Nonexistence of Solutions for Singular Nonlinear Ordinary Inequalities

In this paper we prove nonexistence theorems of nonnegative nontrivial solutions for a singular nonlinear ordinary inequality in bounded domains with singular points on the boundary. The proofs are based on the test function method developed by Mitidieri and Pohozaev. We also give the examples demonstrating that the conditions obtained are sharp in the case of the problem under consideration.

There have been many results on the nonexistence of nonnegative nontrivial solutions for nonlinear differential inequalities (systems), see  and references therein.Tools based on different forms of the maximum principle like the moving planes method or moving spheres method, nonlinear capacitary estimates and Pohozaev type identities, energy methods and Email: xh0535@sina.com

X. Li
Harnack inequality type argument, have been proved to be very successful for solving interesting problems related to applications and to the general theory of partial differential equations.
In the present paper, by modifying the method developed by Mitidieri and Pohozaev in [22] and Galakhov in [16] , we will show nonexistence theorems for the nonlinear differential inequality (1.1) with singular points on the boundary.
We understand solutions to problem (1.1) in the sense of distributions and define the class of admissible solutions to problem (1.1) as We prove the following theorems.
Remark 1.3.For α < p and q > p − 1, a solution of problem (1.1) with a(x) = x −α can be written down explicitly as u(x) = Cx α−p q−p+1 with an appropriate constant C > 0. Thus, the assumption α ≥ p is essential to deal with nonexistence results.

Proofs of Theorems 1.1 and 1.2
In this section, we will prove the two theorems.In doing so we will follow the argument of Theorem 2.1 in [22] and Theorem 3.4 in [16].
To establish a priori estimates of the solutions, we need to define some test functions that will be widely used in the sequel.We consider the test function ξ ∈ C 1 ([0, x 0 ]; [0, 1]) that satisfies where η ∈ (0, x 0 ) is a parameter and c > 0 is a constant.Set where λ > 0 is a parameter to be chosen later according to the nature of the problem.
To prove the main results of this section, we need the following lemma.
Proof.Without loss of generality, we suppose u > 0. If u is allowed to vanish at some points, we consider u δ = u + δ with arbitrary δ > 0 and then pass to the limit as δ → 0 + .Let γ ∈ R be a parameter to be chosen later.Multiplying (1.1) by u γ χ and integrating by parts, we get Applying Young's inequality with exponents l = p p−1 , l = p > 1, ε > 0 to the second integral on the right-hand side of (2.5), we obtain (2.6) Taking ε = γ/2, we have By Hölder's inequality with exponents m = q+γ p−1+γ > 1, m = q+γ q−p+1 > 1 for every γ > 0 to the second integral on the right-hand side of (2.7) (since, by assumption, q > p − 1), we get i.e., . (2.9)