Positive solutions to a fourth-order elliptic problem by the Lusternik–Schnirelmann category

Critical and noncritical regions on the critical hyperbola

“…0 Hereafter in this section we assume all of the hypotheses from Theorem 1.2 and we borrow some arguments from [23]. Without loss of generality, suppose that 0 2…”

“…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.…”

“…To complete the proof of Theorem 1.3 we need some auxiliary results, whose proofs we refer to [23]. Lemma 5.3.…”

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

“…In view of the results proved in [11,12,18], it is natural to ask if, as in (BN λ ) and (V μ ), we have more solutions if Ω has rich topology. In our last result, we give a positive answer to these questions by proving the following multiplicity result: Theorem 1.3.…”