Strong Resonance for Some Quasilinear Elliptic Equations

Abstract

We study the existence of the weak solutions of the nonlinear boundary value problemu s 0 on Ѩ ⍀ , where the nonlinearity g is of the Landesman᎐Lazer type. Our sufficient conditions generalize all previously published results about the strong resonance at the first eigenvalue.

Flows and Critical Points

“…Similar resonance problems have been studied by Perera [6] when [2], and Drábek and Robinson [4] for the special case α ± = const.…”

One‐sided resonance for quasilinear problems with asymmetricnonlinearities

“…But first we give some corollaries and deduce the results of [1,3,8]. In what follows, (u j ) is as in the theorem, that is, ρ j := u j → ∞ and…”

Two nontrivial solutions for strongly resonant nonlinear elliptic equations

“…Namely the PS-condition is not satisfied at all levels. This is in contrast to the works of Arcoya-Orsina [1] and Bouchala-Drabek [3], who consider quasilinear problems at resonance and employ Landesman-Lazer type conditions, which preclude strong resonance at infinity in the sense explained in the introduction. In the next proposition we show that ϕ satisfies the P S c -condition only for certain levels c ∈ R.…”