The effect of Harnack inequality on the existence and nonexistence results for quasi-linear parabolic equations related to Caffarelli-Kohn-Nirenberg inequalities

Abstract

Abstract. In this work we study the problemwhere) and u0 ∈ L 1 (Ω) are positive functions. The main points under analysis are some nonexistence results and complete blow-up in the case p > 2 and γ + 1 > 0 and some examples of existence for (γ + 1) > 0 and 1 < p < 2. These results are interesting as they prove the role of Harnack inequality in this kind of problems and allow to understand better the blow-up behavior.

Multiple solutions for a degenerate elliptic equation involving sublinear terms at infinity

“…Note that the pairs r = 1, α = 2b and r = 2, α = 2b verify relation (2). Consequently, E λ is well defined.…”

“…Due to this fact, problem (P λ ) has been widely studied by several authors, see [1][2][3]7,13] and references therein. Usually, the nonlinear term f : R → R is considered to be superlinear at infinity or simply, f (s) = |s| θ−2 s with θ > 2.…”

“…The presence of the parameter λ ∈ R is essential in our problem; indeed, if beside of (f 1 ), the nonlinear term f is uniformly Lipschitz (with Lipschitz constant L > 0), then problem (P λ ) has only the trivial solution whenever |λ| < (LC 2 2,2b ) −1 ; the constant C 2,2b > 0 is introduced after relation (3). In order to state our main results, we introduce the weighted Sobolev space D 1,2 a (Ω) where the solutions of (P λ ) are going to be sought, which is the completion of C ∞ 0 (Ω) with respect to the norm…”

On quasilinear parabolic equations involving weighted p-Laplacian operators

“…We refer the reader to works [2,3] on Caffarelli-Kohn-Nirenberg inequalities associated to the study quasilinear elliptic equations.…”

Attractors for parabolic equations related to Caffarelli-Kohn-Nirenberg inequalities

“…The existence and properties of solutions to problem type (1.1) have attracted interest in recent years [15][16][17][18][19]. However, to the best of our knowledge, little seems to be known for the longtime behavior of solutions to problem (1.1).…”