Blow-up of solutions of some nonlinear inequalities with singularities on unbounded sets

Unsolvability conditions for some inequalities and systems with functional parameters and singular coefficients on boundary

“…We substitute (34) into (35) and vice versa and move all the terms of form ∫︀ − ( ) 1 ( ) | | 2 ( ) and ∫︀ − ( ) 1 ( ) | | 2 ( ) into the left hand side. Then by (6) and 7, for 7 10 < 1 (which can be ensured by an appropriate choice of the parameters in the Young inequalities) we have…”

“…To prove the nonexistence of solutions by the nonlinear capacity method, we construct test functions with various geometric structure taking into account a specific character of the considered problem. Our first results in this direction were published in [5], [6].…”

Nonexistence results for some nonlinear inequalities with functional parameters

“…To obtain our nonexistence results, we use the test function method (also known as the nonlinear capacity one) suggested in [7] and developed more recently in [4][5][6]. Namely, assuming for contradiction that a solution exists, we multiply both sides of the inequality in question by specially chosen parameter dependent test functions and after partial integration and some algebraic transformations, such as application of the Young inequality, obtain an a priori estimate for a positive functional of the solution.…”

Nonexistence results for some nonlinear nonlocal elliptic inequalities with variable exponents

“…The purpose of this work is to study the nonexistence of nontrivial positive solutions to some classes of nonlinear elliptic inequalities involving the fractional diffusion operator (−Δ) α /2 ,0 < α < 2, and variable exponents. In order to obtain sufficient conditions for the nonexistence of solutions, we make use of the test function method introduced in , and developed in many recent works (see , and the references therein). Our motivation comes from the recent article of Galakhov et al , , where nonexistence results were obtained for some elliptic and parabolic inequalities with functional parameters involving the p ( x )‐Laplacian.…”

“…/˛= 2 , 0 <˛< 2, and variable exponents. In order to obtain sufficient conditions for the nonexistence of solutions, we make use of the test function method introduced in [13], and developed in many recent works (see [5,8,[14][15][16][17][18][19][20][21], and the references therein). Our motivation comes from the recent article of Galakhov et al [16], where nonexistence results were obtained for some elliptic and parabolic inequalities with functional parameters involving the p.x/-Laplacian.…”