Special Issue in honor of Alan C. Lazer

Electronic Journal of Differential Equations, Special Issue 01 (2021), pp. 279–292.

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

BIFURCATION FROM INFINITY WITH OSCILLATORY

NONLINEARITY FOR NEUMANN PROBLEMS

MAYA CHHETRI, NSOKI MAVINGA, ROSA PARDO

Honoring the memory of Alan Lazer

Abstract. We consider a sublinear perturbation of an elliptic eigenvalue
problem with Neumann boundary condition. We give sufficient conditions

on the nonlinear perturbation which guarantee that the unbounded contin-

uum, bifurcating from infinity at the first eigenvalue, contains an unbounded
sequence of turning points as well as an unbounded sequence of resonant solu-

tions. We prove our result by using bifurcation theory combined with a careful

analysis of the oscillatory behavior of the continuum near the bifurcation point.

1. Introduction

We consider the nonlinear elliptic equation with Neumann boundary condition

−∆u = λu+ f(λ, x, u), in Ω

∂u

∂η
= 0, on ∂Ω,

(1.1)

where Ω ⊂ RN is a smooth bounded domain with N ≥ 2, ∂/∂η := η(x) · ∇ denotes
the outer normal derivative on ∂Ω, and λ ∈ R is the bifurcation parameter. Here
the nonlinear perturbation f : R×Ω×R→ R is a Carathéodory function, that is,
f = f(λ, x, s) is measurable in x ∈ Ω, and continuous with respect to (λ, s) ∈ R×R.

Observe that problem (1.1) is a perturbation of the eigenvalue problem

−∆ϕ = λϕ , in Ω

∂ϕ

∂η
= 0 , on ∂Ω .

(1.2)

It is well-known that the eigenvalue problem (1.2) has a sequence of eigenvalues
{λi}∞i=1 with the property that 0 = λ1 < λ2 ≤ · · · ≤ λn · · · → +∞ as n→∞. Each
eigenvalue is of finite multiplicity whose corresponding eigenfunctions {ϕi}∞i=1 are
orthogonal in L2(Ω). The first eigenvalue λ1 = 0 is simple and its corresponding
eigenfunction ϕ1 ≡ const. in Ω and can be normalized so that ϕ1 ≡ 1.

The behavior of a nonlinear perturbation f near zero and/or at infinity greatly
influences the existence/multiplicity results for (1.1) with respect to the parameter

2010 Mathematics Subject Classification. 35B05, 35B40, 35J25.
Key words and phrases. Bifurcation from infinity; oscillatory nonlinearity; turning points;
Neumann boundary condition; resonant solutions.
©2021 This work is licensed under a CC BY 4.0 license.

Published January 3, 2022.

279



280 M. CHHETRI, N. MAVINGA, R. PARDO EJDE/SI/01

λ. In this paper, we are focused on solutions bifurcating from infinity. Therefore,
we assume that f satisfies the following assumptions for large arguments.

(H1) There exist h ∈ Lr(Ω) with r > N/2 and continuous functions Λ : R→ R+

and U : R→ R+ satisfying

|f(λ, x, s)| ≤ Λ(λ)h(x)U (s), ∀(λ, x, s) ∈ R× Ω× R

with lim|s|→∞
U(s)
s = 0.

(H2) There exist a function B ∈ Lr(Ω) with r > N/2, α < 1 and s0 > 0 such
that for s > s0, λ→ 0, and x ∈ Ω, we have

|f(λ, x, s)|
|s|α

≤ B(x) .

(H3) f(λ, x, s) is differentiable in s, and ∂f
∂s (λ, ·, ·) ∈ C(Ω× R) and

sup
|s|≥M

‖∂f
∂s

(λ, ·, s)‖L∞(Ω) → 0 as λ→ 0 and M → +∞ . (1.3)

(H4) For x ∈ Ω,

sup
|s|≥M

|f(λ, x, s)− f(0, x, s)|
|s|α

→ 0 as λ→ 0 and M → +∞ .

Note that (H1) implies that f is sublinear at infinity in the variable s, that is,

lim sup
|s|→∞

|f(λ, x, s)|
|s|

= 0 .

After the pioneering work of Rabinowitz [12], bifurcation from infinity for the sub-
linear perturbation of the linear eigenvalue problem is widely studied. The sub-
linearity assumption guarantees the existence of unbounded branches of solutions
when λ approaches one of the eigenvalues of odd multiplicity. These branches bi-
furcate from infinity in the sense of Rabinowitz, see [11, 12]. For the existence of
unbounded branches of solutions of Dirichlet and nonlinear boundary conditions,
see [1, 3, 4, 10] and references therein.

The focus of this article is to study the weak solutions of (1.1) bifurcating from
infinity. By a weak solution of (1.1), we mean a pair (λ, u) ∈ R×H1(Ω) such that∫

Ω

∇u∇ψ +

∫
Ω

uψ = λ

∫
Ω

uψ +

∫
Ω

f(λ, x, u)ψ, for all ψ ∈ H1(Ω).

Note that by (H1), weak solutions of (1.1) lie in the space W 2,r(Ω), r > N/2,
continuously embedded in C(Ω). Therefore, we consider R×C(Ω) as our underlying
space.

The branch bifurcating from infinity at λ1 = 0 forms a continuum (closed con-
nected set) consisting of elements from the set

{(λ, u) ∈ R× C(Ω) : (λ, u) is a weak solution of (1.1)} .
The set of solutions bifurcating from infinity at λ1 = 0 contains large positive
solutions or large negative solutions (or both) of (1.1). Let D+ ⊂ R × C(Ω)
(resp. D− ⊂ R×C(Ω)) denote the continuum of positive, (resp. negative) solutions
bifurcating at λ1 = 0. It is known (see e.g. [12]) that the solutions in D± can be
expressed as

u = t+ w, where w = o(|t|) as |t| → ∞ . (1.4)



EJDE-2021/SI/01 BIFURCATION FROM INFINITY 281

Our main focus is on the analysis of unbounded continuum D+ bifurcating at
λ1 = 0. In particular, we give sufficient conditions on f which guarantees that D+

is neither subcritical (λ < 0) nor supercritical (λ > 0). This leads to the existence
of unbounded sequences of turning points and unbounded sequence of resonant
solutions at λ = 0 on the continuum D±. We say that (λ∗, u∗) ∈ D+ is a turning
point if there is a neighborhood of (λ∗, u∗) in R × C(Ω) such that there are no
solutions (λ, uλ) close to (λ∗, u∗) for λ > λ∗ or for λ < λ∗.

We note that problem (1.1) is a perturbed eigenvalue problem. Therefore, to
investigate the subcritical or supercritical nature of the continuum D+ bifurcating
from infinity at λ = 0, one must analyze the lower order terms of f(λ, x, s) as λ→ 0
and s→∞. To do this, one defines

F+ :=

∫
Ω

lim inf
(λ,s)→(0,+∞)

sf(λ, ·, s)
|s|1+α

, F+ :=

∫
Ω

lim sup
(λ,s)→(0,+∞)

sf(λ, ·, s)
|s|1+α

. (1.5)

It is known that if F+ > 0, then D+ is subcritical, while if F+ < 0, then D+ is
supercritical, see [7, Thm. 2.1] and [10, Thm. 4.3]. Moreover, if all the unbounded
branches are either subcritical or supercritical then, the resonant problem, that is
when λ = 0, has at least one solution, see [7, Cor. 3.5] and [10, Thm. 5.1].

Therefore, in this article we consider nonlinearities satisfying

F+ < 0 < F+ . (1.6)

This condition means that the bifurcating continuum D+ is neither subcritical
nor supercritical, and hence Landesman-Lazer type conditions do not hold. The
main purpose of this article is to establish the existence of infinitely many resonant
solutions at λ = 0 in the absence of Landesman-Lazer type conditions. We note that
the condition (1.6) reflects the oscillatory behavior of D+ near infinity around the
bifurcation point λ = 0, yielding infinitely many resonant solutions. In particular,
we prove the following result.

Theorem 1.1. Let (H1)–(H4) hold. Suppose there exist two increasing sequences
{tn} and {t′n} that tend to +∞ and satisfy

−∞ < lim
n→+∞

∫
Ω

t′n
f(0, ·, t′n)

|t′n|1+α
< 0 < lim

n→+∞

∫
Ω

tn
f(0, ·, tn)

|tn|1+α
<∞ . (1.7)

Then, the following assertions hold.

(I) There exist two sequences {(λn, un)} and {(λ′n, u′n)} in D+ approaching
(0,∞) as n→∞, with λn < 0 (subcritical), and λ′n > 0 (supercritical).

(II) There is a sequence of turning points {(λ∗n, u∗n)} ∈ D+ such that

λ∗n → 0 and ‖u∗n‖C(Ω) →∞, as n→∞ .

Furthermore, one can choose two subsequences of turning points, one of
them subcritical, λ∗2n+1 < 0, and the other supercritical, λ∗2n > 0.

(III) There is a sequence of resonant solutions, that is, there are infinite solutions
{(0, ûn)} ∈ D+ with ‖ûn‖C(Ω) →∞ as n→∞.

The case for D− can be established in a similar fashion.
We briefly describe how each of the hypotheses (H1)–(H4) and (1.7) play crucial

role in proving Theorem 1.1.

• As discussed earlier, (H1) guarantees that D+ bifurcates from infinity at
λ = 0 and for each (λ, u) ∈ D+, u is given by (1.4).



282 M. CHHETRI, N. MAVINGA, R. PARDO EJDE/SI/01

• Assumption (H2) helps establishing the estimates |λ| = O(tα−1) and |w| =
O(tα) as t→∞ in Proposition 2.3.
• Assumption (H3) ensures that the sign of F+ and F+ can be determined

in terms of integrals involving only the parameter t instead of the solution
variable u in Lemma 2.5.
• The technical assumption (H4) helps in the determination of the location

of λ relative to λ1 = 0. See the end of the proof of part (I).
• The assumption (1.7) determines the oscillatory behavior of the continuum

D+ across the hyperplane λ = 0.

Results such as Theorem 1.1 have been studied in [2, 5] in the case of nonlin-
ear boundary conditions, for bifurcation from infinity or from zero respectively. In
[6] one can find a similar result on the existence of unbounded sequences of stable
solutions, unstable solutions, and turning points, even in the absence of resonant so-
lutions, also for nonlinear boundary conditions. To the best of our knowledge, such
results are not known in the case of Neumann boundary conditions. In [3, 4, 7, 10],
the existence of resonant solutions was established when the nonlinearity satis-
fies some type of Landesman-Lazer conditions. We note that the now ubiquitous
Landesman–Lazer condition that guarantees the existence of a resonant solution
first appeared in a paper by Landesman and Lazer in [9]. We are indebted to their
pioneering work and feel privileged to honor Professor Lazer in this paper.

A motivating example concerning Theorem 1.1 is the oscillatory nonlinearity
function

f(s) := |s|α[sin(|s|β) + C] with β 6= 0 and α < 1.

If β ∈ R and C > 1, or if β < 0 and C > 0, then from definition of F+, see (1.5),
F+ > 0 and the bifurcation from infinity is subcritical. On the other hand if β ∈ R
and C < −1, or if β < 0 and C < 0, then F+ < 0 and the bifurcation from infinity
is supercritical.

Therefore, we consider here the range β > 0 and −1 < C < 1 and note that
Theorem 1.1 applies if

β > 0, α+ β < 1, and − 1 < C < 1.

Therefore, in this range of parameters, there exist unbounded sequences of sub-
critical and supercritical solutions, subcritical and supercritical turning points and
infinite resonant solutions.

The restriction α+β < 1 on the size of β is needed in order to satisfy the condition
(1.3). This restriction means that the “oscillating” nonlinearities f cannot oscillate
very fast.

In Section 2, we discuss some preliminaries, functional framework and prove
technical results associated with assumptions (H1)–(H3) that will be used in the
proof of Theorem 1.1. In Section 3, we prove Theorem 1.1 using bifurcation theory
combined with technical results of Section 2. We also state and prove a corollary
that characterizes the λ-intervals from the bifurcation point to the turning points.

2. Preliminaries and auxiliary results

In this section, we discuss the functional framework and establish few auxiliary
results needed in the proof of Theorem 1.1. Let us start by analyzing the behavior
of a sequence of solutions when we know explicitly that the solutions blow up.



EJDE-2021/SI/01 BIFURCATION FROM INFINITY 283

Proposition 2.1. Let (H1) hold. Let {(λn, un)} ⊂ D+ where λn → λ0, un ≥ 0,
and ‖un‖C(Ω) →∞, then λn → 0, and there exists a subsequence, again denoted by

un, such that

lim
n→∞

un
‖un‖C(Ω)

= 1, in Cµ(Ω) for some µ ∈ (0, 1).

Proof. Let vn = un/‖un‖C(Ω). Since un ∈W 2,r(Ω) (see [8, p. 162]) with r > N/2,

by the compact embedding theorem, we obtain un ∈ Cγ(Ω) for some γ ∈ (0, 1).
Then, since (H1) holds, we obtain that ‖vn‖Cγ(Ω) ≤ C. Using the compact embed-

ding Cγ(Ω) ↪→ Cγ
′
(Ω) for 0 < γ′ < γ, we deduce that there exists a convergent

subsequence (again denoted by vn) such that vn → ϕ in Cγ
′
(Ω). Since vn ≥ 0 and

‖vn‖C(Ω) = 1, it is easy to see that 0 ≤ ϕ 6≡ 0. Moreover, vn satisfies

−∆vn = λnvn +
f(λ, x, un)

‖un‖C(Ω)

, in Ω

∂vn
∂η

= 0, on ∂Ω .

(2.1)

Passing to the limit in the weak formulation of (2.1) and using that f(λ,x,un)
‖un‖C(Ω)

→ 0

in Lr(Ω), we obtain
−∆ϕ = λ0ϕ, in Ω

∂ϕ

∂η
= 0, on ∂Ω,

with 0 ≤ ϕ 6≡ 0. Then necessarily ϕ ≡ 1 and λ0 = 0. �

Next, we will prove that under hypothesis (H2), if u = t + w is a solution as
given in (1.4), then w satisfies

w = O(|t|α) as |t| → ∞ .

We analyze first the linear problem. Let λ ∈ (−∞, λ2) and g(λ, ·) ∈ Lr(Ω) with
r > N/2, and consider the linear problem

−∆u = λu+ g(λ, x), in Ω

∂u

∂η
= 0, on ∂Ω .

(2.2)

Then, (2.2) has a unique solution u ∈ W 2,r(Ω) (see [8, p. 162]) if λ 6= 0. More-
over, since r > N/2, by the compact embedding Theorem u ∈ C(Ω). We observe
that (2.2) is a linear perturbation of the eigenvalue problem. Therefore, to take
advantage of this structure, we decompose

Lr(Ω) = span[ϕ1]⊕ span[ϕ1]⊥ = span[1]⊕
{
φ ∈ Lr(Ω) :

∫
Ω

φ = 0
}
. (2.3)

Then for g(λ, ·) ∈ Lr(Ω), with r > N/2 and g(λ, ·) 6≡ const., there exists a unique
decomposition

g(λ, ·) = a1(λ) + g1(λ, ·),
where a1(λ) (the projection onto span[1]), and g1(λ, ·) (orthogonal to span[1]) are
given by

a1(λ) :=
1

|Ω|

∫
Ω

g(λ, ·) and

∫
Ω

g1(λ, ·) = 0. (2.4)



284 M. CHHETRI, N. MAVINGA, R. PARDO EJDE/SI/01

By the Fredholm Alternative, the linear problem (2.2) has a unique solution if
λ 6= 0 (recall λ1 = 0) and does not have solution if λ = 0 and a1(0) 6= 0. Hence, for
λ 6= 0 the solution u = u(λ) of (2.2) belongs to W 2,r(Ω), (see [8, p. 162]) and hence
to Lr(Ω). Therefore, the solution u has a unique decomposition in Lr(Ω) given by

u =
−a1(λ)

λ
+ w , with

∫
Ω

w = 0 . (2.5)

Moreover, w = w(λ) solves the problem

−∆w = λw + g1(λ, x), in Ω

∂w

∂η
= 0, on ∂Ω ,

(2.6)

where g1 is as defined by (2.4).
On the other hand, if λ = 0, by the Fredholm Alternative and by (2.4), there

exists a function v ∈ W 2,r(Ω) such that v + c solves (2.6) for any c ∈ R. Let us
choose c0 ∈ R such that

∫
Ω
v + c0 = 0 and define w(0) = v + c0. This implies that

w(λ) ∈ span[1]⊥ is well defined for any λ ∈ (−∞, λ2).
The lemma below estimates the C(Ω) norm of the solution of (2.6) if g ∈ Lr(Ω).

Lemma 2.2. For each compact set K ⊂ (−∞, λ2) ⊂ R, there exists a constant
C = C(K), independent of λ ∈ K, such that

‖w(λ)‖C(Ω) ≤ C‖g1(λ, ·)‖Lr(Ω) ,

where w satisfies
∫

Ω
w = 0 and (2.6), and g1 satisfies (2.4).

Proof. We observe that w = w(λ) satisfying (2.5)-(2.6) is well defined for any λ ∈ K
by the discussion above.

We first show that w(λ) is uniformly bounded for any λ in a neighborhood of λ1 =
0. Assume to the contrary that there is a sequence λn → 0 with ‖w(λn)‖C(Ω) →∞.

Then it follows from [7, 10, 11, 12] that

w(λn)

‖w(λn)‖C(Ω)

→ ϕ1 ≡ 1 uniformly (up to a subsequence) in Ω .

This contradicts that
∫

Ω
w(λn) = 0. Therefore, there exist δ > 0 and c > 0 such

that ‖w(λ)‖C(Ω) < c independent of λ for any |λ| < δ.

Second, let λ ∈ K \ (−δ, δ). By the Fredholm Alternative, w(λ) ∈ W 2,r(Ω) is
the unique solution of (2.6). Using the Lr-estimate and the embedding of W 2,r(Ω)
into C(Ω), we obtain

‖w(λ)‖C(Ω) ≤ C‖w(λ)‖W 2,r(Ω) ≤ C‖g1(λ, ·)‖Lr(Ω) <∞ .

To conclude, let λ ∈ K and

T (λ) :
{
g1 ∈ Lr(Ω) :

∫
Ω

g1 = 0
}
→ C(Ω)

be a family of operators defined by T (λ)g1 := w(λ), where w(λ) is the solution of
(2.6). Then, T (λ) is continuous for every λ ∈ K. Moreover, supλ∈K ‖T (λ)g1‖C(Ω) <

∞ from the previous two paragraphs. Therefore, by the Uniform Boundedness
Principle, there exists a constant C = C(K) such that

‖w(λ)‖C(Ω) ≤ C(K)‖g1‖Lr(Ω) for any λ ∈ K,
as desired. �



EJDE-2021/SI/01 BIFURCATION FROM INFINITY 285

Proposition 2.3. Let (H1) and (H2) hold. Then, there exists a neighborhood of
(0,∞) ⊂ R× C(Ω) given by

O := {(λ, u) ∈ R× C(Ω) : |λ| < δ0, u(x) > 0, ‖u‖C(Ω) > M0} ,

for some small δ0 and large M0, such that the following hold:

(i) There exist positive constants C1, C2 (independent of λ) such that if (λ, u) ∈
D+ ∩ O and (λ, u) 6= (0,∞), then

u = t+ w where t > 0,

∫
Ω

w = 0, (2.7)

‖w‖C(Ω) ≤ C1‖B‖Lr(Ω)t
α as t→∞, (2.8)

|λ| ≤ C2t
α−1 as t→∞ . (2.9)

(ii) There exists t0 > 0 such that for all t ≥ t0 there exists (λ, u) ∈ D+ ∩ O
satisfying u = t+ w with

∫
Ω
w = 0.

Proof. Let O be as defined above for δ0 > 0 and M0 > 0. Then, since D+ bifurcates
from infinity at λ = 0, there exist δ0 > 0 and M0 > 0 such that D+ ∩ O 6= ∅.

(i) Let (λ, u) ∈ D+ ∩ O. Because of (2.3), u can be written as u = t + w with∫
Ω
w = 0, hence (2.7) holds. Integrating by parts (1.1) and using the divergence

theorem, we obtain

−λ
∫

Ω

u =

∫
Ω

f(λ, x, u) .

Since u = t+ w and
∫

Ω
w = 0, we obtain

− λ t |Ω| =
∫

Ω

f(λ, x, t+ w) . (2.10)

Now, using (H1) and that w = o(|t|) as |t| → ∞,

|f(λ, x, t+ w)|
|t|

=
|f(λ, x, t+ w)|
|t+ w|

∣∣∣1 +
w

t

∣∣∣→ 0 as t→∞ .

Therefore, by the Lebesgue dominated convergence theorem and (2.10), we obtain
λ→ 0 as t→∞. We note that (H2) yields

|f(λ, x, t+ w)| = |t|α |f(λ, x, t+ w)|
|t+ w|α

|1 +
w

t
|α ≤ |t|αB(x)|1 +

w

t
|α . (2.11)

Therefore, it follows from (2.10) that

|λ| ≤ |t|
α−1

|Ω|

∫
Ω

(
B(x)|1 +

w

t
|α
)
≤ C‖B‖Lr(Ω)|t|α−1 .

This shows (2.9).
By (H1) f(λ, ., u(.)) ∈ Lr(Ω), and hence there exists a unique decomposition

f(λ, x, s) = f1(λ, x, s) +

∫
Ω

f(λ, x, s) ,

where
∫

Ω
f(λ, x, s) is the projection onto span[1] and f1 is orthogonal to span[1],

that is,
∫

Ω
f1(λ, x, s) = 0. By Lemma 2.2, we have

‖w‖L∞(Ω) ≤ C‖f1‖Lr(Ω) ≤ C‖f‖Lr(Ω) .

Hence, from (2.11) and that w = o(|t|), we obtain the estimate (2.8),

‖w‖L∞(Ω) ≤ C‖B‖Lr(Ω)|t|α as t→∞ .



286 M. CHHETRI, N. MAVINGA, R. PARDO EJDE/SI/01

This completes part (i).
(ii) Since D+ bifurcates from infinity at λ = 0, one has that D+∩O, although not

necessarily connected, contains an unbounded connected component S . Therefore,
if (λ, u) ∈ S ⊂ D+ ∩ O, we necessarily have

u = t+ w with

∫
Ω

w = 0 and t =

∫
Ω

u . (2.12)

Using the continuity of the projection t =
∫

Ω
u, we infer that the set

{t ∈ R : (1.1) has a solution satisfying (2.12)}

contains an unbounded connected set. Therefore, part (ii) holds. �

As an immediate consequence of the estimate for w given by (2.8) in Proposi-
tion 2.3, we have the following corollary:

Corollary 2.4. Assume (H1) and (H2) hold. Let {(λn, un)} ⊂ D+ ∩ O be such
that λn → 0 and un = tn + wn with

∫
Ω
wn = 0 and tn =

∫
Ω
un →∞, then

lim
n→∞

un
‖un‖C(Ω)

= 1 uniformly in Ω ,

lim
n→∞

un
tn

= 1 uniformly in Ω ,

lim
n→∞

‖un‖C(Ω)

tn
= 1, uniformly in Ω.

We note that, with minor modification in the proof, the results of Corollary 2.4
remain valid when only (H1) is satisfied.

To guarantee that (1.7) is enough to conclude the existence of subcritical (λ < 0)
and supercritical (λ > 0) solutions in the unbounded continuum D+, we will use
the following result.

Lemma 2.5. Let f satisfy (H3). Suppose there exist α < 1 and a function B1 ∈
L1(Ω) such that for x ∈ Ω, and for all (λ, s) close to the bifurcation point (0,+∞),
we have

f(λ, x, s)

|s|α
≤ B1(x) . (2.13)

Let λn → 0, tn ↑ ∞ and wn ∈ L∞(Ω), such that ‖wn‖L∞(Ω) = O(|tn|α) as n→∞.
Then

lim inf
n→+∞

∫
Ω

(tn + wn)f(λn, ·, tn + wn)

|tn + wn|1+α
≥ lim inf

n→+∞

∫
Ω

tnf(λn, ·, tn)

|tn|1+α
, (2.14)

and

lim sup
n→+∞

∫
Ω

(tn + wn)f(λn, ·, tn + wn)

|tn + wn|1+α
≤ lim sup

n→+∞

∫
Ω

tnf(λn, ·, tn)

|tn|1+α
. (2.15)



EJDE-2021/SI/01 BIFURCATION FROM INFINITY 287

Proof. For any w ∈ L∞(Ω) and t > 0 such that |w| < t/2, using the Mean Value
Theorem, we have (with a constant C that may change from line to line)∫

Ω

|f(λ, ·, t+ w)− f(λ, ·, t)| dx

≤ C‖w‖L∞(Ω)

∫
Ω

∫ 1

0

|∂f
∂s

(λ, ·, t+ τw)| dτ dx

≤ C‖w‖L∞(Ω) sup
τ∈[0,1]

∥∥∂f
∂s

(λ, ·, t+ τw)
∥∥
C(Ω)

.

(2.16)

Then, whenever ‖w‖L∞(Ω) = O(|t|α), using (2.16) and (H3), we obtain∫
Ω

|f(λ, ·, t+ w)− f(λ, ·, t)|
|t|α

dx

≤ C sup
|s|≥M

∥∥∂f
∂s

(λ, ·, s)
∥∥
L∞(Ω)

‖w‖L∞(Ω)

|t|α
→ 0

(2.17)

as λ→ 0 and M →∞.
Now, let λn → 0, tn ↑ ∞ and wn ∈ L∞(Ω), such that ‖wn‖L∞(Ω) = O(|tn|α) as

n→∞. Then, (2.17) yields

lim inf
n→+∞

∫
Ω

tn f(λn, ·, tn + wn)

|tn|1+α

≥ lim
λ→0

n→+∞

∫
Ω

tnf(λ, ·, tn + wn)− tnf(λ, ·, tn)

|tn|1+α
+ lim inf
n→+∞

∫
Ω

tnf(λn, ·, tn)

|tn|1+α

= lim inf
n→+∞

∫
Ω

tnf(λn, ·, tn)

|tn|1+α
.

(2.18)

To establish (2.14), we estimate the left hand side of (2.18) from below. For this,
we note that

tnf(λn, ·, tn + wn)

|tn|1+α
=

(tn + wn)f(λn, ·, tn + wn)

|tn + wn|1+α

∣∣1 +
wn
tn

∣∣α .
Then, using that 1 + wn/tn → 1 in L∞(Ω) and (2.18), we obtain

lim inf
n→+∞

∫
Ω

tnf(λn, ·, tn)

|tn|1+α
≤ lim inf

n→+∞

∫
Ω

tn f(λn, ·, tn + wn)

|tn|1+α

= lim inf
n→+∞

∫
Ω

(tn + wn)f(λn, ·, tn + wn)

|tn + wn|1+α

∣∣1 +
wn
tn

∣∣α
≤ lim inf

n→+∞

∫
Ω

(tn + wn)f(λn, ·, tn + wn)

|tn + wn|1+α
.

The integral on the right-hand side above is well defined by (2.13), hence (2.14)
holds. Similar arguments will establish (2.15). Thus the proof is complete. �

3. Proof of Theorem 1.1

Roughly speaking, if there exist an unbounded sequence of subcritical solutions
and another unbounded sequence of supercritical solutions in the continuum of
solutions, then the connectedness of the continuum guarantees that there are infinite
turning points and hence infinite resonant solutions.



288 M. CHHETRI, N. MAVINGA, R. PARDO EJDE/SI/01

Proof of Theorem 1.1. (I) We observe that conclusions (i)–(iii) of Proposition 2.3
hold for some neighborhood O of the bifurcation point (0,+∞) ∈ R × C(Ω). Let
(λn, un) → (0,+∞) and (λ′n, u

′
n) → (0,+∞) in D+ ∩ O be two sequences. Then,

using (2.12), we have

un = tn + wn and u′n = t′n + w′n

with ∫
Ω

wn = 0 =

∫
Ω

w′n, tn :=

∫
Ω

un, t′n :=

∫
Ω

u′n .

Integrating by parts (1.1) for (λ, u) = (λn, un) and thanks to the divergence Theo-
rem we obtain

−λntn =

∫
Ω

f(λn, x, un) .

Dividing by tn‖un‖α−1

C(Ω)
and using Corollary 2.4 yields

lim inf
n→∞

− λn

‖un‖α−1

C(Ω)

= lim inf
n→∞

∫
Ω

f(λn, x, un)

‖un‖αC(Ω)

.

Moreover,∫
Ω

f(λn, x, un)

‖un‖αC(Ω)

=

∫
Ω

f(λn, x, un)

uαn

( un
‖un‖C(Ω)

)α
=

∫
Ω

f(λn, x, un)

uαn

[( un
‖un‖C(Ω)

)α
− 1
]

+

∫
Ω

f(λn, x, un)

uαn
.

Furthermore, by Corollary 2.4,∫
Ω

∣∣∣f(λn, x, un)

uαn

[( un
‖un‖C(Ω)

)α
− 1
]∣∣∣ ≤ ∫

Ω

B(x)
∣∣∣[( un
‖un‖C(Ω)

)α
− 1
]∣∣∣→ 0,

as n→∞, consequently

lim inf
n→∞

− λn

‖un‖α−1

C(Ω)

≥ lim inf
n→∞

∫
Ω

f(λn, x, un)

uαn
.

Then, utilizing un = tn + wn, we obtain

lim inf
n→∞

0− λn
‖un‖α−1

C(Ω)

≥ lim inf
n→∞

∫
Ω

(tn + wn)f(λn, ·, tn + wn)

|tn + wn|1+α

≥ lim inf
n→+∞

∫
Ω

tnf(λn, ·, tn)

|tn|1+α
(by Lemma 2.5)

= lim inf
n→+∞

∫
Ω

tn[f(λn, ·, tn)− f(0, ·, tn) + f(0, ·, tn)]

|tn|1+α

≥ lim inf
n→+∞

∫
Ω

tn[f(λn, ·, tn)− f(0, ·, tn)]

|tn|1+α
+ lim inf
n→+∞

∫
Ω

tnf(0, ·, tn)

|tn|1+α

= lim inf
n→+∞

∫
Ω

tnf(0, ·, tn)

|tn|1+α
> 0 (by (H4) and (1.7)),

yielding λn < 0 for n sufficiently large. Analogously, we obtain λ′n > 0 for n
sufficiently large. This completes part (I).



EJDE-2021/SI/01 BIFURCATION FROM INFINITY 289

(II) Let {tn} and {t′n} be two sequences of positive real numbers such that
tn, t

′
n → +∞ as n → ∞. Then, up to a subsequence, tn < t′n < tn+1 for all n ≥ 1

and tn, t
′
n ≥ t0, where t0 is as defined in Proposition 2.3 (iii). Then, for tn, t

′
n ≥ t0,

Proposition 2.3 (iii) guarantees (λn, un), (λ′n, u
′
n) ∈ D+ ∩ O such that

un = tn + wn with

∫
Ω

wn = 0 and u′n = t′n + w′n with

∫
Ω

w′n = 0 .

We note that λn < 0 (subcritical) and λ′n > 0 (supercritical) for n sufficiently large,
by part (I).

It follows from Proposition 2.3 (i)-(ii) that if (λ, u) ∈ D+ ∩O and
∫

Ω
u = t > t0

then for t0 sufficiently large, we obtain

‖u‖C(Ω) = ‖t+ w‖C(Ω) ≤ (1 + C1‖B‖Lr(Ω)|t0|α−1)t ≤ 2t . (3.1)

Let

Kn := {(λ, u) ∈ D+ ∩ O :

∫
Ω

u = t, and tn ≤ t ≤ tn+1} . (3.2)

We claim that, for each n ∈ N, Kn is a compact set in R × C(Ω). For this, let
(µk, vk) be a sequence in Kn. Obviously tn ≤

∫
Ω
vk ≤ tn+1 for all k, hence (3.1)

implies that ‖vk‖C(Ω) ≤ 2tn+1 for all k. Moreover, by Proposition 2.3 (i) we have

that |λ| ≤ C1t
α−1 ≤ C1t

α−1
0 . Then, by [10, Thm. 2.4], there exists a constant C,

independent of k, such that

‖vk‖Cα(Ω) ≤ C1

(
1 + ‖vk‖C(Ω)

)
≤ C .

Using the compact embedding Cα(Ω) ↪→ Cβ(Ω) for some β ∈ (0, α), we infer that
there exists u∗ ∈ Cβ(Ω) such that vk → u∗ in Cβ(Ω̄), up to a subsequence. Since
(µk, vk) satisfies

−∆vk = µkvk + f(µk, x, vk), in Ω

∂vk
∂η

= 0, on ∂Ω

and f is Carathéodory, f(µk, ·, vk) → f(µ∗, ·, u∗) pointwise. Then, (H1) and the
Lebesgue dominated convergence theorem imply f(µk, ·, vk)→ f(µ∗, ·, u∗) in Lr(Ω)
as k → ∞. Further, passing to the limit in the weak formulation of the above
equation, we see that u∗ is a weak solution of

−∆u∗ = µ∗u∗ + f(λ∗, x, u∗), in Ω

∂u∗

∂η
= 0, on ∂Ω .

The convergence of (µk, vk) ∈ Kn, and the continuity of the projection P implies
t0 ≤ tn ≤ t∗ =

∫
Ω
u∗ ≤ tn+1. Hence, (µ∗, u∗) ∈ Kn establishing the compactness of

Kn.
Since tn < t′n < tn+1, there exists (λ′n, u

′
n) ∈ Kn with u′n = t′n+w′n with

∫
Ω
w′n =

0 and λ′n > 0 by part (I). Define

λ∗n := sup{λ : (λ, u) ∈ Kn} . (3.3)

Then λ∗n ≥ λ′n > 0. By repeating the limiting argument above combined with the
compactness of Kn, we deduce that there exists u∗n such that (λ∗n, u

∗
n) ∈ Kn.

Using that λ∗n > 0 (supercritical) and tn and tn+1 are associated with λn < 0 and
λn+1 < 0, respectively, we have that tn <

∫
Ω
u∗n < tn+1. We can deduce that there



290 M. CHHETRI, N. MAVINGA, R. PARDO EJDE/SI/01

is no solution (λ, u) nearby (λ∗n, u
∗
n) with λ > λ∗n. Otherwise, by the continuity of

the projection, we have tn <
∫

Ω
u < tn+1. This means (λ, u) ∈ Kn, contradicting

the definition of λ∗n in (3.3). Hence (λ∗n, u
∗
n) is a supercritical turning point.

Similarly, letting

K ′n :=
{

(λ, u) ∈ D+ ∩ O :

∫
Ω

u = t′ and t′n ≤ t′ ≤ t′n+1

}
, (3.4)

λ∗,n := inf{λ : (λ, u) ∈ K ′n} (3.5)

we can show the existence of u∗,n such that (λ∗,n, u∗,n) ∈ K ′n is a subcritical
turning point, that is, λ∗,n < 0. Finally, combining the sequences {λ∗,n} and {λ∗n}
and relabeling, one can choose two subsequences of turning points, one of them
subcritical, λ∗2n+1 < 0, and the other supercritical, λ∗2n > 0. This completes the
proof of part (II).

(III) Here we prove the existence of a sequence of resonant solutions, that is
solutions u corresponding to λ = 0. It suffices to show that there exists n0 ∈ N
large enough such that for each n ≥ n0, both sets Kn and K ′n contain resonant
solutions, that is, solutions of the form (0, u).

We give the proof for the sets Kn. Suppose to the contrary that there exists
a sequence of integers numbers nj → +∞ such that Knj does not contain any
resonant solutions. In that case, the compact sets K+

nj := {(λ, u) ∈ Knj : λ ≥ 0}
can be written as K+

nj := (D+ ∩ O) ∩ {(λ, u) ∈ R × C(Ω) : λ > 0, tnj <
∫

Ω
u <

tnj+1}. Therefore K+
nj contains at least one connected component of D+. This

connected component is nonempty since there exists at least one solution (λ′, u′)
with

∫
Ω
u′ = t′ with t′ ∈ (tnj , tnj+1) and therefore λ′ > 0. By construction, since

(tnj , tnj+1) ∩ (tnj+1 , tnj+2) = ∅, we have that K+
nj ∩ K

+
nj+1 = ∅ for j ∈ N. We

recall that a continuum (a closed connected set) cannot contain two nonempty
disjoint connected components. Therefore, the fact that we constructed a sequence
of nonempty, pairwise disjoint connected components of D+ contradicts that D+

is a continuum in R× C(Ω). Hence, there exists a sequence of resonant solutions,
that is a solution u corresponding to λ = 0.

A similar argument applied to the sets K ′n also results in a sequence of resonant
solutions. This completes the proof of (III), and hence of Theorem 1.1. �

Let Kn,K
′
n, λ
∗
n and λ∗,n be as defined in (3.2), (3.4), (3.3) and (3.5), respectively.

Define the sets

Mn := {λ : λ ≥ 0 and ∃u with (λ, u) ∈ Kn},
M ′n := {λ : λ ≤ 0 and ∃u′ with (λ, u′) ∈ K ′n} .

Then one can prove the following result.

Corollary 3.1. For n sufficiently large, we have

Mn = [0, λ∗n], (3.6)

M ′n = [λ∗,n, 0]. (3.7)

Proof. First, we establish (3.6). By the definition of Kn and λ∗n, one has

Mn ⊆ [0, λ∗n] .

Now, suppose to the contrary that [0, λ∗n] ⊆Mn is not true for n sufficiently large.
Then there exists a sequence nj → +∞ such that [0, λ∗nj ] 6⊆ Mnj . So, there exists



EJDE-2021/SI/01 BIFURCATION FROM INFINITY 291

λnj ∈ [0, λ∗nj ] but λnj /∈ Mnj . Therefore, there is no function unj ∈ C(Ω) with

(λnj , unj ) ∈ Knj . From the proof of part (II) of Theorem 1.1 above, we know that
(λ∗nj , u

∗
nj ) ∈ Knj , and so λ∗nj ∈Mnj . Hence necessarily 0 ≤ λnj < λ∗nj .

Let K̃nj := {(λ, u) ∈ Knj , λ > λnj}. Then K̃nj 6= ∅ since (λ∗nj , u
∗
nj ) ∈ K̃nj .

Now, proceeding as in the proof of part (III) of Theorem 1.1 above, we can show

that K̃nj contains at least one nonempty connected component of D+. As in part
(III) above, we can construct a sequence of nonempty, pairwise disjoint connected
components of D+ for nj large, a contradiction to the fact that D+ is a continuum.
Hence (3.6) holds.

A similar argument establishes (3.7), completing the proof. �

Acknowledgements. R. Pardo was supported by grants PID2019-103860GB-I00
from MICINN Spain, and GR58/08, Grupo 920894 from UCM-BSCH Spain. All
authors acknowledge help from MSRI in bringing this group together for collabo-
ration.

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Maya Chhetri

UNC Greensboro, Greensboro, NC, USA
Email address: m chhetr@uncg.edu



292 M. CHHETRI, N. MAVINGA, R. PARDO EJDE/SI/01

Nsoki Mavinga

Swarthmore College, Swarthmore, PA, USA

Email address: nmaving1@swarthmore.edu

Rosa Pardo

Universidad Complutense de Madrid, Madrid, Spain
Email address: rpardo@ucm.es


	1. Introduction
	2. Preliminaries and auxiliary results
	3. Proof of Theorem ??
	Acknowledgements

	References