Special Issue in honor of Alan C. Lazer

Electronic Journal of Differential Equations, Special Issue 01 (2021), pp. 213–224.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu

INFINITE DIMENSIONAL EXTENSIONS OF THE

LANDESMAN-LAZER THEOREM

MARTIN SCHECHTER

Dedicated to the memory of Alan C. Lazer

Abstract. We show that many of the results obtained for the Landesman-

Lazer type problems can be extended to operators having unbounded essential
spectra stretching from −∞ to +∞. The only requirement is that they have

an isolated eigenvalue of finite multiplicity.

1. Introduction

In their pioneering work, Landesman and Lazer [14] proved the following theo-
rem.

Theorem 1.1. Let g(s) be a continuous function on R such that

g(s)→ g± as s→ ±∞. (1.1)

Let λ` be a simple eigenvalue of the Dirichlet problem

−∆u = λu in Ω, u = 0 on ∂Ω (1.2)

where Ω is a smooth bounded domain in Rn. Then for each h ∈ L2(Ω), a sufficient
condition that there exist a weak solution of

−∆u− λ`u = g(u)− h in Ω, u = 0 on ∂Ω (1.3)

is ∫
Ω

hv dx <

∫
Ω

(g+v
+ − g−v−)dx, v ∈ V \ {0} (1.4)

where v± = max{±v, 0} and V is the eigenspace of (1.2) corresponding to the
eigenvalue λ = λ`.

It was also shown in [14] that (1.4) is also necessary if g− < g(s) < g+ for s ∈ R.
Hypothesis (1.4) excludes the possibility

g+ = g− ≡ g0. (1.5)

2010 Mathematics Subject Classification. 35J35, 47J30, 49J35, 49J40, 58K05.
Key words and phrases. Critical point theory; variational methods; saddle point theory;

semilinear differential equations.
c©2021 This work is licensed under a CC BY 4.0 license.

Published October 6, 2021.

213



214 M. SCHECHTER EJDE/SI/01

On the other hand, if we weaken (1.4) to∫
Ω

hv dx ≤
∫

Ω

(g+v
+ − g−v−) dx, v ∈ V, (1.6)

it is no longer sufficient for a solution of (1.3) to exist. Various authors have given
additional sets of hypotheses that, together with (1.6), will imply the existence of
a solution of (1.3) (cf. the listings in the bibliography and the references quoted in
them.)

In [19], we were able to replace the operator −∆u−λ` with a general self-adjoint
operator A provided the essential spectrum of A is contained in the interval (0,∞).
The reason for this restriction was the fact that we used the following theorem in
the proof.

Theorem 1.2. Let N be a closed subspace of a Hilbert space H and let M = N⊥.
Assume that at least one of the subspaces M,N is finite dimensional. Let G be a
C1 functional on H such that

m1 := inf
w∈M

sup
v∈N

G(v + w) <∞,

m0 := sup
v∈N

inf
w∈M

G(v + w) > −∞.

Then there are a constant c ∈ R and a sequence {uk} ⊂ H such that

m0 ≤ c ≤ m1, G(uk)→ c, G′(uk)→ 0.

The restriction that “at least one of the subspaces M,N is finite dimensional”
causes the restriction that the essential spectrum of A be bounded. In the present
paper we are able to remove this restriction as long as the resolvent of A is not
empty. In order to do this we must replace Theorem 1.2. We treat general boundary
value problems for more general operators.

Let Ω ⊂ Rn be an open set and A a selfadjoint operator on L2(Ω). We assume
that σe(A) is not the whole of R. For convenience, we assume there is an interval
(a, b) containing 0 such that (a, b) ∩ σ(A) = {0}. We let D = D([|A| + 1](1/2)).
With the scalar product (u, v)D = ([|A| + 1](1/2)u, [|A| + 1](1/2)v), it becomes a
Hilbert space. We let

N = E(−∞, a] ∩D, M = E[b,∞) ∩D, Y = N(A),

where E(I) is the spectral projection of A over the interval I. Hence,

N = {v ∈ D : (Av, v) ≤ a‖v‖2},
M = {w ∈ D : (Aw,w) ≥ b‖w‖2},

Y = {y ∈ D(A) : Ay = 0}
are orthogonal invariant subspaces of A with D = N ⊕ Y ⊕M . We assume that
C∞0 (Ω) ⊂ D ⊂ Hm,2(Ω) for some m > 0. In particular,

‖u‖m,2 ≤ C‖u‖D , u ∈ D.
In the more general setting, (1.3) takes on the form

Au = f(x, u), u ∈ D(A) (1.7)

where f(x, t) is a Caratheódory function satisfying

f(x, t)→ f±(x) as t→ ±∞ a.e. in Ω. (1.8)



EJDE-2021/SI/01 LANDESMAN-LAZER THEOREM 215

(In the case of Theorem 1.1, f(x, t) = g(t)− h(x).) Hypothesis (1.4) becomes∫
Ω

(f+v
+ − f−v−)dx > 0, v ∈ N(A) \ {0}. (1.9)

It is a simple matter to show that (1.9) implies both (I) and (II):

(I) infv∈N(A)

∫
Ω
F (x, v)dx > −∞, where

F (x, t) :=

∫ t

0

f(x, s) ds . (1.10)

(II) For each v ∈ N(A) \ {0} there is a w ∈ N(A) such that∫
v>0

f+w dx+

∫
v<0

f−w dx 6= 0. (1.11)

In our first result we show that (I) and (II) are sufficient to ensure the existence
of a solution of (1.7). One can also replace (I) by

(I’) supv∈N(A)

∫
Ω
F (x, v)dx <∞.

Our second step is to replace (1.9) with a hypothesis which will allow

f+(x) = f−(x) ≡ f(x) ∈ N(A)⊥ (1.12)

or even
f(x, t)→ 0 as |t| → ∞. (1.13)

In case of (1.13), a simple sufficient condition is: There are functions F0(x), F1(x) ∈
L1(Ω) such that

F0(x) ≤ F (x, t) ≤ F1(x), x ∈ Ω, t ∈ R (1.14)

and either
F (x, t)→ F0(x) as |t| → ∞ (1.15)

or
F (x, t)→ F1(x) as |t| → ∞. (1.16)

In the case of (1.12) we can replace (1.14)–(1.16) with

F0(x) ≤ F (x, t)− tf(x) ≤ F1(x), (1.17)

F (x, t)− tf(x)→ F0(x) as |t| → ∞, (1.18)

F (x, t)− tf(x)→ F1(x) as |t| → ∞, (1.19)

respectively.
Our main results are stated in Section 2 and proved in Section 3.

2. Semilinear boundary value problems

Let Ω be a domain in Rn, and let A be a selfadjoint operator on L2(Ω) such that

(A) There are constants a < 0 < b such that the essential spectrum σe(A) of A
does not intersect (a, b) and λ = 0 is the only point of (a, b) in the spectrum
σ(A) of A.

(B) There is a function V0(x) > 0 such that multiplication by V0 is a compact
operator from D := D(|A|1/2) to Lp(Ω) for some p ≥ 1.

(C) v 6= 0 a.e. for all v ∈ N(A) \ {0}.
Let f(x, t) be a Caratheódory function on Ω× R such that

(D) |f(x, t)| ≤ V (x) ∈ L2(Ω), x ∈ Ω, t ∈ R,
(E) f(x, t)→ f±(x) a.e. as t→ ±∞,



216 M. SCHECHTER EJDE/SI/01

(F) either

sup
v∈N(A)

∫
Ω

F (x, v)dx <∞ (2.1)

or

inf
v∈N(A)

∫
Ω

F (x, v)dx > −∞, (2.2)

where

F (x, t) :=

∫ t

0

f(x, s)ds. (2.3)

Theorem 2.1. In addition to hypotheses (A)–(F) assume that for each v ∈ N(A)\
{0} there is a w ∈ N(A) such that∫

v>0

f+w +

∫
v<0

f−w 6= 0. (2.4)

Then the problem
Au = f(x, u), u ∈ D(A) (2.5)

has a solution.

Theorem 2.2. In addition to hypotheses (A)–(F), assume that there are functions
F0±(x), F1±(x) in L1(Ω) such that

F0±(x) ≤ F (x, t)− tf±(x) ≤ F1±(x), x ∈ Ω, t ∈ R. (2.6)

Assume also that when (2.1) holds we have

F (x, t)− t f±(x)→ F0±(x) a.e. as t→ ±∞
and when (2.2) holds we have

F (x, t)− tf±(x)→ F1±(x) a.e. as t→ ±∞.
Then (2.5) has at least one solution.

Corollary 2.3. In addition to hypotheses (A)–(E) assume that

f±(x) = f(x) ∈ N(A)⊥.

Assume also that there are functions F0(x), F1(x) in L1(Ω) such that

F0(x) ≤ F (x, t)− tf(x) ≤ F1(x), x ∈ Ω, t ∈ R (2.7)

and either
F (x, t)− tf(x)→ F0(x) a.e. as |t| → ∞ (2.8)

or
F (x, t)− tf(x)→ F1(x) a.e. as |t| → ∞. (2.9)

Then (2.5) has at least one solution.

Corollary 2.4. Assume hypotheses (A)–(E) and

f(x, t)→ 0 a.e. as |t| → ∞. (2.10)

Assume also that there are functions F0±(x), F1±(x) in L1(Ω) such that

F0±(x) ≤ F (x, t) ≤ F1±(x), x ∈ Ω, t ∈ R (2.11)

and either
F (x, t)→ F0±(x) a.e. as t→ ±∞ (2.12)

or
F (x, t)→ F1±(x) a.e. as t→ ±∞. (2.13)



EJDE-2021/SI/01 LANDESMAN-LAZER THEOREM 217

Then (2.5) has at least one solution.

3. Proof of main results

In this section we prove the Theorems of Section 2. We must replace Theorem
1.2. To do so, we introduce a sandwich theory that works in the infinite-dimensional
case.

Let N be a closed, separable subspace of a Hilbert space E. We can define a
new norm | · |w satisfying |v|w ≤ ‖v‖ ∀v ∈ E and such that the topology induced by
this norm is equivalent to the weak topology of N on bounded subsets of N . We
construct the norm so that vj → v weakly in N implies |vj − v|w → 0. Conversely,
if ‖vj‖, ‖v‖ ≤ C for all j > 0 and |vj − v|w → 0, then vj → v weakly in N . This
can be done as follows: Let {ek} be an orthonormal basis for N .Define

(u, v)w =

∞∑
k=1

(u, ek)(v, ek)

2k
, u, v ∈ N.

This is a scalar product. The corresponding norm squared is

|v|2w =

∞∑
k=1

|(v, ek)|2

2k
, v ∈ N.

Then |v|w satisfies |v|w ≤ ‖v‖, v ∈ N . For u ∈ E and Q ⊂ E, we define

dw(u,Q) = inf
v∈Q
|u− v|w.

(Cf. [20]). We denote E equipped with this scalar product and norm by Ew. It is a
scalar product space with the same elements as E. In particular, if (un = vn +wn)

is ‖ · ‖-bounded and un
|·|w→ u, then vn ⇀ v weakly in N , wn → w strongly in N⊥,

un ⇀ v + w weakly in E.
We adjust our assumptions on G for the infinite dimensional case of dimN =∞.

Our requirements on G are given by

Definition 3.1. Let N be a closed separable subspace of a Hilbert space E. A C ′

functionalG on E is called an N-weak-to-weak continuously differentiable functional
on E if |vn − v|w → 0 implies that there is a renamed subsequence satisfying

|G′(vn)−G′(v)|w → 0.

This means that

vn = Pun → v weakly in E, and wn = (I − P )un → w strongly in E

imply that there is a renamed subsequence satisfying

G′(vn + wn)→ G′(v + w) weakly in E,

where P is the projection of E onto N .

The replacement for Theorem 1.2 is as follows.

Theorem 3.2 (Sandwich Theorem). Let N be a closed separable subspace of a
Hilbert space E, and let M = N⊥. For G an N-weak-to-weak continuously differ-
entiable functional on E, assume that

a0 = sup
N
G <∞, b0 = inf

M
G > −∞.



218 M. SCHECHTER EJDE/SI/01

Then there is a sequence {uk} ⊂ E such that

G(uk)→ c, b0 ≤ c ≤ a0, (1 + dw(uk,M))‖G′(uk)‖ → 0. (3.1)

(Cf. [21])

In proving Theorems 2.1 and 2.2, the approach is basically the same for both of
them. We begin by letting

N ′ = ⊕λ<0N(A−λ), N = N ′⊕N(A), M ′ = N⊥∩D, M = M ′⊕N(A). (3.2)

By hypothesis (A), N ′, N(A), N are separable and

D = M ⊕N ′ = M ′ ⊕N. (3.3)

It is easily verified that the functional

G(u) := (Au, u)− 2

∫
Ω

F (x, u)dx (3.4)

is continuously differentiable on D. We take

(u, v)D = ([|A|+ 1](1/2)u, [|A|+ 1](1/2)v), (3.5)

as the scalar product on D. We have

(G′(u), v) = 2(Au, v)− 2(f(x, u), v), u, v ∈ D. (3.6)

Consequently, (2.4) is equivalent to

G′(u) = 0, u ∈ D. (3.7)

To apply Theorem 3.2 we must verify that G(u) is an N -weak-to-weak continu-
ously differentiable functional on D.

Suppose vk = Puk → v, weakly in D, gk = (I − P )uk → g strongly in D, where
P is the projection of D onto N . Since the uk are bounded in D, there is a renamed
subsequence converging to a limit u weakly in D, V uk → V u in L2(Ω) and a.e. in
Ω. Let u′ ∈ D be given. Then f(x, uk(x))u′(x) converges to f(x, u(x))u′(x) a.e.
and is dominated by V |u′| which is in L1(Ω). Consequently, we have∫

Ω

f(x, uk(x))u′(x)dx→
∫

Ω

f(x, u(x))u′(x)dx

as k →∞. Thus,

(G′(uk), u′)/2 = (Auk, u′)−
∫

Ω

f(x, uk(x))u′(x)

→ (Au, u′)−
∫

Ω

f(x, u(x))u′(x)

= (G′(u), u′)/2.

This gives G′(vk +gk)→ G′(v+g) weakly in D. Hence G(u) is an N -weak-to-weak
continuously differentiable functional on D.

Let λ be the largest negative point in the spectrum σ(A) of A, and let λ̄ be the
smallest positive point. Then we have

(Av, v) ≤ λ‖v‖2, v ∈ N ′, (3.8)

λ̄‖w‖2 ≤ (Aw,w), w ∈M ′. (3.9)



EJDE-2021/SI/01 LANDESMAN-LAZER THEOREM 219

Assume that (2.1) holds. By hypothesis (D) and (2.3), G(v) ≤ λ‖v‖2+2‖V ‖·‖v‖ →
−∞ as ‖v‖ → ∞, v ∈ N ′. For w ∈ M we write w = w0 + w′, where w0 ∈ N(A)
and w′ ∈M ′. Since

|F (x,w)− F (x,w0)| ≤ V (x)|w′|
we have

G(w) ≥ λ̄‖w′‖2 − 2

∫
Ω

F (x,w0)dx− 2‖V ‖ · ‖w′‖.

Consequently, (2.1) implies

b0 = inf
M
G > −∞, a0 = sup

N
G <∞. (3.10)

We can now apply Theorem 3.2 to conclude that there is a sequence {uk} ⊂ D such
that

G(uk)→ c, b0 ≤ c ≤ a0, (1 + dw(uk,M))‖G′(uk)‖ → 0.

Let uk = vk + wk + ρky0k where vk ∈ N ′, wk ∈ M ′, y0k ∈ N(A) and ‖y0k‖ = 1,
ρk ≥ 0. We claim that

‖uk‖D ≤ C. (3.11)

To see this we note that (3.1) and (3.6) imply

(Avk, vk)− (f(x, uk), vk) = o(‖vk‖). (3.12)

From this we see that ‖vk‖2 = 0(‖vk‖), and consequently that ‖vk‖ ≤ C. Similarly,
we have

(Awk, wk)− (f(x, uk), vk) = o(‖wk‖) (3.13)

from which we see that ‖wk‖D ≤ C. Suppose ρk → ∞. There is a renamed
subsequence such that y0k → y0 in N(A). Clearly ‖y0‖ = 1. Thus by hypothesis
(C), y0 6= 0 a.e. This means that

|uk| = |vk + wk + ρky0k| → ∞ a.e., (3.14)

f(x, uk)→ q(x) in L2(Ω) (3.15)

where

q(x) =

{
f+(x), y0(x) > 0,

f−(x), y0(x) < 0.
(3.16)

Let u′k = vk + wk. Then u′k ∈ N(A)⊥ and ‖u′k‖D ≤ C. Thus there is a renamed
subsequence such that u′k → u1 weakly in N(A)⊥. Since

(Au′k, h)− (f(x, uk), h) = o(‖h‖D), (3.17)

in the limit we have

Au1 = q. (3.18)

This implies that q ∈ N(A)⊥, i.e., that∫
y0>0

f+v +

∫
y0<0

f−v = 0, v ∈ N(A). (3.19)

But this contradicts (2.4). Hence the ρk are uniformly bounded, and (3.11) holds.
Thus there is a renamed subsequence such that ρky0k → w0 in N(A). By hypothesis
(B) there is a renamed subsequence of V0uk converging in Lp(Ω) and a renamed
subsequence of that converging a.e. in Ω. If we put u = u1 +w0, we see that uk → u
a.e. in Ω. From (3.17) and hypothesis (D) we see that u is a solution of (2.5). This



220 M. SCHECHTER EJDE/SI/01

proves Theorem 2.1 when (2.1) holds. If (2.2) holds, we write v = v0 +v′ for v ∈ N ,
where v0 ∈ N(A) and v′ ∈ N ′. We then have

G(v) ≤ λ‖v′‖2 − 2

∫
Ω

F (x, v0)dx+ 2‖V ‖‖v′‖, v ∈ N. (3.20)

Also

G(w) ≥ λ̄‖w‖2 − 2‖V ‖‖w‖ → ∞ as ‖w‖ → ∞, w ∈M ′ (3.21)

Thus

inf
M ′

G > −∞, sup
N
G <∞. (3.22)

Using the second break up in (3.3) we can apply Theorem 3.2 again to conclude
that there is a sequence satisfying

G(uk)→ c, b0 ≤ c ≤ a0, (1 + dw(uk,M))‖G′(uk)‖ → 0.

We now proceed as before to conclude that (2.5) has a solution. This completes
the proof of Theorem 2.1.

Now we turn to the proof of Theorem 2.2. Assume that (2.1) and (2.7) hold. As
before we get (3.10), and we can apply Theorem 3.2 to the first break up in (3.3)
to obtain a sequence satisfying

G(uk)→ c, b0 ≤ c ≤ a0, (1 + dw(uk,M))‖G′(uk)‖ → 0.

Assume that (3.11) does not hold, i.e., ρk → ∞. We use the same reasoning as
before to show that (3.14) and (3.15) hold, where q(x) satisfies (3.16). This in turn
implies (3.18) and (3.19). Combining (3.17) and (3.18) we have

(A[u′k − u1], h)− (f(x, uk)− q, h) = o(‖h‖D). (3.23)

This implies

u′k → u1 in D. (3.24)

Now (2.7) and (3.14) imply

F (x, uk)− ukq(x)→ Q(x) (3.25)

where

Q(x) =

{
F0+(x), y0(x) > 0,

F0−(x), y0(x) < 0.
(3.26)

Let

a(u, v) = (Au, v), a(u) = (Au, u), u, v ∈ D. (3.27)

By (2.6) we have

G(u) = a(u)− 2

∫
[F (x, u)− uq]dx− 2(u, q)

≤ a(u)− 2a(u, u1)− 2

∫
Q(x)dx

= a(u− u1)− a(u1)− 2

∫
Q(x)dx.

(3.28)

Let P be the (orthogonal) projection onto N ′, and take u = v+w with v ∈ N ′ and
w ∈M . Then

G(v + w) ≤ a(v − Pu1) + a(w − (I − P )u1)− 2

∫
Q(x)dx− a(u1). (3.29)



EJDE-2021/SI/01 LANDESMAN-LAZER THEOREM 221

Hence for each fixed w ∈M ,

G(v + w)→ −∞ as ‖v‖ → ∞, v ∈ N ′. (3.30)

Consequently, for each w ∈M there is a v1 ∈ N ′ such that

G(v1 + w) = max
v∈N ′

G(v + w). (3.31)

Now by (3.25) and (3.28)

G(uk)→ a(u1)− 2(u1, q)− 2

∫
Q(x)dx ≡ c. (3.32)

Take w = (I − P )u1. Then by (3.29) and (3.32),

G(v + w) ≤ a(v − Pu1)− a(u1)− 2

∫
Q(x)dx = a(v − Pu1) + c (3.33)

holds for all v ∈ N ′. Now by Theorem 3.2 and (3.31), there is a v1 ∈ N ′ such that

c ≤ G(v1 + w). (3.34)

In view of (3.32) and (3.33) this implies

c ≤ G(v1 + w) ≤ a(v1 − Pu1) + c. (3.35)

Since a(v1 − Pu1) < 0 unless v1 = Pu1, we must take this value for v1 in (3.34).
Since w = (I − P )u1,

c ≤ G(u1). (3.36)

If we combine this with (3.32) we obtain∫
[F (x, u1)− u1q]dx ≤

∫
Q(x)dx. (3.37)

By (2.6) and (3.26),

Q(x) ≤ F (x, t)− tq(x), x ∈ Ω, t ∈ R. (3.38)

Hence

Q(x) ≤ F (x, u)− uq, x ∈ Ω, u ∈ D. (3.39)

If we combine this with (3.37) we see that

Q(x) ≡ F (x, u1)− u1q. (3.40)

Let

Φ(u) =

∫
Ω

[F (x, u)− uq]dx, u ∈ D. (3.41)

Then

(Φ′(u), h) = (f(x, u)− q, h), h ∈ D. (3.42)

By (3.39) and (3.40),

Φ(u1) ≤ Φ(u), u ∈ D. (3.43)

In view of (3.18) and (3.42), this implies

f(x, u1) = q = Au1. (3.44)

Thus u1 is a solution of (2.5). On the other hand, if (3.11) holds, we can use the
same reasoning as before to conclude that (2.5) has a solution. This proves Theorem
2.2 for the case when (2.1) and (2.7) hold. If (2.2) and (2.8) hold, then we use the
second break up in (3.3), and we prove (3.22) by means of (3.20) and (3.21). Again
we apply Theorem 3.2 to conclude that (3.1) holds for some sequence. If (3.11)



222 M. SCHECHTER EJDE/SI/01

does not hold, this leads to (3.12)–(3.19) as before. We also obtain (3.23)–(3.25)
with (3.26) replaced by

Q(x) =

{
F1+(x), y0(x) > 0,

F1−(x), y0(x) < 0.
(3.45)

In place of (3.28) we have, in view of (2.8),

G(u) ≥ a(u− u1)− a(u1)− 2

∫
Q(x)dx, u ∈ D. (3.46)

This time we let P be the projection onto N (instead of N ′) and we make the break
up u = v + w, v ∈ N,w ∈M ′. Then

G(v + w) ≥ a(v − Pu1) + a(w − (I − P )u1)− 2

∫
Q(x)dx− a(u1). (3.47)

Thus G(v + w) → ∞ as ‖w‖ → ∞ for each fixed v ∈ N . Since G is weakly lower
semicontinuous, for each v ∈ N there is a w1 ∈M such that

G(v + w1) = min
w∈M ′

G(v + w). (3.48)

Again we see that (3.32) holds. We take v = Pu1. Then by (3.47) and (3.32) we
have

G(v+w) ≥ a(w− (I −P )u1)− a(u1)− 2

∫
Q(x)dx = a(w− (I −P )u1) + c (3.49)

for all w ∈M ′. Now by Theorem 3.2, c ≥ G(v +w1) for w, satisfying (3.48). Thus
by (3.32) and (3.49)

c ≥ G(v + w1) ≥ a(w1 − (I − P )u1) + c. (3.50)

This is impossible unless w1 = (I − P )u1. But then

c ≥ G(u1). (3.51)

If we combine this with (3.32), we obtain∫
[F (x, u1)− u1q]dx ≥

∫
Q(x)dx. (3.52)

By (2.6) and (3.45),

F (x, t)− tq(x) ≤ Q(x), x ∈ Ω, t ∈ R (3.53)

which implies

F (x, u)− uq ≤ Q(x), x ∈ Ω, u ∈ D. (3.54)

It now follows form (3.52) and (3.54) that (3.40) holds. If we define Φ(u) by (3.41)
we see that

Φ(u) ≤ Φ(u1), u ∈ D, (3.55)

from which (3.43) and (3.44) follow. Again u1 is a solution of (2.5). If (3.11)
holds, we obtain a solution in the same way as before. This completes the proof of
Theorems 2.1 and 2.2.

Corollaries 2.3 and 2.4 are immediate consequences of Theorem 2.2. In them we
can dispense with hypothesis (F) since it is implied by (2.9) and (2.7). In fact we
have ∫

Ω

F (x, v)dx ≤
∫

Ω

F1(x)dx+

∫
Ω

vf(x)dx.



EJDE-2021/SI/01 LANDESMAN-LAZER THEOREM 223

If v ∈ N(A) the last integral vanishes, and (2.1) holds. Similarly, (2.1) implies both
(2.1) and (2.2).

Editor’s note. We lament reporting that Professor Martin Schechter passed away
on June 7, 2021, a couple of months after this article was accepted for publication.
Professor Schechter was a driving force in Nonlinear Analysis for more than six
decades.

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224 M. SCHECHTER EJDE/SI/01

Martin Schechter

Department of Mathematics, University of California, Irvine, CA 92697-3875, USA

Email address: mschecht@math.uci.edu


	1. Introduction
	2. Semilinear boundary value problems
	3. Proof of main results
	Editor's note

	References