are greater than 1 for a sufficiently small |λ| due to the assumption q > p − 1) to with some constants ε, c 3 (ε), c 4 (ε), c 5 (ε) > 0. Combining (3.2)-(3.5),we have Choosing λ sufficiently close to 0 and taking R → ∞, we obtain a contradiction for n − pq q−p+1 < 0, i.e., p − 1 < q < n(p−1) n−p .The critical case can be treated similarly to the previous theorems.
Further we consider the inequality Theorem 3.2.Let p − 1 < q ≤ n(p−1) n−1 .Suppose that g satisfies assumptions (g1) and (g2).Then inequality (3.6) has no nontrivial nonnegative solutions u ∈ W 1 p,loc (R n ) ∩ W 1 q,loc (R n ).Proof.Multiplying both parts of (3.6) by the test function ϕ R and integrating by parts, we get Using (g1) and (g2), one can estimate the right-hand side of (3.7) from below as On the other hand, applying the Hölder inequality to the left-hand side of (3.7), we obtain (3.9) Combining (3.7)-(3.9),we have with some constants c 1 , c 2 > 0. Taking R → ∞, we obtain a contradiction for n − q q−p+1 < 0. The critical case can be treated similarly to the previous theorems.Remark 3.3.If g satisfies (g'2) instead of (g2), a version of Theorem 3.2 can be proven for a class of solutions that satisfy (2.15) (in particular, u ∈ W 1 p,loc (R n ) ∩ W 1 q (R n )) similarly to Theorems 2.8 and 2.10.

Systems of quasilinear elliptic inequalities
Now consider a system of quasilinear elliptic inequalities where g 1 , g 2 ∈ C 1 (R n ; R n ) are mappings that satisfy (g1) and (g'2).Introduce the quantities Then there holds the following.
Theorem 4.1.Let p, q, p 1 , q 1 > 1, p − 1 < p 1 , q − 1 < q 1 , and min(σ, τ) ≤ 0. Suppose that g 1 and g 2 satisfy assumptions (g1) and (g'2).Then system (4.1) has no nontrivial nonnegative solutions Proof.Assume that there exists (u, v) -a nontrivial nonnegative solution of system (4.1).Let {ϕ R } be the same family of test functions as in Sections 2 and 3. Multiplying the first inequality (4.1) by u λ ε ϕ R and the second one by v λ ε ϕ R , where which can be rewritten as (note that |λ| = −λ since λ < 0) Here and below we omit the argument x if it is the only one in a certain integral, and R n if it is the integration domain.Application of the Young inequality to the right-hand sides of the obtained relations results in where constants c λ and d λ depend only on p, q, and λ.Further, multiplying each inequality (4.1) by ϕ R and integrating by parts, we obtain Note that the integrals on the left-hand sides of these inequalities can be estimated from below by B 2R (0) v q 1 (x) dx and B 2R (0) u p 1 (x) dx (with some positive multiplicative constants) respectively, similarly to the proofs of Theorems 2.7 and 3.1.Thus, combining ( and taking ε → 0, we arrive to a priori estimates B 2R (0) where D λ and E λ > 0 depend only on p, q, and λ.
Apply the Hölder inequality with exponent r > 1 to the first integral on the right-hand side of (4.9): where 1 r Choosing the exponent r so that (λ + p − 1)r = p 1 , from (4.9) and (4.11) we get Applying the Hölder inequality with exponent y > 1 to the last integral on the right-hand side of (4.12), we obtain where 1 y + 1 y = 1.Choosing y in (4.13) so that (1 − λ)(p − 1)y = p 1 and taking into account (4.12), we get the estimate , (4.14) where the exponents r and y are chosen so that (4.15) Note that this choice of r > 1 and y > 1 is possible due to our assumptions on p and p 1 provided that λ < 0 is sufficiently small in absolute value.Similarly, choosing s and z such that and estimating the right-hand side of (4.10) by the Hölder inequality, we get Combining (4.14) and (4.17), we finally arrive at where

.21)
Our assumptions imply that the exponents on the left-hand side of (4.18) and (4.19) are such that Thus from (4.19) we have Taking R → ∞ in (4.22), under the hypotheses of the theorem we arrive at a contradiction, which completes the proof.
We leave the proof to the interested reader.