On Absence of Nonnegative Monotone Solutions for Some Coercive Inequalities in a Half-Space

Abstract

Using the nonlinear capacity method, we investigate the problem of absence of nonnegative monotone solutions for a quasilinear elliptic inequality of type Δpu≥uq in a half-space in terms of parameters p and q.

On critical exponents for weak solutions to the Cauchy problem for one nonlinear equation with gradient nonlinearity

“…Let us also note Galakhov's 9 and Galakhov and Salieva's works. 10 In the present paper, we have obtained the result about the existence of two critical exponents q = 3∕2 and q = 2 such that in wide class of initial data u 0 (x) for 1 < q ⩽ 3∕2, there is no local-in-time weak solution of the Cauchy problem (1), (2). For q > 3∕2, local-in-time weak solution already exists, and for 3∕2 < q ⩽ 2, global-in-time weak solution does not exist.…”

“…In this paper, we will consider the case 𝜌 ext = 0. Introducing the dimensionless variables, we reduce Equation (10) to the form (1).…”

“…In the work of Mitidieri and Pohozaev, 8 a method of nonlinear capacity has been used to obtain results on the role of critical exponents. Let us also note Galakhov's 9 and Galakhov and Salieva's works 10 . In the present paper, we have obtained the result about the existence of two critical exponents $$$ q&amp;amp;#x0003D;3/2 $$$ and $$$ q&amp;amp;#x0003D;2 $$$ such that in wide class of initial data $$$ {u}_0(x) $$$ for $$$ 1&amp;lt;q\leqslant 3/2 $$$, there is no local‐in‐time weak solution of the Cauchy problem (), ().…”