A fourth-order equation with critical growth: the effect of the domain topology

Abstract

In this paper we prove the existence of multiple classical solutions for the fourth-order problemwhere Ω is a smooth bounded domain in R N , N ≥ 8, 2 * = 2N/(N − 4) and µ 1 (Ω) is the first eigenvalue of ∆ 2 in H 2 (Ω) ∩ H 1 0 (Ω). We prove that there exists 0 < µ < µ 1 (Ω) such that, for each 0 < µ < µ, the problem has at least cat Ω (Ω) solutions.

Existence and multiplicity of positive solutions for a fourth-order elliptic equation

“…Roughly speaking, theorem 1.2 says that, differently from the biharmonic version of the Brezis-Nirenberg problem (V μ ), the critical dimension for (P λ ) is only N = 5. Actually, as observed in [13], the notion of critical dimension is also related to the integrability of the L 2 -norm of the gradient of the functions which realize the best constant of the embedding…”

Critical and noncritical regions on the critical hyperbola

“…To prove the results in this paper we consider the equivalent formulation of (S) as the fourth order equation (E). We follow some of the arguments in [23,24], which consider (E) in the particular case of p D 1, i.e., the corresponding problem involving the biharmonic operator. However, in the nonlinear regime of…”

“…In particular, in [24] and [23], the comparison principle for the biharmonic operator under Navier boundary conditions is the key argument to get the positivity of the solutions at the LusternikSchnirelmann critical levels. However, the same procedure seems not suitable in the nonlinear setting and then we use, instead, an energy argument, namely Lemma 5.1, along with the regularity result of Lemma 2.2.…”