Landesman–Lazer condition revisited: the influence of vanishing and oscillating nonlinearities

Landesman–Lazer condition revisited: the influence of vanishing and oscillating nonlinearities

Pavel Drábek

and Martina Langerová

1Department of Mathematics, University of West Bohemia, Univerzitní 8, 306 14 Plze ˇn, Czech Republic

2NTIS, University of West Bohemia, Technická 8, 306 14 Plze ˇn, Czech Republic

Received 7 September 2015, appeared 20 October 2015 Communicated by László Simon

Abstract. In this paper we deal with semilinear problems at resonance. We present a sufficient condition for the existence of a weak solution in terms of asymptotic proper- ties of nonlinearity. Our condition generalizes the classical Landesman–Lazer condition and it also covers the cases of vanishing and oscillating nonlinearities.

Keywords: resonance problem, semilinear equation, Landesman–Lazer condition, sad- dle point theorem, critical points.

2010 Mathematics Subject Classification: 35J20, 35J25, 35B34, 35B38.

1 Introduction

Let Ω ⊆ _Rⁿ be a bounded domain, g: R → _R be a bounded continuous function and f ∈ L²(_Ω). We consider the boundary value problem

−_∆u−λ_ku+g(u) = f inΩ,

u=0 on ∂Ω. (1.1)

Hereλ_k, k≥1, is thek-th eigenvalue of the eigenvalue problem

−_∆u−λu=_{0 inΩ,}

u=0 on∂Ω. (1.2)

By a solutionof (1.1) we understand a functionu∈ H:=W₀^1,2(_Ω)satisfying (1.1) in the weak sense,i.e.,

Z

Ω

∇u∇vdx−λ_k Z

Ω

uvdx+

Z

Ω

g(u)vdx=

Z

Ω

f vdx (1.3)

holds for any test functionv∈ H.

BCorresponding author. Email: pdrabek@kma.zcu.cz

Let m≥ 1 be a multiplicity of λ_k. We arrange the eigenvalues of (1.2) into the increasing sequence:

0<λ₁ <λ₂≤ · · · ≤λ_k−1 <λ_k =· · · =λ_k+m−1 <λ_k+m ≤λ_k+m+1 ≤ · · · →_∞.

The corresponding eigenfunctions,(φ_n), form an orthogonal basis for both L²(_Ω)andH. We assume that everyφ_n is normalized with respect to theL²norm,i.e., kφ_nk₂ =_1, n= 1, 2, . . . . We use the scalar product(u,v) =R

Ω∇u∇vdxand the induced normkuk= R

Ω|∇u|²dx1/2

on H. We split the spaceHinto the following three subspaces spanned by the eigenfunctions of (1.2) as follows:

Hˆ := [φ₁, . . . ,φ_k−1], H¯ := [φ_k, . . . ,φ_k+m−1], H˜ := [φ_k+m,φ_k+m+1, . . .].

ThenH = H^ˆ ⊕H^¯ ⊕H^˜ with dim ˆH= k−1, dim ¯H= m, dim ˜H= ∞. Of course, ifk =1 then m= _{1 (λ1} is a simple eigenvalue) and ˆH = ∅. We split an elementu ∈ H as u = uˆ+u¯+u,˜

ˆ

u∈ H, ¯^ˆ u∈ H^¯ and ˜u∈ H. We split a function^˜ f ∈ L²(_Ω)as f = f^¯+ f^⊥, where R

Ω f^⊥vdx= 0 for anyv∈ H.^¯

The purpose of this paper is to introduce a rather general sufficient condition of the Landesman–Lazer type for the existence of a solution of (1.1).

If(u_n)⊂ H is a sequence such thatku_nk₂ →_∞and there existsφ₀ ∈ H,^¯ _ku^un

nk₂ → φ₀ in L²(_Ω), then

nlim→_∞



 Z

Ω

G(un)dx−

Z

Ω

f u¯ ndx



= ±_∞. (SC)±

Here,G(s) =Rs

0 g(τ)dτis the antiderivative ofg.

Theorem 1.1. Assume that either(SC)+or else(SC)₋holds. Then the problem(1.1)has at least one solution.

Remark 1.2. Note that the sufficient condition which is similar to(SC)₊ but more restrictive than(SC)₊ was introduced recently in [8] where the resonance problem with respect to the Fuˇcík spectrum of the Laplacian was studied. In this paper, we benefit from the fact that the resonance occurs at the eigenvalue which allows us to split the underlying function space H into the sum of orthogonal subspaces. In contrast with [8], where such splitting is impossible, we can get rid of the f^⊥-part of the right-hand side f in (SC)_±. This makes our conditions more general and geometrically more transparent.

In the following we interpret our conditions(SC)_±in historical context. We first consider a bounded continuous nonlinear functiong: R→_Rwith finite limitsg(±_∞):=lim_s→±_∞g(s). Example 1.3. Let us assume that

g(∓_∞)

Z

Ω

φ⁺dx−g(±_∞)

Z

Ω

φ⁻dx<

Z

Ω

f¯φdx

< g(±_∞)

Z

Ω

φ⁺dx−g(∓_∞)

Z

Ω

φ⁻dx

(LL)_±

holds for all eigenfunctions φ associated with λ_k. This is the classical Landesman–Lazer condition (see [14]). Assumeku_nk₂ → _{∞and k}^un

unk₂ →φ₀ for some eigenfunctionφ₀. Then by

l’Hospital’s rule we have

nlim→_∞

1 ku_nk₂



 Z

Ω

G(u_n)dx−

Z

Ω

f u¯ _ndx



= lim

n→_∞ Z

Ω

G(u_n) u_n − f^¯

u_n ku_nk₂ ^dx

=

Z

Ω

g(+_∞) + f^¯

φ⁺₀ dx−

Z

Ω

g(−_∞) + f^¯ φ₀⁻dx.

The last expression is either positive or negative due to (_LL)_± and hence conditions (_SC)_± hold. In other words, we proved that(LL)_±imply(SC)_±.

Assume, moreover,g(−_∞)<0<g(+_∞)(think, for example, aboutg(s) =arctans). Then problem (1.1) has a solution for all f which belong to the “strip” (given by inequalities(LL)+) around the linear subspace

L²(_Ω)^⊥:=







f ∈ L²(_Ω): Z

Ω

fφdx=0 for allφ∈ H^¯





 of L²(_Ω).

We note that the conditions (LL)_± are empty if g(−_∞) = g(+_∞). However, we prove existence results even in this case.

Example 1.4. It follows from Theorem 1.1 that the problem (1.1) with g(s) = _(e+|s|)^sgn_ln(^s_e+|s|) (eis Euler’s number) has at least one solution for f ∈L²(_Ω)^⊥. Indeed,

|slim|→_∞G(s) = lim

|s|→_∞ln(ln(e+|s|)) =_∞

implies that (SC)₊ holds true. Hence, our conditions(SC)_± cover the case ofvanishing non- linearitiesg(±_∞) =0 (see [7]). It should be emphasized, that in contrast with previous works on vanishing nonlinearities our approach does not require any kind of symmetry or sign condition about g(cf. [2–4,9,11,13]). At the same time, it generalizes the results from [10,12].

The verification of (SC)_± does not require the existence of limits g(±_∞) at all. See the following example.

Example 1.5. Consider g(s) = arctans+c·coss with an arbitrary constant c ∈ _{R. An easy} calculation yields that (SC)₊ is satisfied, and hence, according to Theorem1.1, problem (1.1) has at least one solution for any f ∈ L²(_Ω)satisfying

Z

Ω

fφdx

< ^π 2

Z

Ω

|φ|dx (1.4)

for anyφ∈ H. On the other hand, the conditions^¯ (LL)_± and various generalizations of(LL)_± (see, e.g. [5,6]) do not apply if |c| ≥ ^π₂, due to the fact that these conditions are vacuous in this case.

Remark 1.6. In fact, the above mentioned case g(s) = _arctans+c·_coss is covered by the so called potential Landesman–Lazer condition:

G^∓ Z

Ω

φ⁺dx−G^± Z

Ω

φ⁻dx <

Z

Ω

f¯φdx< G^± Z

Ω

φ⁺dx−G^∓ Z

Ω

φ⁻dx (PLL)±

where G^± := lim_s→±_∞ ^G(s)

s . Indeed, l’Hospital’s rule implies G⁻ = −^π₂, G⁺ = ^π₂ and the condition (PLL)+ reduces to (1.4). For the use of (PLL)_± see, e.g. the papers [1,18–22].

The conditions(PLL)_± eliminate the influence of the boundedoscillating termc·coss which disappears “in an average” as|s| →_∞.

However, the conditions (PLL)_± do not cover the case like g(s) = ^s

1+s² +c·coss, where c∈_Ris an arbitrary constant. Indeed, both conditions are empty due to the factG^± =_{0. On} the other hand, it follows from our Theorem1.1 that (1.1) with g given above has a solution for any f ∈ L²(_Ω)^⊥. This fact illustrates that our conditions(SC)_±refine also the conditions (_PLL)_±and complement the results from [15] and [16].

In the following example we treat rather general nonlinearity.

Example 1.7. It follows from our Theorem1.1that the boundary value problem

−_∆u−λ_ku+ ^u

(e+u²)ln(e+u²)^1/2 +c·cosu= f inΩ, u=0 on∂Ω,

(1.5)

has a solution for arbitraryc∈_Rand for any f ∈ L²(_Ω)satisfying Z

Ω

fφdx =0

for allφ∈ H. Indeed, since^¯

|slim|→_∞G(s) = lim

|s|→_∞

h ln

ln(e+s²)^1/2+c·sinsi

= _∞

the condition(SC)+is satisfied. The existence result for problems of type (1.5) does not follow from any work published in the literature so far.

Examples and remarks presented above justify the novelty of our work and show that our conditions(SC)_±are new and recover previously published results.

2 Preliminaries

In this section we stress some helpful facts used in the proof of Theorem1.1.

Lemma 2.1. There exist c₁>0, c₂ >0such that for any u∈ H we have Z

Ω

|∇uˆ|²dx−λ_k Z

Ω

(uˆ)²dx≤ −c₁kuˆk² (2.1)

and

Z

Ω

g(u)uˆdx−

Z

Ω

fuˆdx

≤c₂kuˆk_{. (2.2)}

Proof. The inequality (2.1) follows from the variational characterization of λ_k, (2.2) follows from the Hölder inequality, the boundedness ofgand the fact f ∈ L²(_Ω).

Lemma 2.2. There exist c₃>0, c₄>0such that for any u∈ H we have Z

Ω

|∇u˜|²dx−λ_k Z

Ω

(u˜)²dx≥c₃ku˜k² _(2.3)

and

Z

Ω

g(u)u˜dx−

Z

Ω

fu˜dx

≤ c₄ku˜k. (2.4)

Proof. The inequality (2.3) is also a consequence of the variational characterization of λ_k, and (2.4) follows similarly as (2.2).

Lemma 2.3. There exist c5>0such that for any u∈ H we have

Z

Ω

G(u)dx−

Z

Ω

f udx

≤c₅kuk₂. (2.5)

Proof. The inequality (2.5) follows from the Hölder inequality, the boundedness of g and the fact f ∈ L²(_Ω).

3 Proof of Theorem 1.1

We define theenergy functionalassociated with (1.1),E: H→_{R, by} E(u):= ¹

2 Z

Ω

|∇u|²dx− ^λk 2

Z

Ω

(u)²dx+

Z

Ω

G(u)dx−

Z

Ω

f udx, u∈H. Obviously, all critical points ofE satisfy (1.3) and vice versa.

We apply the Saddle Point Theorem due to P. Rabinowitz [17] to prove the existence of a critical point ofE.

Theorem 3.1. LetE ∈C¹(H,R)and H=H⁻⊕H⁺,dimH⁻ <_∞,dimH⁺= ∞. Assume that (a) there exist a bounded neighborhood D of o in H⁻and a constantα∈_Rsuch thatE

∂D ≤α;

(b) there exists a constant β> αsuch thatE

H⁺ ≥β;

(c) E satisfies (PS)condition, i.e., if(E(u_n))⊂ _Ris a bounded sequence and∇E(u_n)→ o in H, then there exist a subsequence(un_k)⊂(un)and an element u∈ H such that un_k →u in H.

Then the functionalE has a critical point in H.

At first we verify the Palais–Smale condition.

Lemma 3.2. Let us assume(_SC)_±. ThenE satisfies(PS)condition.

Proof. In the first step we prove that (un) is bounded in L²(_Ω). Assume the contrary, i.e., ku_nk₂ →_{∞. Set}v_n:= _ku^un

nk₂. Then E(u_n)

kunk²₂ ^:= ¹ 2

Z

Ω

|∇vn|²dx− ^λk 2

Z

Ω

(vn)²dx+

Z

Ω

G(u_n)

kunk²₂ ^dx− ¹ kunk₂

Z

Ω

f vndx →0. (3.1)

The second term is equal to −^λ₂^k since kv_nk₂ = 1, the last two terms go to zero due to Lemma2.3.

Then it follows from (3.1) that(v_n)is a bounded sequence inH. Passing to a subsequence, if necessary, we may assume that there exists v ∈ H such that v_n * v (weakly) in H and vn →vin L²(_Ω).

For arbitraryw∈ H,

0← (∇E⁰(un),w) ku_nk₂ =

Z

Ω

∇vn∇wdx−λ_k Z

Ω

vnwdx

+ ¹

kunk₂

Z

Ω

g(u_n)wdx− ¹ kunk₂

Z

Ω

f wdx.

(3.2)

We haveR

Ω∇v_n∇wdx →R

Ω∇v∇wdx byv_n *v in H,R

Ωv_nwdx →R

Ωvwdx byv_n →v in L²(_Ω), _ku¹

nk₂

R

Ω f wdx → 0 and _ku¹

nk₂

R

Ωg(un)wdx → 0 by f ∈ L²(_Ω), the boundedness of g and by our assumptionku_nk₂ →_∞. Then it follows from (3.2) that

Z

Ω

∇v∇wdx−λ_k Z

Ω

vwdx=0

holds for arbitrary w ∈ H, i.e., v = φ₀ ∈ H^¯ is an eigenfunction associated with λ_k. That is,

un

kunk₂ →φ₀in L²(_Ω).

Now, by the assumption∇E(u_n)→oand the orthogonal decomposition of H, we have o(kuˆnk) = (∇E(un), ˆun) =

Z

Ω

|∇uˆn|²dx−λ_k Z

Ω

(uˆn)²dx+

Z

Ω

g(un)uˆndx−

Z

Ω

fuˆndx. (3.3) By Lemma2.1, it follows from (3.3) that

o(1)≤ −c₁kuˆ_nk+c₂

withc₁,c₂ >0 independent ofn. Hencekuˆ_nkis a bounded sequence.

Similarly, we also have o(ku˜_nk) = (∇E(u_n), ˜u_n) =

Z

Ω

|∇u˜_n|²dx−λ_k Z

Ω

(u˜_n)²dx+

Z

Ω

g(u_n)u˜_ndx−

Z

Ω

fu˜_ndx. (3.4) By Lemma2.2, it follows from (3.4) that

o(₁)≥c₃ku˜_nk −c₄

withc3,c₄> 0 independent ofn. Henceku˜nkis a bounded sequence. Let us split nowE(un) as follows

E(u_n) = ¹ 2

Z

Ω

|∇uˆ_n|²dx−^λk 2

Z

Ω

(uˆ_n)²dx

| {z }

A

+¹ 2

Z

Ω

|∇u˜_n|²dx−^λk 2

Z

Ω

(u˜_n)²dx

| {z }

B

+

Z

Ω

G(un)dx−

Z

Ω

f u¯ ndx

| {z }

C

−

Z

Ω

f^⊥uˆndx−

Z

Ω

f^⊥u˜ndx

| {z }

D

.

The boundedness of kuˆ_nk and ku˜_nk implies that A,B and D are bounded terms. On the other hand, (SC)+ forces C → +_∞ and (SC)− forces C → −∞. In particular, we conclude E(u_n) → ±_∞ which contradicts the assumption of the boundedness of (E(u_n)). We thus proved that(u_n)is a bounded sequence inL²(_Ω).

In the second step we select a strongly convergent subsequence (in H) from (un). Let us examine again the terms in

E(u_n):= ¹ 2

Z

Ω

|∇u_n|²dx− ^λk 2

Z

Ω

(u_n)²dx+

Z

Ω

G(u_n)dx−

Z

Ω

f u_ndx.

By the assumption (E(u_n)) is bounded. The boundedness of the sequence (u_n) in L²(_Ω) implies thatR

Ω(u_n)²dx,R

ΩG(u_n)dx andR

Ω f u_ndxare bounded independently of n, as well.

Therefore, kunk² =R

Ω|∇un|²dxmust be also bounded. Hence, we may assume, without lost of generality, that u_n*u inH for someu∈ H, andu_n→uin L²(_Ω). Then

0←(∇E(u_n),u_n−u) =

Z

Ω

∇u_n∇(u_n−u)dx−λ_k Z

Ω

u_n(u_n−u)dx +

Z

Ω

g(u_n)(u_n−u)dx−

Z

Ω

f(u_n−u)dx.

Since

−λ_k Z

Ω

u_n(u_n−u)dx+

Z

Ω

g(u_n)(u_n−u)dx−

Z

Ω

f(u_n−u)dx→0,

we conclude that _Z

Ω

∇u_n∇(u_n−u)dx→0, as well. So,

Z

Ω

|∇u_n|²dx−

Z

Ω

∇u_n∇udx→0 which together with

Z

Ω

∇u_n∇udx− ku_nk² →₀ (due to the weak convergenceu_n*u) yields

kunk → kuk.

The uniform convexity of H then implies that u_n → u in H. Hence E satisfies the condition (c)in Theorem3.1.

Now we prove that also the hypotheses(_a)_and(_b)hold. To this end we have to consider separately the case(SC)+ and(SC)₋.

1. Let us assume that(_SC)₊ holds. We set

H⁻:= H,^ˆ H⁺ := H^¯ ⊕H.^˜ It follows from Lemmas2.1and2.3that

kulimˆk→_∞E(uˆ):= lim

kuˆk→_∞



 1 2 Z

Ω

|∇uˆ|²dx− ^λk 2

Z

Ω

(uˆ)²dx+

Z

Ω

G(uˆ)dx−

Z

Ω

fuˆdx



= −_∞ (3.5)

for ˆu∈ H.^ˆ

On the other hand, we prove that there existsβ∈_Rsuch that

uinf∈H⁺E(u)≥ β.

Assume the contrary,i.e., there exists a sequence(_un)⊂ H⁺such that

nlim→_∞E(un) =−_{∞. (3.6)}

Thenku_nk₂ →_{∞, and for}v_n:= _ku^un

nk₂ (v_n∈ H⁺) we have 0≥_{lim sup}

n→_∞

E(u_n)

ku_nk²₂ ^:=_{lim sup}

n→_∞



 1 2

Z

Ω

|∇v_n|²dx− ^λk 2

Z

Ω

(v_n)²dx

+

Z

Ω

G(u_n) ku_nk²₂ ^dx−

Z

Ω

f v_n ku_nk₂^dx



.

(3.7)

Clearly, by Lemma2.3, we have Z

Ω

G(u_n) ku_nk²₂ ^dx−

Z

Ω

f v_n

ku_nk₂^dx→0. (3.8)

It follows from (3.7) and (3.8) that kv_nkis a bounded sequence. Passing to a subsequence, if necessary, we may assume that there exists v ∈ H⁺ such that vn * v in H and vn → v in L²(_Ω). Moreover,

lim inf

n→_∞ Z

Ω

|∇v_n|²dx≥

Z

Ω

|∇v|²dx (3.9)

by the weak lower semicontinuity of the norm in H. We deduce from (3.7)–(3.9) that Z

Ω

|∇v|²dx−λ_k Z

Ω

(v)²dx ≤0,

and hence, from Lemma 2.2, it follows that v = φ₀ ∈ H^¯ is an eigenfunction associated with λ_k. That is,

u_n

ku_nk₂ →φ₀ in L²(_Ω).

By Lemma 2.2, by the properties of the orthogonal decomposition of H⁺ and f and by the condition(SC)₊, we have foru_n ∈ H⁺

nlim→_∞E(u_n):= lim

n→_∞



 1 2

Z

Ω

|∇u_n|²dx− ^λk 2

Z

Ω

(u_n)²dx +

Z

Ω

G(u_n)dx−

Z

Ω

f u_ndx





= lim

n→_∞



 1 2

Z

Ω

|∇u˜_n|²dx−^λk 2

Z

Ω

(u˜_n)²dx+

Z

Ω

G(u_n)dx−

Z

Ω

f u¯ _ndx−

Z

Ω

f^⊥u˜_ndx





≥ lim

n→_∞

h

c₃ku˜_nk²− kf^⊥k₂ku˜_nk₂ⁱ+ lim

n→_∞



 Z

Ω

G(u_n)dx−

Z

Ω

f u¯ _ndx





= +_∞.

This contradicts (3.6). By (3.5) there exists R> 0 such that for D:= {u ∈ H⁻ :kuk ≤ R}the following inequality holds

sup

u∈∂D

E(u)<α:= β−1.

Hence, we proved that the hypotheses (a) and (b) in Theorem3.1hold.

2. Let us assume that(_SC)₋ holds. In this case we set H⁻:= H^ˆ ⊕H,^¯ H⁺ := H.^˜ Letu∈ H⁺. Then by Lemmas2.2and2.3, we have

E(u):= ¹ 2

Z

Ω

|∇u|²dx− ^λk 2

Z

Ω

(u)²dx+

Z

Ω

G(u)dx−

Z

Ω

f udx

≥c₃kuk²−c₅kuk₂≥ c₃kuk²−c₆kuk. Hence, there exists β∈_{Rsuch that}E(u)≥ βfor allu∈ H⁺.

On the other hand, we prove that

kuk→lim_∞,u∈H⁻

E(u) =−_∞. (3.10)

Notice, that dimH⁻<_∞implies that the normsk · kandk · k₂are equivalent onH⁻. Assume, by contradiction, that (3.10) does not hold,i.e., there exist a sequence(u_n)⊂ H⁻and a constant c∈_Rsuch thatku_nk₂→_∞and

E(un)≥ c. (3.11)

Set vn := _ku^un

nk₂. Due to dimH⁻ < _∞ we may assume that there exists v ∈ H⁻ such that vn→vboth in HandL²(_Ω). Then

0≤ lim inf

n→_∞

E(u_n) ku_nk²₂

= lim inf

n→_∞



 1 2

Z

Ω

|∇v_n|²dx−^λk 2

Z

Ω

(v_n)²dx+

Z

Ω

G(u_n) ku_nk²₂ ^dx−

Z

Ω

f v_n ku_nk₂ ^dx





= ¹ 2

Z

Ω

|∇v|²dx−^λk 2

Z

Ω

(v)²dx,

(3.12)

by Lemma2.3. According to Lemma2.1, (3.12) impliesv = φ₀ ∈ H, an eigenfunction associ-^¯ ated with λ_k. Hence _ku^un

nk₂ → φ₀ in L²(_Ω). It follows from the orthogonal decomposition of H⁻and f, Lemma2.1 and(_SC)_{− that for}u_n∈ H⁻

nlim→_∞E(u_n):= lim

n→_∞



 1 2

Z

Ω

|∇u_n|²dx −^λk 2

Z

Ω

(u_n)²dx+

Z

Ω

G(u_n)dx−

Z

Ω

f u_ndx





= lim

n→_∞



 1 2

Z

Ω

|∇uˆ_n|²dx−^λk 2

Z

Ω

(uˆ_n)²dx+

Z

Ω

G(u_n)dx−

Z

Ω

f u¯ _ndx−

Z

Ω

f^⊥uˆ_ndx





≤ lim

n→_∞

−c₁kuˆnk²+c₂kuˆnk+ lim

n→_∞



 Z

Ω

G(un)dx−

Z

Ω

f u¯ ndx





=−_∞.

This contradicts (3.11),i.e., (3.10) holds true. Let us choose again D:= {u ∈ H⁻ : kuk ≤ R}. Then, forR>0 large enough, we have

sup

u∈∂D

E(u)< α:=β−1.

Thus, the hypotheses (a) and (b) in Theorem3.1are satisfied.

Recall that the hypothesis (c) in Theorem3.1is proved in Lemma3.2for both cases(_SC)_±. It then follows from Theorem3.1that under the assumptions(SC)_±there exists a critical point ofE. Since this is also a solution of (1.1), the proof of Theorem1.1 is finished.

Acknowledgements

This research was supported by the Grant 13-00863S of the Grant Agency of Czech Republic and by the project LO1506 of the Czech Ministry of Education, Youth and Sports.

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