The blow-up of critical anisotropic equations with critical directions

Abstract

Abstract. We investigate blow-up theory for doubly critical anisotropic problems in bounded domains of the Euclidean space. Mathematics Subject Classification (2000). 35J60 Nonlinear PDE of elliptic type.

Nonexistence of nontrivialweak solutions for anisotropic elliptic problems

“…In recent years, there have been a large number of papers and an increasing interest in the results of existence for anisotropic elliptic problems, with no hope of being complete, let us mention Alves-El Hamidi [1], Di Castro-Montefusco [7], El Hamidi-Rakotoson [8], Figueiredo-Santos Júnior-Suárez [9], Fragalà-Gazzola-Kawohl [10] and Vétoiss [14]. However, to the best of our knowledge, very little is known about nonexistence of nontrivial weak solutions of the problem (1), we only refer to Di Castro-Montefusco [7] and Fragalà-Gazzola-Kawohl [10], so our present paper is a new contribution in this direction.…”

A New Method to Deal with the Stability of the Weak Solutions for a Nonlinear Parabolic Equation

“…By the way, we hope the method introduced in this paper can be beneficial to the wellposedness problems for the evolutionary ( )-Laplacian equations, overdetermined anisotropic elliptic equations, and the infiltration equations. As we know, there are many papers devoted to these equations; one can refer to [11][12][13][14][15][16][17][18][19][20][21][22][23] and the references therein.…”

On certain anisotropic elliptic equations arising in congested optimal transport: Local gradient bounds

“…One of the first paper to address regularity issues for them has been [17]. Recently, they started to attract an increasing interest, due to their applications in biology and physics (see for example [18] and the references therein). We also cite the papers [3,8], where some spectral properties of this operator are investigated.…”