Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity

Multiple Nontrivial Solutions of Resonant and Nonresonant Asymptotically Linear Problems

“…We indicate briefly how the methods used to prove Theorem 1 can be used to obtain multiple nontrivial solutions of the resonance-type problem (10) à.u + Xiu + g0(u) = 0, u\dü, where the function g0 is bounded. The existence of solutions of (10) when go(0) -0 has recently been investigated by Bartolo, Benci and Fortunato in [5]. The following theorem appears to be a new result for this type of problem.…”

The existence of a nontrivial solution to a nonlinear elliptic problem of linking type without the Ambrosetti-Rabinowitz condition

“…The idea to weaken the (PS) condition to the (C) c condition has existed in some papers (see e.g. [2,22] and references therein). To obtain the linking theorem we need, we imitate the framework given in [18].…”

“…Lemma 2.6 below) is very crucial in the process of the whole proof. This type of deformation lemma under the (C) c condition had appeared in [2], but the form given in [2] is not the form we need. The linking theorem given in [18] is obtained from a general minimax theorem.…”

Two nontrivial solutions for quasilinear periodic equations

“…We assume that the equation is strongly resonant at the first (zero) eigenvalue of the negative ordinary scalar p-Laplacian with periodic boundary conditions (i.e., f (t, x) → 0 as |x| → ∞ and the potential F (t, x) = x 0 f (t, r)dr has finite limits as |x| → ∞, i.e., the potential has a small rate of increase as |x| → ∞). The term "strong resonance" (describing the situation just mentioned) was coined by Bartolo-Benci-Fortunato [2]. Our approach is variational and uses smooth critical point theory (see Chang [3] and Mawhin-Willem [12]) and the so-called "second deformation theorem" (see Chang [3], p.23).…”