USA-Chile Workshop on Nonlinear Analysis, Electron. J. Diff. Eqns., Conf. 06, 2001, pp. 155–171. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)

Double resonant problems which are locally non-quadratic at infinity ^{∗}

Marcelo F. Furtado & Elves A. B. Silva

Abstract

We establish the existence of a nontrivial solution for a double resonant elliptic problem under a local non-quadraticity condition at infinity and pointwise limits. We also study the existence of a nonzero solution when there is resonance at the first eigenvalue. The first result is obtained as an application of an abstract theorem that establishes the existence of a nontrivial critical point for functionals of class C^{2} on real Hilbert spaces.

1 Introduction

The main goal of this article is to establish sufficient conditions for the existence of a nontrivial solution for a double resonant semi-linear elliptic problem under a local non-quadraticity condition and pointwise limits. We also study the existence and the multiplicity of solutions when there is resonance at the first eigenvalue.

To achieve our main objective, we prove a generalization of a critical point theorem due to Lazer-Solimini [14]. Their theorem establishes the existence of a nontrivial critical point for a functional of class C^{2} defined on a real Hilbert space, under the hypotheses of Rabinowitz’s saddle point theorem [19].

As it is well known, minimax theorems and related results are based on the existence of a linking structure and on deformation lemmas [2, 3, 20, 17, 29, 6, 26]. In general, to be able to derive such deformation results, it is supposed that the functional satisfies a compactness condition. In this article, we assume the (SCe) condition introduced by Silva in [25] and defined below (see Definition 2.1).

Denoting by m(I, u) [\overline{m}(I, u)] the Morse index [augmented Morse index] of the functional I ∈ C^{2}(E, ℝ) at the point u, we prove the following result:

∗Mathematics Subject Classifications: 35J65, 58E05.

Key words: Resonance, non-quadratic at infinity, Morse index.

\begin{matrix}c \\ ◯ \end{matrix}2001 Southwest Texas State University.

Published January 8, 2001.

The first author was supported by CAPES/Brazil.

The second author was partially supported by CNPq/Brazil.

Theorem 1.1 Let E = V ⊕W be a real Hilbert space with V finite dimensional and W = V^{⊥}. Suppose I ∈ C^{2}(E, ℝ) satisfies (SCe) and

(I_{1}) there exists β ∈ ℝ such that I(v) ≤ β, for all v in V,

(I_{2}) there exists γ ∈ ℝ such that I(w) ≥ γ, for all w in W,

(I_{3}) the origin is a critical point of I, I′′(0) is a Fredholm operator and either dim V < m(I, 0) or \overline{m}(I, 0) < dim V.

Then I possesses a nonzero critical point.

The above result is a generalization of the Theorem 1.1 in [14] since the condition (I_{1}) does not imply the anti-coercivity of I on the subspace V. Theorem 1.1 also complements a recent result by Perera-Schechter [18]. Note that in [18] it is assumed the Palais-Smale compactness condition (PS) which is stronger than condition (SCe) and may not be true under the hypotheses of our application.

As in [14], our proof of Theorem 1.1 is based on the infinite dimensional Morse theory. To compensate the lack of anticoercivity of I on V, we use a deformation result, due to Silva [25] (see also [23, 24]), that sends V ∩ ∂B_{R}(0), for R > 0 sufficiently large, below the level surface γ of I, preserving the linking between V ∩ ∂B_{R}(0) and the subspace W.

As observed above, Theorem 1.1 is motivated by the semilinear elliptic problem,

−Δu = f(x, u) in Ω, u = 0 on ∂Ω,  (1.1)

where Ω is a bounded smooth domain in ℝ^{N} (N ≥ 1) and the nonlinearity f ∈ C^{1}(\overline{Ω} × ℝ, ℝ) satisfies f(x, 0) ≡ 0 and the subcritical growth condition:

(f_{1}) there are constants a_{1}, a_{2} > 0 such that

|f_{s}(x, s)| ≤ a_{1}|s|^{σ−2} + a_{2},
for all x ∈ Ω, s ∈ ℝ where σ > 2 (2 < σ < 2N(N − 2)^{−1} if N ≥ 3).

Standard arguments show that the associated functional I : H_{0}^{1}(Ω) → ℝ given by

I(u) = \frac{1}{2} ∫_{Ω} |∇u|^{2} dx − ∫_{Ω} F (x, u) dx,  (1.2)

with F (x, s) = ∫_{0}^{s} f(x, t) dt, is of class C^{2} and that the classical solutions of (1.1) are the critical points of I. We also note that u ≡ 0 is a (trivial) solution for the problem (1.1). Thus, our first objective is to establish the existence of a nonzero critical point for I.

Denoting by λ_{k} the k^{th} eigenvalue of −Δ on Ω with zero boundary conditions and considering the limits

L(x) = lim inf_{|s|→∞} \frac{2F (x, s)}{s^{2}} and K(x) = lim sup_{|s|→∞} \frac{2F (x, s)}{s^{2}},  (1.3)

taken in the pointwise sense, we suppose (F_{1}) λ_{j} ≤ L(x) ≤ K(x) ≤ λ_{j+1}, a.e. x ∈ Ω. Furthermore, if L(x) ≡ λ_{j}, there

exists D_{+} ∈ L^{1}(Ω) such that

F (x, s) ≥ \frac{λ_{j}}{2} s^{2} + D_{+}(x), ∀ s ∈ ℝ, a.e. x ∈ Ω,  (1.4)

and

(F_{2}) there exist q > 1 (q ≥ \frac{N}{2} , if N ≥ 3), A ∈ L^{q}(Ω) and B ∈ L^{1}(Ω) such that

|F (x, s)| ≤ A(x)s^{2} + B(x), ∀ s ∈ ℝ, a.e x ∈ Ω.

We note that (F_{1}) characterizes (1.1) as a double resonant problem (see [7, 8, 9, 4, 5]) and that the conditions (F_{1}) and (F_{2}) are related to the geometry of the saddle point theorem.

The necessary Morse index estimates for the functional I at u ≡ 0 are provided by a local condition for f(x, s) at the origin:

(f_{2}) f_{s}(x, 0) ≰ λ_{j} or λ_{j+1} ≰ f_{s}(x, 0).

Here, we write f_{s}(x, 0) ≰ (≱)λ_{k} to indicate that f_{s}(x, 0) ≤ (≥)λ_{k} with strict inequality holding on a set of positive measure.

In this article, we also consider a local non-quadraticity condition at infinity on the primitive F. More specifically, setting H(x, s) = f(x, s)s − 2F (x, s), we suppose

(NQ)_{+} there exist Ω_{0} ⊂ Ω with positive measure and C_{+} ∈ L^{1}(Ω) such that

(i) lim_{|s|→∞} H(x, s) = ∞, a.e. x ∈ Ω_{0},
(ii) H(x, s) ≥ C_{+}(x), ∀ s ∈ ℝ, a.e. x ∈ Ω.

We note that condition (NQ)_{+} with Ω_{0} = Ω has been assumed in the works [7, 5] (see also [21, 22]). In this article we show that the hypothesis (NQ)_{+}, for Ω_{0} sufficiently large, and weaker versions of (F_{1}) and (F_{2}) provide the necessary compactness for the functional I.

Now, denoting by μ(U) the Lebesgue measure of a measurable set U ⊂ ℝ^{N}, we state our application of Theorem 1.1:

Theorem 1.2 There exists 0 < α < μ(Ω) such that, if f ∈ C^{1}(\overline{Ω} × ℝ, ℝ) satisfies f(x, 0) ≡ 0, (f_{1}), (f_{2}), (F_{1}), (F_{2}) and (NQ)_{+} with μ(Ω_{0}) > α, then the problem (1.1) possesses a nonzero solution.

It is worthwhile mentioning that in [7, 9, 4, 5] the authors establish the existence of solution for (1.1) (without supposing f(x, 0) ≡ 0). Actually, applying a version of the saddle point theorem due to Silva [25] and the technical results proved in this article, we are able to study (1.1) in this setting when f ∈ C(\overline{Ω} × ℝ, ℝ) and satisfies (f̂_{1}) there are constants a_{3}, a_{4} > 0 such that

|f(x, s)| ≤ a_{3}|s|^{σ−1} + a_{4},
for all x ∈ Ω, s ∈ ℝ where σ > 1 (1 < σ < 2N(N − 2)^{−1} if N ≥ 3).

Noting that on this case the functional I is of class C^{1}, and that the critical points of I are weak solutions of (1.1), we obtain

Theorem 1.3 There exists 0 < α < μ(Ω) such that, if f ∈ C(\overline{Ω}×ℝ, ℝ) satisfies (f̂_{1}), (F_{1}), (F_{2}) and (NQ)_{+} with μ(Ω_{0}) > α, then the problem (1.1) possesses a weak solution.

In this work, we also present versions of Theorems 1.2 and 1.3 under conditions that are dual to (NQ)_{+} and (F_{1}) (see Theorems 3.7 and 4.1).

In our final result, we study the existence of a nonzero solution for the problem (1.1) under a resonant condition at the first eigenvalue. On this case, we may replace the condition (F_{2}) by

(F̂_{2}) there exist q > 1 (q ≥ \frac{N}{2} if N ≥ 3), A ∈ L^{q}(Ω) and B ∈ L^{1}(Ω) such that

F (x, s) ≤ A(x)s^{2} + B(x), ∀ s ∈ ℝ, a.e x ∈ Ω,

and consider the following local condition for the primitive F

(F_{3}) there exists r_{1} > 0 such that

F (x, s) ≥ 0, ∀ 0 < s < r_{1}, a.e. x ∈ Ω.

Setting

L_{0}(x) = lim inf_{s→0^{+}} \frac{2F (x, s)}{s^{2}},

with the limit taken in the pointwise sense, we also suppose the following generalization of the local condition (f_{2}):

(f̂_{2}) L_{0} ∈ L^{1}(Ω), L_{0}(x) ≥ λ_{1}, a.e. x ∈ Ω. Furthermore, if L_{0}(x) ≡ λ_{1}, there exists r_{2} > 0 such that

F (x, s) ≥ \frac{λ_{1}}{2} s^{2}, ∀ 0 < s < r_{2}, a.e. x ∈ Ω.  (1.5)

On this case, we may assume (NQ)_{+} without any restriction on the measure of the set Ω_{0} obtaining

Theorem 1.4 Suppose f ∈ C(\overline{Ω}×ℝ, ℝ) satisfies (f̂_{1}), (f̂_{2}), (F̂_{2}), (F_{3}), K(x) ≤ λ_{1} and (NQ)_{+}. Then the problem (1.1) possesses a nonzero weak solution.

There is a vast literature for the resonance at the first eigenvalue. Here, we cite [12, 11, 10] which are closely related to the above result. We also observe that our argument provides the possibility of considering the nonautonomous problem with pointwise limits in those articles.

Double resonant problems were treated first by Figueiredo-Berestick [4]. In [9], Costa-Oliveira assumed (F_{1}) and a related double resonant condition for f. Later, Costa–Magalhães [7] replaced this last hypothesis by the non-quadraticity condition (NQ)_{+} with Ω_{0} = Ω. We note that in [9, 7] the authors do not allow L(x) ≡ λ_{j} and they also assume uniform limits on (1.3) with respect to Ω. In [5], Carrião–Gonçalves–Pádua assumed the non-quadraticity on Ω and pointwise limits on (1.3) with some extra hypotheses. We should also mention the articles by Silva [25] and Costa–Magalhães [8] where it is considered, under the non-quadraticity condition, the problem of existence of critical points for strongly indefinite functionals and applications. Finally, we observe that in [18] an abstract theorem related to ours is used to study the existence of a nontrivial solution for problem (1.1) under a nonresonant condition and pointwise limits.

The article has the following structure: in Section 2, after presenting some preliminary results, we prove Theorem 1.1. There, we also state the version of Silva’s result [25] that we use to prove Theorem 1.3. In Section 3, we prove Theorem 1.2. In Section 4, we present the proofs of Theorems 1.3 and 1.4.

2 Abstract Result

Let E be a real Hilbert space with norm ∥·∥ derived from the inner product ⟨·, ·⟩. This symbol will also represent the pairing between E and its dual E^{∗}.

Given a functional I of class C^{1} on E, and γ, β, c ∈ ℝ, we set I^{β} = {u ∈ E | I(u) ≤ β}, I_{γ} = {u ∈ E | I(u) ≥ γ}, K = {u ∈ E | I′(u) = 0} and K_{c} = {u ∈ K | I(u) = c}.

Throughout this article we use a generalization of the classical Palais-Smale condition which has been introduced by Silva [25]:

Definition 2.1 The functional I ∈ C^{1}(E, ℝ) satisfies the strong Cerami condition at the level c ∈ ℝ [(SCe)_{c}] if for any sequence (u_{n}) ⊂ E such that I(u_{n}) → c and I′(u_{n}) → 0 we have either

(i) (u_{n}) is bounded and possesses a convergent subsequence, or

(ii) for every (u_{n_{j}}) ⊂ (u_{n}) such that ∥u_{n_{j}}∥ → ∞, we have

lim_{n_{j}→∞} ∥I′(u_{n_{j}})∥∥u_{n_{j}}∥ → ∞.

If I satisfies (SCe)_{c} for every c ∈ [a, b] or for every c ∈ ℝ, we say that I satisfies (SCe) on [a, b] or (SCe), respectively.

Since the condition (SCe) is stronger than the Cerami condition we have (see [27]) the following version of the deformation lemma due to Chang [6]:

Lemma 2.2 Suppose I ∈ C^{1}(E, ℝ) satisfies (SCe) on [a, b], a is the only critical value of I on [a, b) and K_{a} have only pointwise connected components. Then I^{a} is a strong deformation retract of I^{b} \ K_{b}.

Remark 2.3 We may have b = ∞ in Lemma 2.2. On this case, I^{a} is a strong deformation retract of E.

For the proof of Theorem 1.1 we also use a technical result proved in [25]:

Lemma 2.4 Let E = V ⊕ W be a real Hilbert space with W = V^{⊥}. Suppose I ∈ C^{1}(E, ℝ) satisfies (I_{1}), (I_{2}) and (SCe) on [γ, β]. Then, given \overline{ε} > 0, there exist ε ∈ (0, \overline{ε}), R > 0 and a continuous function η : [0, 1] × E → E such that

(η_{1}) η(t, u) = u, for every u ∈ W,

(η_{2}) η(t, ·) : E → E is a homeomorfism, for every t ∈ [0, 1],

(η_{3}) η(1, u) ∈ I^{γ−ε}, for every u ∈ V ∩ ∂B_{R}(0),

(η_{4}) I(η(t, u)) ≤ I(u), for every u ∈ E, t ∈ [0, 1].

Given two topological spaces A and X, with A a subspace of X, let H_{n}(X, A) denote the n^{th} relative singular homology group with coefficient group equal to the real numbers. The next result is based on [14].

Proposition 2.5 Let E = V ⊕ W be a real Hilbert space with V finite dimensional and W = V^{⊥}. Suppose I ∈ C^{2}(E, ℝ) satisfies (I_{1}), (I_{2}), (SCe) and has only a finite number of critical points, all of which are nondegenerate. Then I possesses a critical point u such that m(I, u) = dim V.

Proof. Let k = dim V. When k = 0, the condition (I_{2}) implies that I is bounded from below and consenquently, by (SCe), the infimum of I is attained at a point u ∈ E (see [3, 20]). Since u is nondegenerate, m(I, u) = 0.

Now, let us consider k ≥ 1. Arguing by contradiction, we suppose that I has no critical point with Morse index equals to k.

First, we choose d_{0} < γ such that I has no critical points on I^{d_{0}}. Noting that I satisfies (I_{2}) with γ replaced by d_{0}, we may apply Lemma 2.4 to obtain ε ∈ (0, 1), R > 0 and η ∈ C([0, 1] × E, E) satisfying (η_{1}), (η_{2}), (η_{4}) and

S̃ = η(1, S) ⊂ I^{d_{0}−ε} ⊂ I^{d_{0}} ⊂ E \ W,  (2.1)

where S = V ∩ ∂B_{R}(0).

Moreover, by (η_{1}) and (η_{2}), we have that η(1, E \ W) = E \ W. Thus, considering the inclusions i : S̃ → E \ W and l : S → E \ W, we obtain the following commutative diagram of homomorphisms

H_{∗}(S) ⟶^{l_{∗}} H_{∗}(E \ W)
ψ_{1∗} ↓  ↓ ψ_{2∗}
H_{∗}(S̃) ⟶^{i_{∗}} H_{∗}(E \ W),

where ψ_{1} = η(1, ·)|_{S} and ψ_{2} = η(1, ·)|_{E\W}. Invoking (η_{2}) one more time, we have that ψ_{1∗} and ψ_{2∗} are isomorphisms. Therefore, since S is a strong deformation retract of E \ W, we conclude that

i_{∗} : H_{∗}(S̃) → H_{∗}(E \ W)

is an isomorphism.

Now, let j : S̃ → I^{d_{0}} and h : I^{d_{0}} → H \ W be the inclusion maps. Since, by (2.1), i = h ∘ j, we have that i_{∗} = h_{∗} ∘ j_{∗}. Hence, h_{∗} is surjective and

dim H_{k−1}(I^{d_{0}}) ≥ dim H_{k−1}(E \ W) = dim H_{k−1}(S).

Therefore,

dim H_{k−1}(I^{d_{0}}) ≥ \{ \begin{matrix}1, & if k > 1, \\ 2, & if k = 1. \end{matrix}  (2.2)

Now, consider

c_{1} < c_{2} < ⋯ < c_{m},

the possible critical levels of I, and take d_{1}, …, d_{m} real numbers such that d_{j−1} < c_{j} < d_{j}, for j = 1, …, m. If I does not possess critical values, set d_{1} = d_{m} > d_{0}. Recalling that dim H_{k}(I^{d_{j}}, I^{d_{j−1}}) is equal to the number of critical points of I with Morse index k at the critical level c_{j} (see [17]), we have

dim H_{k}(I^{d_{j}}, I^{d_{j−1}}) = 0, ∀ j = 1, …, m.  (2.3)

The exactness of the sequence

⋯ ⟶H_{k} (I^{d_{j−1}}, I^{d_{0}}) ⟶H_{k} (I^{d_{j}}, I^{d_{0}}) ⟶H_{k} (I^{d_{j}}, I^{d_{j−1}}) ⟶ ⋯

implies that

dim H_{k} (I^{d_{j}}, I^{d_{0}}) ≤ dim H_{k} (I^{d_{j}}, I^{d_{j−1}}) + dim H_{k} (I^{d_{j−1}}, I^{d_{0}}) .

Consequently, by the above inequality and (2.3), we obtain

dim H_{k}(I^{d_{m}}, I^{d_{0}}) ≤ ∑_{j=1}^{m} dim H_{k}(I^{d_{j}}, I^{d_{j−1}}) = 0.  (2.4)

Considering the exact sequence of the pair (I^{d_{m}}, I^{d_{0}}), we conclude that

dim H_{k−1}(I^{d_{0}}) ≤ dim H_{k}(I^{d_{m}}, I^{d_{0}}) + dim H_{k−1}(I^{d_{m}}).

Since I^{d_{m}} is a strong deformation retract of E (see Remark 2.3), by (2.4) and the above expression, we have

dim H_{k−1}(I^{d_{0}}) ≤ dim H_{k−1}(E) = \{ \begin{matrix}0, & se k > 1, \\ 1, & se k = 1, \end{matrix}

which contradicts (2.2) and concludes the proof of Proposition 2.5. ♢

Proposition 2.5 is a key ingredient for the proof of Theorem 1.1. We also need the following version of a result by Marino-Prodi [15] (see also [28]).

Lemma 2.6 Suppose I ∈ C^{2}(E, ℝ) satisfies (SCe), u_{0} is an isolated critical point of I and I′′(u_{0}) is a Fredholm operator. Then, given ε > 0, there exists J ∈ C^{2}(E, ℝ) such that J satisfies (SCe), I(u) = J(u) for ∥u − u_{0}∥ ≥ ε, J has only a finite number of critical points all of which are nondegenerate in the open ball B_{ε}(u_{0}), and ∥I^{(j)}(u) − J^{(j)}(u)∥ < ε, for j = 0, 1, 2 and u ∈ E.

Remark 2.7 In [15], the authors consider the above lemma with the hypothesis that I satisfies the Palais-Smale condition. Noting that for bounded sequences the (SCe) condition is equivalent to the Palais-Smale condition and that the functionals I and J are equals on the complement of B_{ε}(u_{0}), we see easily that the result in [15] holds with the weaker condition (SCe).

We are ready to prove our abstract theorem. The following argument is due to Lazer–Solimini [14] and it will be presented for the sake of completness.

Proof of Theorem 1.1. Without loss of generality we may assume that 0 is an isolated critical point. Since I′′(0) is a Fredholm operator, we can use the spectral theory to obtain a constant b > 0 and an orthogonal decomposition E = E^{0} ⊕ E^{+} ⊕ E^{−}, where E^{0} is the kernel of I′′(0), E^{+} and E^{−} are closed and invariants under I′′(0), and

⟨I′′(0)u, u⟩ ≥ b ∥u∥^{2} , ∀ u ∈ E^{+},  (2.5)
⟨I′′(0)u, u⟩ ≤ −b ∥u∥^{2} , ∀ u ∈ E^{−}.  (2.6)

Now, let J be the functional given by Lemma 2.6, with ε > 0 such that 0 < ε < \frac{b}{3} and ∥I′′(u) − I′′(0)∥ < \frac{b}{3}, if ∥u∥ < ε. It is clear that J also satisfies (I_{1}) and (I_{2}) for appropriated constants.

To prove Theorem 1.1 it suffices to show that J possesses a critical point on the complement of B_{ε}(0). Suppose, by contradiction, that this is not the case. Then, by Proposition 2.5, there exists a critical point \overline{u} of J such that m(J, \overline{u}) = dim V.

By our choice of ε and Lema 2.6, we obtain

∥J′′(\overline{u}) − I′′(0)∥ ≤ ∥J′′(\overline{u}) − I′′(\overline{u})∥ + ∥I′′(\overline{u}) − I′′(0)∥ < \frac{2b}{3}.

Hence, by (2.5) and (2.6), we get

⟨J′′(\overline{u})u, u⟩ ≥ \frac{b}{3} ∥u∥^{2} , ∀ u ∈ E^{+},

⟨J′′(\overline{u})u, u⟩ ≤ −\frac{b}{3} ∥u∥^{2} , ∀ u ∈ E^{−}.

Consequently,

m(I, 0) = dim E^{−} ≤ m(J, \overline{u}) ≤ dim (E^{0} ⊕ E^{−}) = \overline{m}(I, 0).

The last expression and the hypotheses of Theorem 1.1 show that m(J, \overline{u}) ≠ dim V. This concludes the proof of Theorem 1.1. ♢

By the above argument and repeated applications of Lemma 2.6, we get Proposition 2.8 Let E = V ⊕ W be a real Hilbert space with V finite dimensional and W = V^{⊥}. Suppose I ∈ C^{2}(E, ℝ) satisfies (SCe), (I_{1}), (I_{2}) and

(Î_{3}) I possesses a finite number of critical points {u_{j}}_{j=1}^{m} such that I′′(u_{j}) is a Fredholm operator and either dim V < m(I, u_{j}) or \overline{m}(I, u_{j}) < dim V, for

j = 1, …, m.

Then there exists a critical point u of I with u ≠ u_{j}, j = 1, …, m.

Finally, we state the version of the saddle point theorem, due to Silva [25] (see also [23, 24]), that will be used in the proof of Theorem 1.3.

Theorem 2.9 Let E = V ⊕W be a real Hilbert space with V finite dimensional and W = V^{⊥}. Suppose I ∈ C^{1}(E, ℝ) satisfies (I_{1}), (I_{2}) and (SCe)_{b} for every b ≥ γ. Then I possesses a critical value b ∈ [γ, β].

3 Proof of Theorem 1.2

We denote by | · |_{p} the L^{p}(Ω)-norm (1 ≤ p ≤ ∞) and by ∥·∥ the norm in E = H_{0}^{1}(Ω) induced by the inner product

⟨u, v⟩ = ∫_{Ω} ∇u · ∇v dx, ∀ u, v ∈ H_{0}^{1}(Ω).

As mentioned previously, the condition (f_{1}) implies that the functional I given by (1.2) is of class C^{2}, and the classical solutions of (1.1) are the critical points of I. A standard argument [20, 6] shows that for φ, ψ ∈ H,

I′′(0)(φ, ψ) = ∫_{Ω} ∇φ · ∇ψ dx − ∫_{Ω} f_{s}(x, 0)φψ dx .  (3.1)

We denote by E_{k} the subspace of E spanned by {ϕ_{1}, …, ϕ_{k}}, where {ϕ_{1}, ϕ_{2}, …} is the orthonormal basis of E formed by the eigenfunctions associated to the eigenvalues 0 < λ_{1} < λ_{2} ≤ … ≤ λ_{n} ≤ … of −Δ on Ω with zero boundary conditions.

Given N ≥ 3, consider 2^{∗} = \frac{2N}{N−2} and set

S = S(N) = inf_{u∈E, u≢0} \frac{∫_{Ω} |∇u|^{2} dx}{(∫_{Ω} |u|^{2^{∗}} dx)^{2/2^{∗}}} > 0,

the best constant for the Sobolev embedding H_{0}^{1}(Ω) ↪ L^{2^{∗}}(Ω). Note that S depends only on the dimension N [29].

We start with a technical result

Lemma 3.1 Suppose f ∈ C(\overline{Ω} × ℝ, ℝ) satisfies K(x) ≤ λ_{j+1}. Then, given ε > 0, there exists Ω̃ ⊂ Ω and a constant M_{1} = M_{1}(ε) > 0 such that μ(Ω\Ω̃) < ε and

F (x, s) ≤ \frac{1}{2}(ε + λ_{j+1})s^{2} + M_{1}, ∀ s ∈ ℝ, a.e. x ∈ Ω̃.  (3.2)

Proof. Effectively, given ε > 0, we apply the Egorov’s Theorem for the sequence

G_{n}(x) = sup \{\frac{2F (x, s)}{s^{2}} | |s| ≥ n\} ,

to obtain Ω̃ ⊂ Ω such that μ(Ω \ Ω̃) < ε and

lim sup_{|s|→∞} \frac{2F (x, s)}{s^{2}} ≤ λ_{j+1}, unif. for a.e. x ∈ Ω̃.

The above expression provide r = r(ε) > 0 such that

F (x, s) ≤ \frac{1}{2}(ε + λ_{j+1})s^{2}, ∀ |s| ≥ r, a.e. x ∈ Ω̃.  (3.3)

Now, by the continuity of F (x, s) on \overline{Ω} × [−r, r],

F (x, s) ≤ M_{1}, ∀ |s| ≤ r, a.e. x ∈ Ω.

The above expression and (3.3) prove the statement (3.2). ♢

Remark 3.2 If we suppose that for every r > 0 there exists B_{r} ∈ L^{1}(Ω) such that

F (x, s) ≤ B_{r}(x), ∀ |s| ≤ r, a.e. x ∈ Ω,

we obtain a version of the above result with M_{1} ∈ L^{1}(Ω) without to assume the continuity of f.

Now, we prove the compactness condition for the functional I.

Lemma 3.3 There exists 0 < α < μ(Ω) such that, if f ∈ C(\overline{Ω} × ℝ, ℝ) satisfies (f̂_{1}), (F̂_{2}), K(x) ≤ λ_{j+1} and (NQ)_{+} with μ(Ω_{0}) > α, then the functional I satisfies (SCe).

Proof. Let (u_{n}) ⊂ E be such that I(u_{n}) → c ∈ ℝ and I′(u_{n}) → 0. If (u_{n}) is bounded, hypothesis (f̂_{1}) implies that (u_{n}) possesses a convergent subsequence. Consequently, by Definition 2.1, it suffices to verify that ∥I′(u_{n_{j}})∥∥u_{n_{j}}∥ → ∞, for every sequence (u_{n_{j}}) ⊂ (u_{n}) such that ∥u_{n_{j}}∥ → ∞.

Arguing by contradiction, we suppose that there exists a subsequence, which we will denote by (u_{n}), such that ∥u_{n}∥ → ∞ and ∥I′(u_{n})∥ ∥u_{n}∥ is bounded. This assumption and the fact that I(u_{n}) → c provide M ∈ ℝ such that

lim inf ∫_{Ω} H(x, u_{n}) dx = lim inf [2I(u_{n}) − I′(u_{n})u_{n}] ≤ M.  (3.4)

On the other hand, given ε > 0, for n sufficiently large, we have

\frac{1}{2} ∥u_{n}∥^{2} ≤ (ε + c) + ∫_{Ω} F (x, u_{n}) dx.  (3.5)

Now, we apply Lemma 3.1 to obtain M_{1} > 0 and Ω̃ ⊂ Ω such that μ(Ω \ Ω̃) < ε and

F (x, u_{n}) ≤ \frac{1}{2}(ε + λ_{j+1})u_{n}^{2} + M_{1}, a.e. x ∈ Ω̃.

Considering N ≥ 3, we may use the above expression, (3.5), Hölder inequality, (F̂_{2}) and the Sobolev Embeding Theorem, to obtain

\frac{1}{2} ∥u_{n}∥^{2} ≤ (c + ε) + \frac{1}{2}(ε + λ_{j+1})|u_{n}|_{2}^{2} + M_{1}μ(Ω)
+ ∫_{Ω\Ω̃} [Au_{n}^{2} + B] dx
≤ \frac{1}{2}(ε + λ_{j+1})|u_{n}|_{2}^{2} + S^{−1}|A|_{L^{\frac{N}{2}} (Ω\Ω̃)} ∥u_{n}∥^{2} + M̃,

where M̃ = M̃(ε) = c + ε + M_{1}μ(Ω) + |B|_{1}, or equivalently

(\frac{1}{2} − S^{−1}|A|_{L^{\frac{N}{2}} (Ω\Ω̃)}) ∥u_{n}∥^{2} ≤ \frac{1}{2}(ε + λ_{j+1})|u_{n}|_{2}^{2} + M̃.  (3.6)

Defining ũ_{n} = \frac{u_{n}}{∥u_{n}∥} we may suppose that ũ_{n} ⇀ ũ weakly in E, ũ_{n} → ũ strongly in L^{2}(Ω) and ũ_{n}(x) → ũ(x) for almost everywhere x ∈ Ω. Thus, dividing (3.6) by ∥u_{n}∥^{2}, taking n → ∞, ε → 0, we conclude that

1 ≤ λ_{j+1}|ũ|_{2}^{2}.  (3.7)

At this point, we claim that there exists a measurable set Ω_{1} ⊂ Ω_{0} with positive measure such that

ũ(x) ≠ 0, a.e. x ∈ Ω_{1}.  (3.8)

Supposing the claim, we use (NQ)_{+} and Fatou’s lemma to conclude that

lim inf ∫_{Ω} H(x, u_{n}) dx ≥ ∫_{Ω} lim inf H(x, u_{n}) dx = ∞,

which contradicts (3.4).

To prove the claim we set

α = μ(Ω) − (\frac{S}{λ_{j+1}} )^{N/2} > 0,

and suppose, by contradiction, that the claim is false. Then, by (3.7), we have

1 ≤ λ_{j+1} ∫_{Ω} |ũ(x)|^{2} dx ≤ λ_{j+1} [∫_{Ω\Ω_{0}} |ũ(x)|^{2^{∗}} dx]^{2/2^{∗}} μ(Ω \ Ω_{0})^{2/N}
≤ λ_{j+1}S^{−1} \left∥{ \atop ũ}\right∥^{2} μ(Ω \ Ω_{0})^{2/N} ≤ λ_{j+1}S^{−1}μ(Ω \ Ω_{0})^{2/N} < 1.

For N = 1 or N = 2, we use the Sobolev embedding H_{0}^{1}(Ω) ↪ L^{r}(Ω), 1 < r < ∞, and the above argument for μ(Ω \ Ω_{0}) sufficiently small. ♢ Remark 3.4 We note that for K(x) ≤ λ_{1} the above result is valid with α = 0. Effectively, on this case, we verify (3.8) with Ω_{1} = Ω_{0} using (3.7) and the Poincaré’s inequality

1 ≤ λ_{1}|ũ|_{2}^{2} ≤ \left∥{ \atop ũ}\right∥^{2} ≤ 1.

The last expression shows that ũ is a λ_{1}-eigenfunction and therefore ũ(x) ≠ 0 a.e. x ∈ Ω.

The following lemma is a version of a technical result due to Costa-Magalhães [7].

Lemma 3.5 Suppose f satisfies (NQ)_{+} and K(x) ≤ λ_{j+1}. Then

F (x, s) − \frac{1}{2}K(x)s^{2} ≤ −\frac{C_{+}(x)}{2}, ∀ s ∈ ℝ, a.e x ∈ Ω.

Proof. Let g(x, s) = f(x, s) − K(x)s and G(x, s) = F (x, s) − \frac{1}{2}K(x)s^{2}. Then we have

g(x, s)s − 2G(x, s) = f(x, s)s − 2F (x, s) .

Taking s > 0 and using (NQ)_{+}, we obtain

\frac{d}{ds} [\frac{G(x, s)}{s^{2}}] = \frac{g(x, s)s − 2G(x, s)}{s^{3}} ≥ \frac{C_{+}(x)}{s^{3}}.  (3.9)

Integrating the above expression on [s, t] ⊂ (0, ∞), we get

\frac{G(x, s)}{s^{2}} ≤ \frac{G(x, t)}{t^{2}} − \frac{C_{+}(x)}{2} [\frac{1}{s^{2}} − \frac{1}{t^{2}} ] .

Since K(x) ≤ λ_{j+1}, we have that lim sup_{t→∞} t^{−2}G(x, t) ≤ 0 for almost everywhere x ∈ Ω, and therefore

G(x, s) ≤ −\frac{C_{+}(x)}{2}, ∀ s > 0, a.e. x ∈ Ω.

The proof for s < 0 is similar. ♢

The above lemma is used to verify the condition (I_{2}) of Theorem 1.1. The next result is a version of Proposition 2 in [9] (see also [16]). It provides estimates on subsets of Ω and it will be used to verify the hypothesis (I_{1}) of Theorem 1.1.

Lemma 3.6 Suppose (F_{1}) and L(x) ≱ λ_{j}. Then there exist δ_{1}, ε > 0 such that, for every subset Ω̃ ⊂ Ω with μ(Ω \ Ω̃) < ε, we have

∥u∥^{2} − ∫Ω̃ L(x)u^{2} dx ≤ −δ_{1} ∥u∥^{2} , ∀ u ∈ E_{j}.

Proof. By Proposition 2 in [9], the lemma holds for Ω̃ = Ω and δ̂_{1} > 0. Let u ∈ E_{j}, Ω̃ ⊂ Ω. Supposing N ≥ 3, using Hölder inequality and the Sobolev Embeding Theorem, we obtain

∥u∥^{2} − ∫Ω̃ L(x)u^{2} dx = ∥u∥^{2} − ∫_{Ω} L(x)u^{2} dx + ∫_{Ω\Ω̃} L(x)u^{2} dx
≤ −δ̂_{1} ∥u∥^{2} + |L|_{∞}S^{−1}μ(Ω \ Ω̃)^{\frac{2}{N}} ∥u∥^{2}
= ∥u∥^{2} (−δ̂_{1} + |L|_{∞}S^{−1}μ(Ω \ Ω̃)^{\frac{2}{N}} ) .

Hence, the lemma holds for δ_{1} = δ̂_{1}/2 and ε > 0 sufficiently small. The argument for the cases N = 1 or N = 2 is similar. ♢

Proof of Theorem 1.2 Without loss of generality, we may suppose that the origin is an isolated critical point of I. Taking V = E_{k} and W = V^{⊥}, by Lemma 3.5, we have

I(w) = \frac{1}{2}( ∥w∥^{2} − ∫_{Ω} K(x)w^{2} dx) − ∫_{Ω} [F (x, w) − \frac{K(x)}{2}w^{2}] dx ≥ −\frac{|C_{+}|_{1}}{2} ,

for every w ∈ W. Consenquently, I satisfies (I_{2}). To verify (I_{1}), we first suppose that L(x) ≡ λ_{j}. Then, using (1.4), we have

I(v) ≤ \frac{1}{2} (∥v∥^{2} − λ_{j}|v|_{2}^{2}) + |D_{+}|_{1} ≤ |D_{+}|_{1}, ∀ v ∈ V.

Thus, I satisfies (I_{1}) on this case.

Now, let us consider L(x) ≱ λ_{j}. Consider δ_{1}, ε > 0 given by Lemma 3.6. Using the same argument employed in the proof of Lemma 3.1, we obtain M_{1} = M_{1}(ε) > 0 such that

2F (x, v) ≥ (L(x) − ε)v^{2} − M_{1}, ∀ v ∈ V, a.e. x ∈ Ω̃,

where Ω̃ ⊂ Ω satisfies μ(Ω \ Ω̃) < ε. Taking N ≥ 3, the above expression, Lemma 3.6, (F_{2}), Hölder inequality and the Sobolev Embeding Theorem imply

2I(v) ≤ ∥v∥^{2} − ∫Ω̃ L(x)v^{2} dx + ε|v|_{2}^{2} + M_{1}μ(Ω) + 2 ∫_{Ω\Ω̃} [Av^{2} + B] dx
≤ (−δ_{1} + \frac{ε}{λ_{1}} + 2S^{−1}|A|_{L^{\frac{N}{2}} (Ω\Ω̃)}) ∥v∥^{2} + M_{1}μ(Ω) + 2|B|_{1},

for every v ∈ V. Considering ε smaller if necessary, we conclude that I(v) → −∞, when ∥v∥ → ∞. Thus, (I_{1}) also holds on this case. The proof for N = 1 and N = 2 is similar.

Since f satifies (f_{1}), I′′(0) is a Fredholm operator. To establish the Morse Index estimates, we suppose first that f_{s}(x, 0) ≱ λ_{j+1}. Using (3.1) and the same argument employed on the proof of the Lemma 3.6, we obtain δ_{1} > 0 such that

I′′(0)(u, u) = ∥u∥^{2} − ∫_{Ω} f_{s}(x, 0)u^{2} dx ≤ −δ_{1} ∥u∥^{2} , ∀ u ∈ E_{j+1}.

Consequently, m(I, 0) ≥ j + 1 > dim V. An analogous argument shows that, \overline{m}(I, 0) ≤ j − 1 < dim V, whenever f_{s}(x, 0) ≰ λ_{j}. Hence, I satisfies (I_{3}). Now, we invoke Lemma 3.3 and Theorem 1.1 to obtain a nonzero solution for (1.1). ♢

We also observe that the problem (1.1) can be considered under conditions that are dual to (NQ)_{+} and (F_{1}). More specifically, assuming

(F̂_{1}) λ_{j} ≤ L(x) ≤ K(x) ≤ λ_{j+1}, a.e. x ∈ Ω. Furthermore, if K(x) ≡ λ_{j+1}, there exist D_{−} ∈ L^{1}(Ω) such that

F (x, s) ≤ \frac{λ_{j+1}}{2} s^{2} + D_{−}(x), ∀ s ∈ ℝ, a.e. x ∈ Ω,

and

(NQ)_{−} there exist Ω_{0} ⊂ Ω with positive measure and C_{−} ∈ L^{1}(Ω) such that

(i) lim_{|s|→∞} H(x, s) = −∞, a.e. x ∈ Ω_{0},
(ii) H(x, s) ≤ C_{−}(x),  ∀ s ∈ ℝ, a.e. x ∈ Ω,

an argument similar to the one employed in the proof of Theorem 1.2 provides the following theorem.

Theorem 3.7 There exists 0 < α < μ(Ω) such that, if f ∈ C^{1}(\overline{Ω} × ℝ, ℝ) satisfies f(x, 0) ≡ 0, (f_{1}), (f_{2}), (F̂_{1}), (F_{2}) and (NQ)_{−} with μ(Ω_{0}) > α, then the problem (1.1) possesses a nonzero solution.

4 Proofs of Theorems 1.3 and 1.4

We start by observing that, under hypothesis (f̂_{1}), the functional I given by (1.2) is of class C^{1} in E = H_{0}^{1}(Ω) and that the weak solutions of (1.1) are the critical points of I.

Proof of Theorem 1.3 The same arguments employed in the proof of Theorem 1.2 show that I satisfies (I_{1}) and (I_{2}). Thus, we may invoke Lemma 3.3 and Theorem 2.9 to derive the existence of a critical point for I. ♢

Considering the dual conditions (NQ)_{−} and (F̂_{1}), we may apply the above argument to prove

Theorem 4.1 There exist 0 < α < μ(Ω) such that, if f ∈ C(\overline{Ω} × ℝ, ℝ) satisfies (f̂_{1}), (F̂_{1}), (F_{2}) and (NQ)_{−} with μ(Ω_{0}) > α, then the problem (1.1) possesses a weak solution.

Finally, let us establish the multiplicity of solutions for the resonance at the first eigenvalue.

Proof of Theorem 1.4 Using (f̂_{1}), (F̂_{2}), K(x) ≤ λ_{1} and Remark 3.4, we conclude that I satisfies (SCe). The Lemma 3.5 and the argument used in the proof of Theorem 1.2 imply that I is bounded from bellow. Thus, by (SCe), the infimun of I is attained at a critical point u ∈ E. Hence, to prove Theorem 1.4, it suffices to verify that I(tϕ_{1}) ≤ 0, for some t > 0, where ϕ_{1} is the first eigenfunction of −Δ on Ω with zero boundary conditions.

Suppose first that L_{0}(x) ≡ λ_{1}. The regularity of ϕ_{1} implies that, for t > 0 sufficiently small, we have 0 < tϕ_{1}(x) < r_{2} for a.e. x ∈ Ω, with r_{2} given by (f̂_{2}). Thus, we may use (1.5) to obtain the desired inequality

I(tϕ_{1}) = \frac{1}{2} ∫_{Ω} λ_{1}(tϕ_{1})^{2} dx − ∫_{Ω} F (x, tϕ_{1}) dx ≤ 0 .

Now, let us consider L_{0}(x) ≱ λ_{1}. Then, given ε > 0, we may use the definition of L_{0}(x) and the same argument of the proof of Lemma 3.1, to obtain Ω̃ ⊂ Ω such that μ(Ω \ Ω̃) < ε and

F (x, s) ≥ \frac{1}{2}(L_{0}(x) − ε)s^{2}, 0 < s < r, a.e. x ∈ Ω̃,  (4.1)

for some constant r = r(ε) > 0. Furthermore, we may suppose that ε and r are so that

∫Ω̃(L_{0}(x) − λ_{1})ϕ_{1}^{2} dx ≥ α > 0,  (4.2)

and r < r_{1}, with r_{1} given by (F_{3}). This last relation implies that

m = inf_{0<s<r} F (x, s) ≥ 0.  (4.3)

Now, using the regularity of ϕ_{1} one more time, we have that 0 < tϕ_{1}(x) < r, for t > 0 sufficiently small and for almost everywhere x ∈ Ω. Considering (4.1)–(4.3), we obtain

2I(tϕ_{1}) ≤ ∫_{Ω} λ_{1}(tϕ_{1})^{2} dx − ∫Ω̃(L_{0}(x) − ε)(tϕ_{1})^{2} dx
− ∫_{Ω\Ω̃} F (x, tϕ_{1}) dx
≤ −t^{2} [α − ε (|ϕ_{1}|_{2}^{2} + λ_{1}|ϕ_{1}|_{∞}^{2})] .

Taking ε smaller, if necessary, we conclude that I(tϕ_{1}) < 0, for every t > 0 sufficiently small. This concludes the proof of Theorem 1.4. ♢

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Marcelo F. Furtado Dpto. Matemática, Universidade de Brasília 70910 Brasilia, DF Brazil e-mail: furtado@mat.unb.br Elves A. B. Silva Dpto. Matemática, Universidade de Brasília 70910 Brasilia, DF Brazil e-mail: elves@mat.unb.br