The effect of domain topology on the number of positive solutions for singular elliptic problems involving the Caffarelli–Kohn–Nirenberg inequalities

Abstract

In this paper we consider the critical singular equation involving the Caffarelli-Kohn-Nirenberg inequalities of the typeHere Ω is a bounded domain with smooth boundary in R N and contains 0 in its interior, 0 μ < ( √μ − a) 2 ,μ = ( N −2 2 ) 2 , N 3, a b < a + 1, a d < a + 1, p = p(a, b), λ is a positive parameter and 2 q < D. By Lusternik-Schnirelmann category theory, we prove that the problem (P 1 ) has at least cat(Ω) positive solutions.

Singular and Degenerate Boundary Value Problems Related to the Electricity Theory

“…As for the methods utilized to overcome these inconveniences, they depend on the problem. Many papers discuss the existence of solutions for (1) when the weight has the particular form ( ) = | | , so that, under some appropriate hypotheses on , a Caffarelli-Kohn-Nirenberg inequality can be applied; see, for example, [3][4][5] and the references therein.…”

Multiplicity of solutions to the weighted critical quasilinear problems

“…For a singular potential, the existence of infinitely many small solutions which converge to zero was obtained by He and Zou [15]. Yang and Shen [30] studied the critical singular equation involving the Caffarelli-Kohn-Nirenberg inequalities with the special case f (x, u) by using the Lyusternik-Schnirelman Category Theory and obtained the existence of at least cat(Ω) positive solutions. Some existence results for problem (1.1) with λ = 1, were obtained by Huang and Wu [17] using variational methods and analysis techniques.…”