A Nonlinear Schrödinger Equation Resonating at an Essential Spectrum

We consider the nonlinear Schrödinger equation −Δu + f(u) = V(x)u in R. The potential function V satisfies that the essential spectrum of the Schrödinger operator −Δ − V is [0, +∞) and this Schrödinger operator has infinitely many negative eigenvalues accumulating at zero. The nonlinearity f satisfies the resonance type condition lim |t|→∞ f(t)/t = 0. Under some additional conditions on V and f, we prove that this equation has infinitely many solutions.


Introduction and Statement of Results
In this paper, the following nonlinear Schrödinger problem in R  ( ≥ 3) is considered: The nonlinearity  satisfies the resonance type condition: Here,  ess () denotes the essential spectrum (see [1,Chapter 7] or [2,Chapter 7.4]) of the Schrödinger operator  defined as follows: Equation (1) arises in quantum mechanics and is related to the study of the nonlinear Schrödinger equation for a particle in an electromagnetic field and it has attracted considerable attention from researchers in recent decades.One can see [3][4][5][6] and the references therein.However, there are very few results on (1) with the resonance type condition (2).
The following nonlinear elliptic problem with resonance type conditions in bounded domain has been studied by many authors and numerous existence and multiplicity results have been obtained in the past forty years: −Δ = ℎ (, ) in Ω,  = 0 on Ω.
Here, Ω is a bounded domain in R  , and  satisfies the resonance type condition: ]  is the th eigenvalue of −Δ with 0-Dirichlet boundary condition on Ω.We refer to [7][8][9][10] and references therein for more detailed discussions of some historical results.This problem has deeply inspired developments in critical point theory in the last forty years, such as the Landesman-Lazer type conditions (e.g., [8,10]), Morse theory (e.g., [9]), and the variational reduction method (e.g., [11]).The difficulty of this problem lies in the proof of the boundedness of the Palais-Smale sequence or the ()  sequence (see Definition 5 in Section 2) of its corresponding functional.
The resonant Schrödinger problem (1) is much more difficult than the bounded domain case and it has fewer studies.Unlike the case of bounded domains for which the linear operators −Δ in bounded domain are compact, there are continuous spectra of the linear operator .Moreover, the Advances in Mathematical Physics proofs of boundedness and compactness of the ()  sequence of the corresponding functional of (4) are greatly different from and more difficult than the case of bounded domain (see [4,5,12]).
(ii) A typical example for  which satisfies Here, , and  > 0 in  −1 .
Definition 5 (see [16]).Let  be a Banach space.Let  ∈  1 (, R) and  ∈ R. One can call the fact that  satisfies the Cerami condition at , denoted by ()  condition, if for any there exists  ∈  such that, up to a subsequence, ‖  − ‖ → 0 as  → ∞.If  satisfies the ()  condition for every  ∈ R, then  is said to satisfy the Cerami condition.A sequence {  } ⊂  satisfying (32) is called a ()  sequence of .

Proof of Theorem 1
For the proof of Theorem 1, the following abstract theorem, which is a corollary of [13, Theorem 2.4], is used.
Let  be a real Hilbert space with norm ‖ ⋅ ‖.Suppose that  is a  1 -functional defined in  and satisfies the following conditions: (A 1 )  satisfies the ()  condition for every  > 0, and (0) ≥ 0.
Each   has been repeated in the sequence according to its finite multiplicity.Then, by Lemma 3, the following is true: Let Here,  0 is the constant in (10).From this definition, Let {  |  ∈ R} denote the spectral family of the operator  defined by (3).Let Here, id :  2 (R  ) →  2 (R  ) is the identity map and It is easy to see that the codimension of  equals  * − 1.
Proof of Theorem 1.We are going to prove that the functional Φ satisfies the conditions (A 1 )-(A 4 ) in Theorem 9.By Lemma 8, the functional Φ satisfies the condition (A 1 ).By (f 4 ), Φ satisfies the condition (A 2 ).
By (10) and (11), it can be deduced that, for any  > 0, there exists   > 0 such that Here, Let  be the space defined by (98).It has a finite codimension.By Lemma 10 and (110), we have, for any  ∈ , where the positive constant   comes from the Sobolev inequality ‖‖   ≤ (  ) Therefore, Φ satisfies the condition (A 4 ) of Theorem 9.
Because Φ satisfies the conditions (A 1 )-(A 4 ) of Theorem 9, by Theorem 9, it has infinitely many critical points.It follows that (6) has infinitely many solutions.