Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)8804Research ArticleNonexistence of Positive Solutions to Semilinear Elliptic Inequalities for Polyharmonic OperatorTsegawB BDepartment of Mathematical Analysis and Theory of Functionsbirilewb@yahoo.comPeoples’ Friendship University of Russia150420134243208092016Copyright © 2013,2013In this paper, we study the nonexistence of positive solution for some higher-order semilinear elliptic inequality particularly involving polyharmonic operator: Δku(x) ≥ x1 α1 x2 α2… xn αnuq(x), where k ∈ ℕ,q 1, x = (x1,x2,…,xn) and αi ∈ ℝ,i = 1,2,…,n. The purpose of this paper is to establish conditions on values of αi,i = 1,2,…,n for the nonexistence of positive solution to this problem in a bounded and unbounded domain. The main tools are a priori estimates and integral inequalities. Using the test function method, we derive ﬁrst a priori estimates for solutions of the inequality based on integral inequalities and on the weak formulation of the problem with an optimal choice of test functions and then we formulate the nonexistence condition of the solution of the problem. The choice of such functions is determined by the nonlinear characters of the problem and depends on the concept of solutions that we are dealing with.semilinear elliptic inequalitiesanisotropic singularitiespolyharmonic operatorsapriori estimates and nonexistence of solutionsполулинейные эллиптические неравенстваанизотропные особенностиполигармонический оператораприорные оценки и отсутствие решений