Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 15, pp. 1–9.

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

DOI: 10.58997/ejde.2024.15

A BIHARMONIC EQUATION WITH DISCONTINUOUS

NONLINEARITIES

EDUARDO ARIAS, MARCO CALAHORRANO, ALFONSO CASTRO

Abstract. We study the biharmonic equation with discontinuous nonlinear-

ity and homogeneous Dirichlet type boundary conditions

∆2u = H(u− a)q(u) in Ω,

u = 0 on ∂Ω,

∂u

∂n
= 0 on ∂Ω,

(1)

where ∆ is the Laplace operator, a > 0, H denotes the Heaviside function, q
is a continuous function, and Ω is a bounded domain in RN with N ≥ 3.

Adapting the method introduced by Ambrosetti and Badiale (The Dual

Variational Principle), which is a modification of Clarke and Ekeland’s Dual
Action Principle, we prove the existence of nontrivial solutions to (1). This

method provides a differentiable functional whose critical points yield solutions

to (1) despite the discontinuity of H(s− a)q(s) at s = a.
Considering Ω of class C4,γ for some γ ∈ (0, 1), and the function q con-

strained under certain conditions, we show the existence of two non-trivial
solutions. Furthermore, we prove that the free boundary set Ωa = {x ∈ Ω :

u(x) = a} has measure zero when u is a minimizer of the action functional.

1. Introduction

The main objective of this work is to study the existence of solutions to the PDE

∆2u = H(u− a)q(u) in Ω,

u = 0 on ∂Ω,

∂u

∂n
= 0 on ∂Ω,

(1.1)

where ∆ is the Laplace operator, a > 0, H denotes the Heaviside function, q ∈ C(R),
and Ω is a domain of RN with N ≥ 3.

The action functional associated with (1.1) is given by

J(u) =

∫
Ω

(
(∆u)2 −Q(u)

)
dx ∀u ∈ H2

0 (Ω), (1.2)

2020 Mathematics Subject Classification. 31B30, 35J60, 35J65, 58E05.
Key words and phrases. Biharmonic equation; nonlinear discontinuity; critical point;

dual variational principle; free boundary problem.
©2024. This work is licensed under a CC BY 4.0 license.

Submitted October 15, 2022. Published February 6, 2024.

1



2 E. ARIAS, M. CALAHORRANO, A. CASTRO EJDE-2024/15

where Q(t) :=
∫ t

0
H(s − a)q(s) ds, and H2

0 (Ω) denotes de Sobolev space of square
integrable functions having square integrable first and second order partial deriva-
tives and vanishing in ∂Ω together with its first order partial derivatives. Since H
is not continuous at s = a, Q need not be differentiable at s = a, and, therefore,
J need not be differentiable. We bypass this difficulty using the Dual Variational
Principle introduced by Ambrosetti and Badiale (1989) which yields a differentiable
functional even when Q is not continuous.

2. Preliminaries

Throughout this article we assume that q is a continuous function and that

q(s) ≥ 0 for all s ≥ 0, q is non-decreasing; (2.1)

q(s) ≤ α|s|+ c0, with 0 < α < µ1 and c0 a constant, (2.2)

where µ1 is the first eigenvalue of the biharmonic operator with homogeneous
Dirichlet boundary conditions.

Let us consider the multivalued function q̂ defined by

q̂(s) :=


q(s) if s > a,

[0, q(a)] if s = a,

0 if s < a.

Definition 2.1. A function u : Ω→ R is called a multi valued solution of the PDE
(1) if u ∈ H2

0 (Ω) ∩H4(Ω) and u satisfies

∆2u ∈ q̂(u), a.e. in Ω.

Definition 2.2. Let u a solution of (1). The set

Ωa = {x ∈ Ω : u(x) = a}
is called the free boundary.

Letting p(s) = H(s− a)q(s), we rewrite (1) as

∆2u = p(u) in Ω,

u = 0 on ∂Ω,

∂u

∂n
= 0 on ∂Ω.

(2.3)

Definition 2.3. A function u : Ω → R is called a solution to the PDE (2.3) if
u ∈ H2

0 (Ω) ∩H4(Ω) and u satisfies

∆2u = p(u) a.e. in Ω.

Let us define pm(s) := p(s) + ms. Note that, for m > 0, the function pm is
strictly increasing and (2.3) is equivalent to

∆2u+mu = pm(u) in Ω,

u = 0 on ∂Ω,

∂u

∂n
= 0 on ∂Ω.

(2.4)

Let us consider the multivalued function p̂ defined by

p̂(s) :=

{
pm(s) if s 6= a,

[ma,ma+ q(a)] if s = a,



EJDE-2024/15 A BIHARMONIC EQUATION WITH DISCONTINUOUS NONLINEARITIES 3

where b = q(a).
Let p∗ denote the generalized inverse of p̂ given by

p∗(w) = s ⇐⇒ w ∈ p̂(s).

Remark 2.4. The function p∗ is a continuous though p̂ is a multivalued function,
and

p∗(w) = a ⇐⇒ ma ≤ w ≤ pm(a) = ma+ q(a).

Defining P ∗(w) :=
∫ w

0
p∗(s) ds, we see that P ∗ ∈ C1(R). Also, from (2.2),

w

m+ α
− c0 + q(a)

m
≤ p∗(w) ≤ w

m
for all w ∈ R. (2.5)

From the above inequalities we obtain

P ∗(w) ≥ 1

2

w2

m+ α
− c0 + q(a)

m
|w| for all w ∈ R, (2.6)

P ∗(w) ≤ w2

2m
for all w ∈ R. (2.7)

Assuming that Ω of class C2, for every w ∈ L2(Ω) the problem

(∆2 +m)v = w in Ω,

v = 0 on ∂Ω,

∂v

∂n
= 0 on ∂Ω

has a unique weak solution v ∈ H2
0 (Ω) ∩ H4(Ω). Defining v = G(w), elliptic

regularity theory implies that G is a continuous linear operator from L2(Ω) into
H2

0 (Ω) ∩H4(Ω)). Moreover,∫
Ω

w(x)G(w)(x)dx ≤ 1

m+ µ1

∫
Ω

w2(x)dx. (2.8)

Next we define f : L2(Ω)→ R by

f(w) :=

∫
Ω

(
P ∗(w)− 1

2
wG(w)

)
dx.

Since P ∗ is a differentiable function, f ∈ C1(L2(Ω)).

3. Main results

Lemma 3.1. If w ∈ L2(Ω) is a critical point of f , then u := G(w) is a solution to
(2.3) in the sense that u ∈ H2

0 (Ω) ∩H4(Ω) and ∆2u = p(u) a.e. in Ω.

Proof. Let w ∈ L2(Ω) be such that f ′(w) = 0, then p∗(w) = G(w) a.e. in Ω. Hence
u := G(w) ∈ H2

0 (Ω) ∩ H4(Ω) and satisfies (∆2 + m)u = w. This implies that
p∗(w) = u a.e. in Ω, and from the definition of p∗ we obtain that w ∈ p̂(u), and
hence

∆2u+mu ∈ p̂(u) a.e. in Ω.

For x ∈ Ω\Ωa, i.e., when u(x) 6= a we have p̂(u(x)) = mu(x) +p(u(x)) and then
∆2u(x) = p(u(x)) a.e. x ∈ Ω \ Ωa.

Since u is constant a.e. in Ωa, ∆2u = 0 a.e. in Ωa. Therefore,

∆2u+ pm(u(x)) = mu(x) +H(0)q(a) = ma a.e. in Ω.

Thus ∆2u = p(u) a.e. in Ωa. These show that u is a solution of (2.3). �



4 E. ARIAS, M. CALAHORRANO, A. CASTRO EJDE-2024/15

Next we apply the direct method of the calculus of variations to prove the exis-
tence of a solution (2.3).

Theorem 3.2 (First existence theorem). There exists w0 ∈ L2(Ω) such that

f(w0) = min
w∈L2(Ω)

f(w).

Fixing u0 := G(w0), where u0 is a solution of (2.3), the set

Ωa = {x ∈ Ω : u0(x) = a}
has zero measure.

Proof. For w ∈ L2(Ω), from (2.8) and (2.6),

f(w) ≥ 1

2

[ 1

m+ α
− 1

m+ µ1

]
‖w‖2L2(Ω) − C‖w‖L2(Ω). (3.1)

The hypothesis 0 < α < µ1 and the inequality (3.1) implies

lim
‖u‖L2(Ω)→+∞

f(u) = +∞. (3.2)

That is, f is coercive. Let m̂ = infw∈L2(Ω) f(w). From the coercivity of f , we have
m̂ > −∞. This and the compactness of G imply that f attains its global minimum
at some w0. Let u0 = G(w0) be a solution of (2.3).

Let χ denote the characteristic function of Ωa. This results in

d

dε
f(w0 + εχ) =

∫
Ω

(p∗(w0 + εχ)− εG(χ)−G(w0))χdx

=

∫
Ωa

p∗(w0 + εχ) dx− ε
∫

Ω

χG(χ) dx−
∫

Ωa

u0 dx

for every ε ∈ R. From G(w0) = u0 and ∆2u0 = 0 a.e. in Ωa, it follows that w0 = ma
a.e. in Ωa. Hence, taking 0 < ε < b, one finds that

ma ≤ w0 + εχ ≤ ma+ b = ma+ q(a)

a.e. in Ωa. Then p∗(w0(x) + εχ(x)) = a a.e. in Ωa and∫
Ωa

p∗(w0 + εχ) dx =

∫
Ωa

a dx = a|Ωa| =
∫

Ωa

u0 dx.

Since χ ∈ L2(Ω) by the definition of G there exists z ∈ H2
0 (Ω) ∩H4(Ω) such that

z = G(χ), it follows that

(G(χ) | χ) =

∫
Ω

(z∆2z +mz2) dx.

The above equalities imply

d

dε
f(w0 + εχ) = −ε

(∫
Ω

(∆z)2 dx +m‖z‖2L2(Ω)

)
.

If |Ωa| > 0, it follows that
d

dε
f(w0 + εχ) < 0

a contradiction, because w0 is the global minimum of f . �

We note that the last arguments of the proof are valid for any local minimum of
f . The next lemma and Lemma 3.5 prove that the graph f satisfies the geometric
hypotheses of the Mountain-Pass theorem.



EJDE-2024/15 A BIHARMONIC EQUATION WITH DISCONTINUOUS NONLINEARITIES 5

Lemma 3.3. For each a > 0 and m > 0, there exists ε > 0 and γ > 0 such that
if ‖u‖L2(Ω) ≤ ε then f(u) ≥ γ‖u‖2L2(Ω). Hence f attains a strict local minimum at
u = 0.

Proof. Let α1 ∈ (α, µ1). Since p∗(s) = ms for all s ∈ (−∞, a], P ∗(s) = s2

2m for any
s ∈ (−∞,ma]. Also, from (2.2), there exists c1 ≥ ma such that

P ∗(s) ≥ 1

2(m+ α1)
s2 for s ≥ c1. (3.3)

For v ∈ L2(Ω)\{0}, let W = {x ∈ Ω;ma ≤ v(x) ≤ c1}, v1 = χΩ\W v and v2 = χW v,
where χS denotes the characteristic function of the set S. Thus,

∫
Ω

P ∗(v1)dx ≥ 1

2(m+ α1)

∫
Ω

v2
1(x)dx. (3.4)

Letting |W | denote the Lebesgue measure of the set W , we have

|W | ≤
‖v2‖2L2(Ω)

m2a2
=
‖v2‖2L2(W )

m2a2
. (3.5)

Since p∗(ma) = a, for s ∈ [ma, c1] we have P ∗(s) ≥ a
2c1
s2. Therefore

a

2c1

∫
W

v2
2(x)dx ≤

∫
W

P ∗(v2(x))dx ≤ c21
2m
|W | ≤ c21

2m3a2

∫
W

v2
2(x)dx. (3.6)

From the definition of µ1, we have
∫

Ω
G(v1)v1dx ≤ 1

m+µ1

∫
Ω
v2

1dx. By regularity

properties of elliptic operators, there exist p > 2 and K > 0 such that

‖G(u)‖Lp(Ω) ≤ K(p)‖u‖L2(Ω) for all u ∈ L2(Ω). (3.7)

Hence, for i = 1, 2, see (3.5),

∫
Ω

v2(x)G(vi(x))dx =

∫
W

v2(x)G(vi(x))dx

≤ ‖v2‖L2(Ω)

(∫
W

(G(vi))
2(x)dx

)1/2

≤ ‖v2‖L2(Ω)

(∫
W

(G(vi))
p(x)dx

)1/p

|W |(p−2)/2p

≤ K(p)‖v2‖L2(Ω)‖vi‖L2(Ω)|W |(p−2)/2p

≤ K(p)

(ma)(p−2)/p
‖v2‖2(p−1)/p

L2(Ω) ‖vi‖L2(Ω).

(3.8)



6 E. ARIAS, M. CALAHORRANO, A. CASTRO EJDE-2024/15

Therefore,∫
Ω

v(x)G(v(x))dx

=

∫
Ω

(v1G(v1) + v2G(v1) + v1G(v2) + v2G(v2))dx

≤ 1

m+ µ1
‖v1‖2L2(Ω) +

∫
Ω

(2v2G(v1) + v2G(v2))dx

=
1

m+ µ1
‖v1‖2L2(Ω) +

∫
W

(2v2G(v1) + v2G(v2))dx

≤ 1

m+ µ1
‖v1‖2L2(Ω)

+
K(p)

(ma)(p−2)/p
‖v2‖2(p−1)/p

L2(Ω)

(
2‖v1‖L2(Ω) + ‖v2‖L2(Ω)

)
≤ 1

m+ µ1
‖v1‖2L2(Ω) + C‖v2‖2(p−1)/p

L2(Ω)

(
‖v1‖L2(Ω) + ‖v2‖L2(Ω)

)
,

(3.9)

with C > 0 independent of v. Combining (3.4), (3.6), and (3.9), we have

f(v) =

∫
Ω

[
P ∗(v(x))− 1

2
v(x)G(v(x))

]
dx

≥ 1

2(m+ α1)
‖v1‖2L2(Ω) +

a

2c1
‖v2‖2L2(Ω) −

1

2(m+ µ1)
‖v1‖2L2(Ω)

− C‖v2‖2(p−1)/p
L2(Ω)

(
‖v1‖L2(Ω) + ‖v2‖L2(Ω)

)
≥ µ1 − α1

4(m+ α1)(m+ µ1)
‖v1‖2L2(Ω) +

a

2c1
‖v2‖2L2(Ω)

− C‖v2‖2(p−1)/p
L2(Ω)

(
‖v1‖L2(Ω) + ‖v2‖L2(Ω)

)
≥ γ1‖v‖2L2(Ω) − 2C‖v‖1+2(p−1)/p

L2(Ω)

≥ γ1‖v‖2L2(Ω)

(
1− 2C

γ1
‖v‖(3p−2)/p

L2(Ω)

)
,

(3.10)

where

γ1 = min{ µ1 − α1

4(m+ α1)(m+ µ1)
,
a

2c1
}.

Since p > 2, (3p− 2)/p > 0. Hence taking ε = (γ1/(4C))p/(3p−2) and γ = γ1/2, the
lemma is proven. �

The next lemmas show that, under suitable conditions on Ω and an appropriate
relationship between a and q(a), f possesses a pair of non-trivial critical points: a
negative global minimum and a positive Mountain-Pass critical point.

Definition 3.4. Let U be a domain in RN , k ∈ N, γ ∈ [0, 1), and ε > 0. We say
that U is ε-close in Ck,γ-sense to the unit ball B if there exists a surjective mapping
g ∈ Ck,γ(B;U) such that

‖g − Id‖Ck,γ(B;U) ≤ ε.

In 2020 Grunau and Sweers[13] show that there is εN > 0 such that if Ω is ε-close
in C4,γ-sense to the unitary ball B with ε < εN , then the first eigenfunction ϕ1 for



EJDE-2024/15 A BIHARMONIC EQUATION WITH DISCONTINUOUS NONLINEARITIES 7

the first eigenvalue µ1 of

∆2ϕ = µϕ in Ω,

ϕ = 0 on ∂Ω,

∂ϕ

∂n
= 0 on ∂Ω

is unique (up to normalization), and ϕ1 > 0 in Ω.

Lemma 3.5. Let Ω be ε-close in Ck,γ-sense to the unit ball B. If

q(a)

a
=
b

a
> 2µ1

‖ϕ1‖L1(Ω)

‖ϕ1‖2L2(Ω)

, (3.11)

then f(bϕ1) < 0.

Proof. Since 0 < bϕ1(x) ≤ b and p∗(w) ≤ a, for 0 ≤ w ≤ b, it follows that

f(bϕ1) =

∫
Ω

P ∗(bϕ1) dx− 1

2
b2
∫

Ω

G(ϕ1)ϕ1 dx

≤ ba‖ϕ1‖L1(Ω) −
b2

2µ1
‖ϕ1‖2L2(Ω).

This and (3.11) imply f(bϕ1) < 0. �

Finally, we prove that f satisfies a weak form of (PS) condition.

Lemma 3.6. Let {wk}k∈N in L2(Ω) be such that {f ′(wk)}k∈N converges to 0 and
{f(wk)}k∈N converges to a real number c, then there exists w ∈ L2(Ω) with f(w) =
c, f ′(w) = 0, and wk ⇀ w.

Proof. The coercivity of the functional f implies, up to subsequences, the existence
of w ∈ L2(Ω) such that wn ⇀ w in L2(Ω). From f ′(wk)→ 0 and the compactness
of G, it follows that G(wn) → v := G(w), strongly in L2(Ω), and a.e. in Ω. Let
Γ = {x ∈ Ω : v(x) = a} and Ω1 = Ω \ Γ.

Let us begin studying the convergence in Ω1. Since p ∈ C(R\{a}) and p∗(wk)→ v
a.e. in Ω, hence wk → p(v) a.e. in Ω1. Clearly, |w| ≤ C1|p∗(w)|+ C2; this and the
convergence of {p∗(wk)}k∈N in L2(Ω) imply that there exists h ∈ L2(Ω) such that
|wk| ≤ h for every k ∈ N. Applying the Lebesgue dominated convergence theorem:
wk → p(v) a.e. in L2(Ω1). From the uniqueness of the weak limit, one infers that
w = p(v) in L2(Ω1). Since p∗ is asymptotically linear, it follows that

p∗(wk)→ p∗(w) in L2(Ω1), and

∫
Ω1

P ∗(wk) dx→
∫

Ω1

P ∗(w) dx. (3.12)

On the other hand, for a.e. x ∈ Γ, one has w(x) = mv(x) = ma and hence
p∗(w(x)) = p∗(ma) = a = v(x). This jointly with (3.12) imply p∗(w) = v, which in
turnf ′(w)v = 0, hence f ′(w) = 0. In a similar way, from (3.12) and the definition
of P ∗(s) for s ∈ [ma,ma+ b], one finds that∫

Ω

P ∗(wk) dx→
∫

Ω

P ∗(w) dx.

Letting c =
∫

Ω
[P ∗(w)− 1

2wG(w)] dx it follows that f(w) = c, which completes the
proof. �



8 E. ARIAS, M. CALAHORRANO, A. CASTRO EJDE-2024/15

Theorem 3.7. Assume that the domain Ω is ε-close in Ck,γ-sense to the unit
ball B. Suppose that (2.1), (2.2), and (3.11) hold. Then the problem (2.3) has
two distinct solutions u0 6= u1, and one of these solutions, obtained through the
minimizer, has a free boundary set of measure zero.

Proof. Let w0 be the global minimum of f given by Theorem 3.2. By Lemma 3.5,
f(w0) < 0. Hence w0 6= 0 and u0 = G(w0) is a non-trivial solution of (2.3) and the
free boundary Ωa(u0) = {x ∈ Ω : u0(x) = a} has zero measure.

Taking ρ = ε/2 > 0 and β = γε/2 > 0 in Lemma 3.3 we see that f(u) ≥ β > 0 for
‖u‖L2(Ω) = ρ > 0. This Lemmas 3.5, and 3.6 allow us to apply the Mountain-Pass
Theorem (see [5]), yielding a second non-trivial critical point w1, with f(w1) ≥
β > 0. Hence u1 = G(w1) 6= 0 is a second non-trivial solution of (2.3). Since
f(w0) < 0 < f(w1), w0 6= w1 and as a consequence u0 6= u1.

Finally, the zero measure of Ωa(u0) follows from the fact that u0 minimizes f
over all functions with zero measure on the set Ωa(u0), as proven in Theorem 3.2.
However, it is possible for the free boundary of u1 to have positive measure.

Therefore, by Lemma 3.1, problem (2.3) has two different solutions u0 6= u1,
with the free boundary of u0 having zero measure. �

Acknowledgments. We would like to thank the Escuela Politécnica Nacional for
providing financial support through the proyecto semilla PIS-17-01 during the de-
velopment of this work.

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[14] A. Szulkin; Ljusternik-Schnirelmann theory on C1-manifolds, Annales de l’Institut Henri

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Eduardo Arias

Departamento de Matemática, Escuela Politécnica Nacional, Quito PO-Box 17-01-2759,
Ecuador

Email address: marcelo.arias@epn.edu.ec, eduardo.arias.94@outlook.es

Marco Calahorrano

Departmento de Matemática, Escuela Politécnica Nacional, Quito PO-Box 17-01-2759,

Ecuador
Email address: marco.calahorrano@epn.edu.ec

Alfonso Castro
Department of Mathematics, Harvey Mudd College, Claremont, CA 91711, USA

Email address: castro@g.hmc.edu


	1. Introduction
	2. Preliminaries
	3. Main results
	Acknowledgments

	References