Singular elliptic problems with lack of compactness

Abstract

Abstract. We consider the following nonlinear singular elliptic equationwhere g belongs to an appropriate weighted Sobolev space, and p denotes the Caffarelli-KohnNirenberg critical exponent associated to a, b, and N . Under some natural assumptions on the positive potential K(x) we establish the existence of some λ 0 > 0 such that the above problem has at least two distinct solutions provided that λ ∈ (0, λ 0 ). The proof relies on Ekeland's Variational Principle and on the Mountain Pass Theorem without the Palais-Smale condition, combined with a weighted variant of the Brezis-Lieb Lemma.

On semilinear elliptic equations involving concave–convex nonlinearities and sign-changing weight function

“…(E f ) possesses infinitely many solutions. Furthermore, Cîrstea and Rȃdulescu [5], Cao and Zhou [6], and Ghergu and Rȃdulescu [10] have been investigated the analogue Eq.…”

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

“…When singular problems with inhomogeneous term were concerned, existence results had also been obtained by Ghergu and Rãdulescu in [26]. Very recently, Kang, Li and Peng obtained positive solutions for problem (P 1 ) in [24] and extended the results obtained by Brezis and Nirenberg [1], there they also discussed the critical dimensions of problem (P 1 ).…”

On G-invariant solutions of a singular biharmonic elliptic system involving multiple critical exponents in R N $R^{N}$

“…Moreover, let us also mention that when μ = 0 and the right-hand side nonlinearity term |x| -s u 2 * (s)-1 in (1.2) is substituted by u q-1 with 1 < q ≤ 2 * , there have been a variety of remarkable results on G-invariant solutions in [9][10][11]. Furthermore, for other results about this aspect, see [12] with singular Lane-Emden-Fowler equations, [13] with singular p-Laplacian equations, [14] with biharmonic operators and [15] with p(x)-biharmonic operators [16], and monograph [17] with generalized Lane-Emden-Fowler equations or Gierer-Meinhardt systems involving singular nonlinearity. For the systems of singular elliptic equations involving critical exponents, a wide range of works concerning the solutions structures have been presented in recent years.…”