Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 60, pp. 1–14.

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

DOI: 10.58997/ejde.2025.60

NEHARI MANIFOLD FOR DEGENERATE LOGISTIC PARABOLIC

EQUATIONS

JULIANA FERNANDES, LILIANE MAIA

Communicated by Giovanni Molica Bisci

Abstract. In this article we analyze the behavior of solutions to a degenerate logistic equation

with a nonlinear term b(x)f(u) where the weight function b is non-positive. We use variational
techniques and the comparison principle to study the evolutionary dynamics. A crucial role

is then played by the Nehari manifold, as we note how it changes as the parameter λ in the

equation or the function b vary, affecting the existence and non-existence of stationary solutions.
We describe a detailed picture of the positive dynamics and also address the local behavior of

solutions near a nodal equilibrium, which sheds some further light on the study of the evolution

of sign-changing solutions.

1. Introduction

Our goal in this article is to study the solutions of the semilinear parabolic equation

∂tu = ∆u+ λu+ b(x)|u|ν−1u, (x, t) ∈ Ω× (0,+∞),

u|∂Ω = 0, t ∈ (0,+∞)

u|t=0 = u0(x), x ∈ Ω,

(1.1)

where Ω is an open smooth bounded domain in RN , N ≥ 2, λ is a real positive parameter,
1 < ν < 2∗ − 1, where 2∗ = +∞ if N = 2, or 2∗ = 2N/(N − 2) if N ≥ 3, and b is a continuous
function satisfying b(x) ≤ 0 and b(x) = 0 in a smooth proper subdomain Ω0 of Ω, with positive
Lebesgue measure and smooth boundary.

The main purpose is to apply variational methods and comparison principle in order to analyze
the behavior of solutions as the initial data varies in the phase space. The Nehari manifold N con-
tained in the abstract space, associated to the energy functional of the stationary elliptic problem,
will be used to locate the stationary solutions, and identify convergence regions of evolutionary
trajectories. We note how the different ranges of λ and the sign of the weight function b(x) deeply
affect the global dynamics picture. To our knowledge, this is the first time the so-called Nehari
approach is applied to address the asymptotic analysis of solutions to the logistic problem.

From the Population Dynamics point of view, this class of equations appears as an interplay of
two well known classical laws: Malthusian and Verhulst (logistic) growth. In that context, u(x, t)
models the evolution of the distribution of a single species in the inhabiting area Ω, λ is related
to the growth rate of the population u, b(x) translates the crowding effects within the region
Ω \Ω0. The region Ω0 is also referred to as the favorable region, as it represents the region where
u is allowed to enjoy exponential growth, as expected by the Malthus law. The analysis of the
evolution of this problem may be found, for instance in the very complete and interesting work by
Julián López-Gómez [18].

2020 Mathematics Subject Classification. 35A01, 35B40, 35K58, 35A15.

Key words and phrases. Logistic equation; variational methods; degenerate problem; parabolic equation.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted May 13, 2025. Published June 10, 2025.

1



2 J. FERNANDES, L. MAIA EJDE-2025/60

The analysis of the asymptotic behavior of positive solutions for this class of degenerate logistic
equations was also done in [3]; see also references therein for some other related contributions.

Although the interest in non-negative solutions justifies itself from the biological motivation
given above, sign-changing solutions have also been investigated for a wide class of reaction-
diffusion equations. In that direction, some work has already been done for the analysis of solu-
tions. Under very general dissipative conditions on the nonlinearity, it was obtained in [25] the
existence of two extremal equilibria, which gives bounds for the asymptotic dynamics and there-
fore for the related global attractor. The result can also be applied to a large class of degenerate
logistic equations. Also stability of such solutions is addressed in [17]. Regarding the stationary
problem, some existence questions where treated in [5] for a more general problem, having (1.1)
as a particular case. The study of sign changing solutions in one-dimensional spatial variable
settings is a relatively recent development initiated by [19, 20, 21, 22, 23]. They characterized the
existence of such solutions in important cases, providing both analytical and numerical examples.
These examples illustrate that the set of solutions with one interior node can have either one or
two components.

Recent observations highlight that analyzing the structure of solutions even with just one inte-
rior node is much more complex than examining positive solutions. Numerical analysis and recent
multiplicity results [22, 7] suggest that the number of nodal solutions increases with the number
of wells in the function b(x). In a more recent paper [8], it is conducted a sharp numerical study
regarding the structure of the set of solutions with one interior node for specific classes of weight
functions.

The existence of a sign-changing solution for the stationary scalar problem of type (2.1), as-
sociated with (1.1), is closely related with the segregation phenomena of two populations with
diffusion and large interaction. This analysis has been performed in a series of papers by Dancer
and others [9, 10, 11] (see also references therein), for a class of systems of competing species

−∆u = au− u2 − cuv, in Ω,

−∆v = dv − v2 − euv, in Ω,

u|∂Ω = v|∂Ω0,
(1.2)

where a, c, d, and e are positive constants. A study of the asymptotic behaviour of the positive
solution of (1.2), when both parameters c and e go to infinity, shows that the pairs (u, v) of positive
solutions, u > 0 and v > 0 in Ω, approach in some sense a non-negative solution (u0, v0), with
disjoint supports, which in turn gives a nontrivial solution which changes sign w0 := w+

0 − w−
0 =

u0 − v0 of the elliptic equation −∆w0 = w+
0 (a− α−1w0)− w−

0 (d+ w0), with c
−1e → α ∈ (0,∞).

In this sense, a sign-changing solution of the latter might provide a solution pair of the system
(1.2), in a configuration of segregated populations. Likewise, a more general system with variable
parameters and general powers |u|ν−1u and |v|ν−1v rather than u2 and v2, multiplied by a variable
weight function b(x), with similar large coupling terms, models the steady state of two competing
species u and v co-existing in a region Ω. Henceforth, the sign-changing solution of the associated
scalar equation would be connected to the segregation of the species in a limiting scenario of highly
strong competition between the two populations.

On the analysis of the behavior of solutions to parabolic PDE’s, some alternative tools have
been used. Besides the usual comparison principle and sub/super solution, variational methods
have also been applied in the theory. More precisely, the Nehari manifold N was proved to be
important also for the study of the evolution dynamics, as it can be used as a borderline separating
regions of global existence and blow-up.

More recently, the authors proposed in [14] an analysis of the interplay of variational methods
and dynamical systems, using the Nehari manifold to give a rather complete picture of the abstract
phase space, for reaction-diffusion equations with asymptotically linear growth. The study was
inspired by the ideas developed in [15] and [13], for semilinear heat equations with the presence
of finite time blow-up.

The related semilinear stationary elliptic equation has been extensively addressed using different
approaches. General questions on existence, multiplicity, and nonexistence of positive solutions



EJDE-2025/6- DEGENERATE LOGISTIC PARABOLIC EQUATIONS 3

have been discussed. The main tools that have been applied are bifurcation, sub and super-
solutions or minimization and linking methods [1, 12, 24]. The existence problem for sign-changing
solutions is a bit more delicate with just a few results in this setting to be found in the literature
[24, 12]. We tackle the existence problem of such solutions using Ambrosetti and Rabinowitz [2]
Mountain Pass Theorem on N .

The article is organized as follows. In Section 2 we present the stationary problem, and the
geometric features of the associated Nehari manifold. In Section 3, we prove existence, nonexis-
tence of a positive solution under this new approach and obtain a sign-changing solution. Finally,
Section 4 is devoted to the parabolic dynamics. We get uniform boundedness of trajectories, and
local behavior near the nodal solution.

2. Stationary problem

Firstly, we discuss existence and multiplicity of positive and nodal solutions for the stationary
problem, as it will help us to get a better idea of the global dynamic structure of solutions on the
phase space.

We consider the Hilbert space H1
0 (Ω) with its standard scalar product and norm

⟨u, v⟩ :=
∫
Ω

∇u · ∇v, ∥u∥t :=
(∫

Ω

|∇u|2
)1/2

,

and the associated stationary elliptic problem

−∆u = λu+ b(x)|u|ν−1u, x ∈ Ω,

u|∂Ω = 0, u ∈ H1
0 (Ω).

(2.1)

Recall that the eigenvalues for the negative Laplacian in a bounded domain, with Dirichlet
boundary condition, are given by 0 < λ1(Ω) < λ2(Ω) < · · · < λn(Ω) < · · · , λj(Ω) → ∞ as j →
∞, with eigenfunctions ϕj . Similarly, if Ω0 ̸= ∅, we denote by λ1(Ω0) the first eigenvalue of −∆
in the smooth subdomain Ω0 also with Dirichlet boundary condition.

Using the method of sub and super solution, together with bifurcation arguments, Ouyang in
[24] obtained a complete description on existence and non-existence of positive solutions of the
elliptic nonlinear problem. Notice that, since b(x) ≤ 0 and not identically zero, the condition∫
Ω
b(x)ϕν+1

1 dx < 0 assumed in [1] is automatically satisfied. We summarize in the next theorem
the classical results found in [24] and [1] about existence and uniqueness of positive solutions of the
elliptic problem. Using variational methods, we prove in Lemma 2.2 the nonexistence statement,
based in [1].

Theorem 2.1. Assume that b ≤ 0 ( ̸≡ 0) is a continuous function on Ω, with Ω0 = {x ∈ Ω :
b(x) = 0}, and

f(λ, x, u) := λu+ b(x)|u|ν−1u

Then it holds that,

(i) If Ω0 = ∅, then for every λ > λ1(Ω) there exists a unique positive solution u of problem
(2.1).

(ii) If Ω0 ̸= ∅, then for every λ1(Ω) < λ < λ1(Ω0) there exists a unique positive solution u of
problem (2.1), and for every λ ≥ λ1(Ω0) or λ ≤ λ1(Ω), problem (2.1) admits no positive
solution.

Lemma 2.2. Problem (2.1) does not admit a positive solution for any λ ≥ λ1(Ω0).

The proof is standard by now and is found, for instance, in [1, Lemma 2.2]
To apply some variational arguments, we consider the functional associated with the equation

in (2.1), I : H1
0 (Ω) → R, given by

I(u) =
1

2
∥u∥2 − 1

2

∫
Ω

λu2 − 1

ν + 1

∫
Ω

b(x)|u|ν+1, (2.2)



4 J. FERNANDES, L. MAIA EJDE-2025/60

which is of class C2, with derivative

I ′(u)v =

∫
Ω

∇u · ∇v −
∫
Ω

λuv −
∫
Ω

b(x)|u|ν−1uv, u, v ∈ H1
0 (Ω). (2.3)

The functional J : H1
0 (Ω) → R given by

J(u) := I ′(u)u =

∫
Ω

|∇u|2 −
∫
Ω

λu2 −
∫
Ω

b(x)|u|ν+1 (2.4)

is of class C1 and defines the so-called Nehari manifold

N := {u ∈ H1
0 (Ω) : J(u) = 0}.

We also consider the complementary sets

N+ := {u ∈ H1
0 (Ω) : u ̸= 0, J(u) > 0}, (2.5)

N− := {u ∈ H1
0 (Ω) : u ̸= 0, J(u) < 0}. (2.6)

So if u ∈ N , substituting J(u) = 0 in the functional I, gives

I(u)

∫
Ω

(|∇u|2 − λu2) dx =
(1
2
− 1

ν + 1

)∫
Ω

b(x)|u|ν+1 dx. (2.7)

The points in the Nehari manifold N correspond to critical points of the maps

ϕu : t 7→ I(tu) :=
t2

2

∫
Ω

|∇u|2 −
∫
Ω

F (λ, x, tu) , (2.8)

where F (λ, x, u) =
∫ u

0
f(λ, x, s)ds, and so it is natural to divide N into three subsets corresponding

to local minima, local maxima and points of inflexion of fibrering maps ϕu. Notice that

ϕ′′u(t) =

∫
Ω

(|∇u|2 − fu(λ, x, tu)u
2) dx.

It is then naturally defined the following sets,

S+ = {u ∈ N :

∫
Ω

(|∇u|2 − fu(λ, x, u)u
2) dx > 0},

S− = {u ∈ N :

∫
Ω

(|∇u|2 − fu(λ, x, u)u
2) dx < 0},

S0 = {u ∈ N :

∫
Ω

(|∇u|2 − fu(λ, x, u)u
2) dx = 0}.

Moreover, it follows that they can be rewritten as

S+ = {u ∈ N : (1− ν)

∫
Ω

b(x)|u|ν+1 dx > 0}

= {u ∈ N :

∫
Ω

b(x)|u|ν+1 dx < 0}.
(2.9)

Similarly, S− = {u ∈ N :
∫
Ω
b(x)|u|ν+1 dx > 0} and S0 = {u ∈ N :

∫
Ω
b(x)|u|ν+1 dx = 0}.

Remark 2.3. If u ∈ N , from (2.7) and (2.9) it holds that u ∈ S+ if and only if I(u) < 0. One
similarly gets that u ∈ S− and u ∈ S0, if and only if, I(u) > 0 and I(u) = 0, respectively.

In what follows, based in [4], we need to consider the subsets

L+ := {u ∈ H1
0 (Ω) : ∥u∥ = 1,

∫
Ω

(|∇u|2 − λu2) dx > 0}

and similarly L− and L0, replacing > by < and =, respectively. We also define

B+ := {u ∈ H1
0 (Ω) : ∥u∥ = 1,

∫
Ω

b(x)|u|ν+1 dx > 0}

and B− and B0 analogously. The next proposition explores the role played by b(x) ≤ 0 on this
setting.



EJDE-2025/6- DEGENERATE LOGISTIC PARABOLIC EQUATIONS 5

Proposition 2.4. (i) If λ1(Ω) < λ < λ1(Ω0), then L− ∩B0 = ∅, and
(ii) if λ1(Ω0) < λ, then L− ∩B0 ̸= ∅.

Proof. Suppose w ∈ L− ∩B0, so by definition w ̸= 0 and

0 =

∫
Ω

−b(x)|w|ν+1 ≥
∫
Ω0

−b(x)|w|ν+1. (2.10)

We claim that the support of w is contained in the closure of Ω0. Indeed, without lost of generality
assume w is continuous, and there is x0 ∈ Ω\Ω0 such that |w(x0)| = δ > 0. Then, there is a small
ball Bε(x0) ⊂ Ω \ Ω0 inside which |w(x)| ≥ δ/2 and thus, by (2.10)

0 =

∫
Ω

−b(x)|w|ν+1 ≥
∫
Bε(x0)

−b(x)|w|ν+1 ≥
∫
Bε(x0)

−b(x)|δ
2
|ν+1 > 0,

which is impossible. So supp{w} ⊂ Ω0. But then, if λ < λ1(Ω0),

0 <

∫
Ω0

(λ1(Ω0)− λ)w2 ≤
∫
Ω0

|∇w|2 − λw2 =

∫
Ω

|∇w|2 − λw2 ≤ 0,

since w ∈ L−, which gives a contradiction, and hence L− ∩B0 = ∅.
In case λ1(Ω0) < λ, let ϕ01 be the positive (normalized) eigenfunction associated with the first

eigenvalue λ1(Ω0). Then, the support of ϕ01 is equal to Ω0 and
∫
Ω
b(x)|ϕ01|ν+1 =

∫
Ω0
b(x)|ϕ01|ν+1 =

0, and so ϕ01 ∈ B0. Moreover, ϕ01 ∈ L− since∫
Ω

|∇ϕ01|2 − λ(ϕ01)
2 = (λ1(Ω0)− λ)∥ϕ01∥L2(Ω0) < 0.

□

We present the next result found in [4, Theorem 4.2], with minor adaptations. We include a
proof for the sake of completeness.

Theorem 2.5. Suppose L− ∩B0 = ∅. Then

(i) S0 = {0};
(ii) S− = ∅;
(iii) S+ is bounded;

Proof. (i) Suppose u0 ∈ S0 \ {0}. Then u0

∥u0∥ ∈ L0 ∩ B0 ⊂ L− ∩ B0 = ∅, which is impossible.

Hence S0 = {0}.
(ii) Since b(x) ≤ 0, then by (2.9) S− = ∅.
(iii) Suppose S+ is unbounded. Then, there exists a sequence {un} ∈ S+ such that∫

Ω

(|∇un|2 − λu2n) dx =

∫
Ω

b(x)|un|ν+1 dx < 0 (2.11)

and ∥un∥ → ∞. Take vn := un

∥un∥ , then vn ⇀ v in H1
0 (Ω) and, by Sobolev compact embedding,

vn → v in Lq(Ω), for 2 ≤ q < 2∗, and vn(x) → v(x) a.e. in Ω. Dividing (2.11) by ∥un∥2 yields∫
Ω

|∇vn|2 − λv2n dx =

∫
Ω

b(x)|vn|ν+1∥un∥ν−1 dx. (2.12)

Since the left hand side is uniformly boundedand ∥un∥ → ∞, we have limn→∞
∫
Ω
b(x)|vn|ν+1 dx =

0, and hence
∫
Ω
b(x)|v|ν+1 dx = 0. Now we claim that vn → v strongly in H1

0 (Ω). Indeed, if vn
does not converge strongly to v,∫

Ω

|∇v|2 − λv2 < lim inf
n→∞

∫
Ω

|∇vn|2 − λv2n ≤ 0. (2.13)

Hence v ̸= 0 and v
∥v∥ ∈ L−. We conclude that v

∥v∥ ∈ L− ∩ B0 ⊂ L− ∩ B0, contradicting the

assumption. So vn → v, ∥v∥ = 1, and v ∈ B0. Moreover, by (2.12)∫
Ω

|∇v|2 − λv2 = lim inf
n→∞

∫
Ω

|∇vn|2 − λv2n ≤ 0,



6 J. FERNANDES, L. MAIA EJDE-2025/60

which implies v ∈ L−. Thus v ∈ L− ∩B0, which is an absurd. This completes the proof. □

The delicate study of the different subsets in the complement of N is fundamental to our
developments later in the parabolic setting. We follow the literature and denote, for k ∈ R,
Ik(u) = {u ∈ H1

0 (Ω) : I(u) < k}.

Lemma 2.6. Suppose L− ∩ B0 = ∅. Then for any k > 0, it holds that Ik ∩ N+ is bounded in
H1

0 (Ω).

Proof. Case 1. Let u ∈ N+ and suppose there exists tu > 0 such that tuu ∈ S+, which means∫
Ω
b|tuu|ν+1 < 0. Then

∫
Ω
b|u|ν+1 < 0 , i. e. u

∥u∥ ∈ B−. Moreover, since J(u) > 0, then tu < 1,

because I(tu) is decreasing in the variable t up to t = tu.
Since S+ is bounded, by Theorem 2.5(iii), there exists Mλ > 0 such that ∥tuu∥ < Mλ. If we

show that there is T > 0, uniform in u in this case, such that T < tu < 1, then ∥u∥ < Mλ/T
and we conclude the proof. If not, there is a sequence (un) ⊂ N+, such that tnun ∈ S+ and
tn → 0. Since ∥tnun∥ < Mλ, it follows that ∥un∥ may go to infinity. Assume, by contradiction,
that this is the case so that there exists a sequence (un) ⊂ Ik ∩ N+, for which there exist tn < 1
satisfying tnun ∈ S+, and such that ∥un∥ → +∞, and take vn := un

∥un∥ . Then vn ⇀ v in H1
0 (Ω)

and, by Sobolev compact embedding, vn → v in Lq(Ω), for 2 ≤ q < 2∗, and vn(x) → v(x) a.e. in
Ω. Moreover, since J(tun

un) = 0,

k > I(un) = I(un)−
1

(ν + 1) tν+1
un

J(tun
un)

=
1

2

∫
Ω

|∇un|2 − λu2n − 1

ν + 1

∫
Ω

b(x)|un|ν+1 −
( 1

(ν + 1) tν+1
un

t2un

∫
Ω

|∇un|2 − λu2n

− 1

(ν + 1) tν+1
un

tν+1
un

∫
Ω

b(x)|un|ν+1
)

= (
1

2
− 1

(ν + 1) tν−1
un

)

∫
Ω

|∇un|2 − λu2n.

(2.14)

By (2.14) and −1/tν−1
n → −∞, for n sufficiently large we have∫

Ω

|∇un|2 − λu2n = tν−1
n

∫
Ω

buν+1
n < 0,

which yields vn ∈ L− ∩B−. Then, dividing (2.14) by ∥un∥2, we obtain

k

∥un∥2
>

(1
2
− 1

(ν + 1) tν−1
n

)∫
Ω

|∇vn|2 − λv2n > 0.

Now, if vn does not converge to v,∫
Ω

|∇v|2 − λv2 < lim inf
n→∞

∫
Ω

|∇vn|2 − λv2n = 0.

Similarly,

k > I(un) = I(un)−
1

2 t2un

J(tun
un)

=
1

2

∫
Ω

|∇un|2 − λu2n − 1

ν + 1

∫
Ω

b(x)|un|ν+1

−
( 1

2 t2un

t2un

∫
Ω

|∇un|2 − λu2n − 1

2 t2un

tν+1
u

∫
Ω

b(x)|un|ν+1
)

=
( tν−1

un

2
− 1

ν + 1

)∫
Ω

b(x)|un|ν+1.

(2.15)

Moreover, since tn → 0 and
∫
Ω
b(x)|un|ν+1 < 0, for n sufficiently large, by (2.15)

k

∥un∥ν+1
> (

tν−1
n

2
− 1

ν + 1
)

∫
Ω

b(x)|vn|ν+1 > 0,



EJDE-2025/6- DEGENERATE LOGISTIC PARABOLIC EQUATIONS 7

so

lim
n→∞

∫
Ω

b(x)|vn|ν+1 =

∫
Ω

b(x)|v|ν+1 = 0.

Hence v ̸= 0 and v
∥v∥ ∈ L− ∩ B0, which contradicts the hypothesis. Consequently, vn → v and

∥v∥ = 1. Moreover ∫
Ω

|∇v|2 − λv2 = 0 =

∫
Ω

b(x)|v|ν+1,

so v
∥v∥ ∈ L0 ∩B0 ⊂ L− ∩B0, again a contradiction since L− ∩B0 = ∅. Hence 0 < T < tn and the

boundedness of u in N+ is true in this case.

Case 2. Let u ∈ N+ ∩ Ik and u non-projectable in S+, which implies u ∈ L+ ∩B−. Then

k > I(u) > I(u)− 1

(ν + 1)
J(u) > (

1

2
− 1

(ν + 1)
)

∫
Ω

|∇u|2 − λu2 > 0. (2.16)

Suppose, by contradiction, that in this case there exists a sequence (un) ⊂ Ik ∩ N+, such that
∥un∥ → +∞, and take vn := un

∥un∥ . Then vn ⇀ v in H1
0 (Ω) and, by Sobolev compact embedding,

vn → v in Lq(Ω), for 2 ≤ q < 2∗, and vn(x) → v(x) a.e. in Ω. Then, dividing (2.16) by ∥un∥2, we
obtain

k

∥un∥2
> (

1

2
− 1

(ν + 1)
)

∫
Ω

|∇vn|2 − λv2n > 0

and taking the limit as n→ ∞, if vn does not converge to v,∫
Ω

|∇v|2 − λv2 < lim inf
n→∞

∫
Ω

|∇vn|2 − λv2n = 0. (2.17)

Hence v ̸= 0 and v
∥v∥ ∈ L−, and since u/∥u∥ ∈ B−,

∫
Ω
b(x)|v|ν+1 ≤ 0. On the other hand,

k

∥un∥2
>

1

∥un∥2
I(un) =

1

2

∫
Ω

|∇vn|2 − λv2n − 1

(ν + 1)∥un∥2

∫
Ω

b(x)|un|ν+1

≥ 1

2

∫
Ω

|∇vn|2 − λv2n − 1

(ν + 1)

∫
Ω

b(x)|vn|ν+1.

(2.18)

Taking the limit in (2.18), as n → ∞, and using (2.17) and the fact that vn → v in Lν+1(Ω), it
follows that 0 ≤ 1

(ν+1)

∫
Ω
b(x)|v|ν+1. Therefore,

∫
Ω
b(x)|v|ν+1 = 0, and then v

∥v∥ ∈ L− ∩ B0 ⊂
L− ∩ B0, giving again a contradiction since L− ∩ B0 = ∅. So, vn → v strongly, ∥v∥ = 1, and it
holds ∫

Ω

|∇v|2 − λv2 =

∫
Ω

b(x)|v|ν+1 = 0,

which means v
∥v∥ ∈ L0 ∩B0 ⊂ L− ∩B0, again a contradiction since L− ∩B0 = ∅. This completes

the proof. □

Lemma 2.7. Suppose L− ∩ B0 = ∅. If u ∈ N− then u is projectable on S+, i.e. there exists tu
such that tuu ∈ S+. Moreover, I(u) < 0 and the set N− is bounded in H1

0 (Ω).

Proof. Since u ∈ N−, we have

J(u) =

∫
Ω

|∇u|2 − λ|u|2 −
∫
Ω

b(x)|u|ν+1 < 0.

If supp{u} ⊂ Ω0, then
u

∥u∥ ∈ B0, and also
∫
Ω
|∇u|2 − λ|u|2 < 0, which implies u

∥u∥ ∈ L−. This is

a contradiction with the assumption L− ∩ B0 = ∅. Hence supp{u} ∩ (Ω \ Ω0) ̸= ∅. In this case,
for t > 0, we take J(tu) in (2.4), since the coefficients of t2 is negative and of tν+1 is positive,
there exists a unique tu > 1, such that tuu ∈ S+. By Theorem 2.5 (iii), S+ is bounded, so there is
C > 0 such that ∥tuu∥ < C. Therefore, ∥u∥ < C/tu < C. Moreover, using I(u) < I(u)− 1/2J(u),
then

I(u) =
1

2

∫
Ω

|∇u|2 − λu2 − 1

ν + 1

∫
Ω

b(x)|u|ν+1 <

∫
Ω

(
1

2
− 1

ν + 1
)b(x)|u|ν+1 < 0.

□



8 J. FERNANDES, L. MAIA EJDE-2025/60

Remark 2.8. It follows that, if L− ∩ B0 = ∅, then the set of functions (I0 ∩ N+) ∪ S+ ∪ N− is
bounded in H1

0 (Ω).

The next result provides a characterization for projectable functions on N \{0}, which coincides
with S+ if λ1(Ω) < λ < λ1(Ω0). In other words, since b(x) ≤ 0, we describe the only possible
geometry for having a turning point for the fibrering map.

Proposition 2.9. Let λ1(Ω) < λ < λ1(Ω0) and u0 ∈ H1
0 (Ω) \ {0}. There exists ᾱ > 0 such that

ᾱu0 ∈ S+ if and only if u0

∥u0∥ ∈ L− ∩B−.

Proof. First let us prove that u0

∥u0∥ ∈ L− ∩B− is a sufficient condition. Since
∫
Ω
|∇u0|2 − λu20 and∫

Ω
b(x)|u0|ν+1 are both negative, the fibrering map ϕu0

has exactly one turning point at

t(u0) =
[∫

Ω
|∇u0|2 − λu20∫

Ω
b(x)|u0|ν+1

] 1
ν−1

,

that is t(u0)u0 ∈ S+.
Conversely,we suppose there exists ᾱ > 0 such that ᾱu0 ∈ S+. Then ᾱu0

∥ᾱu0∥ ∈ L−, and hence
u0

∥u0∥ ∈ L−. Since L− ∩ B0 = ∅, by Proposition 2.4, and B+ = ∅, then u0

∥u0∥ ∈ L− ∩ B−. So, the

proof of this lemma is complete. □

3. Existence and nonexistence results

The next theorem improves the classical results in [24, 1, 4] on the existence of a positive
solution of (2.1), for a better interval of values of the parameter λ. By using projections on Nehari,
combined with Lemma 2.2, it provides a sharp upper bound for the range of λ. Alternatively, this
result was proved in [3, Theorem 4.2] for subsets Ω0 ⊂ Ω that may not be smooth, using a different
method which is more involved due to the generality of their setting.

We define d := infu∈S+ I(u), with −∞ ≤ d. By Remark 2.3, it holds d < 0 if S+ is not empty.

Theorem 3.1. If λ1(Ω) < λ < λ1(Ω0), then I is bounded from below in H1
0 (Ω), and there exists

a minimizer φ > 0 such that I(φ) = d.

Proof. Take any u ̸= 0 in H1
0 (Ω). If there exists t > 0 such that tu ∈ N , then tu ∈ S+ by

Theorem 2.5, and hence I(u) ≥ I(tu) ≥ d. On the other hand, if there exists no such t, we claim

that I(tu) ≥ 0 for all t > 0. Indeed, by Proposition 2.9, u
∥u∥ ∈ L+ ∪B0. Therefore, there are two

cases. If u
∥u∥ ∈ L+, then I(tu) ≥ 0 for all t > 0. If u

∥u∥ ∈ B0, then u
∥u∥ ∈ L+, since L− ∩ B0 = ∅.

In this case I(tu) ≥ 0 for all t > 0. We conclude that I(u) ≥ d, for any u ∈ H1
0 (Ω). Moreover,

by [4, Theorem 4.4 ], since ϕ1 ∈ L−, B+ = ∅ and L− ∩ B0 = ∅, it follows that there exists a
minimizer φ of I(u) on S+ which is also a minimizer in the whole space, because N is a natural
constraint. W.l.o.g. φ ≥ 0, since I(φ) = I(|φ|), then |φ| is an interior minimum and so also a
solution. Suppose φ(x0) = 0 for some x0 ∈ Ω. By the Hopf Lemma this is impossible, hence
φ > 0, and by the uniqueness of the positive solution it is the Ouyang solution. □

Remark 3.2. Observe that since f(λ, x, u) = λu + b(x)|u|ν−1u ≤ C(1 + |u|ν), for ν + 1 ≤ 2∗,
then we can apply the essential Brezis-Kato Lemma and obtain that a weak solution u of (2.1) is

in C1,α
loc (Ω), for any α < 1. If ∂Ω ∈ C2, then u ∈ C1,α(Ω), and additionally, if b ∈ C0,α(Ω), then

u ∈ C2,α(Ω) ∩ C(Ω) is a classical solution of problem (2.1).

The next result relies on Remark 3.2 and can be found in [17, Lemma 5.2], and suits our settings.

Lemma 3.3. Let λ1(Ω) < λ < λ1(Ω0). The unique positive stationary solution φ is isolated from
other stationary solutions with respect to the H1

0 (Ω) topology. Similarly for the negative solution
−φ.

Regarding the trivial solution, it was proved in [25, Theorems 4.2 and 4.5] that it is an isolated
equilibrium point and it is known to be unstable in the subset of nonnegative initial data for
λ1(Ω) < λ, see for instance [3].



EJDE-2025/6- DEGENERATE LOGISTIC PARABOLIC EQUATIONS 9

To obtain another solution, we employ the Mountain Pass Theorem of Ambrosetti and Rabi-
nowitz [2]. Recall that a sequence (un) in H

1
0 (Ω) is said to be Palais Smale at c for I, and denoted

by (PS)c, if I(un) → c and I ′(un) → 0.

Theorem 3.4. Let Ĩ(u) := I(u)− d. For φ > 0 and −φ < 0 local minima on S+, it holds that

(i) Ĩ(φ) = 0;

(ii) there exists ρ and δ > 0 such that Ĩ(u) ≥ δ > 0, for any u ∈ Bρ(φ) ∩N ;

(iii) Ĩ(−φ) = 0 and ρ < ∥φ− (−φ)∥ = 2∥φ∥;
that is, Ĩ satisfies the geometrical hypotheses of the Mountain Pass Theorem on N . Moreover Ĩ
satisfies (PS)c condition at

c = inf
γ∈Γ

max
0≤t≤1

I(γ(t)),

where Γ = {γ : [0, 1] → N : γ(0) = φ, γ(1) = −φ}, and so there exists a nontrivial solution u∗ of
(2.1)satisfying I(u∗) = c > d.

To prove Theorem 3.4, we need the following lemma.

Lemma 3.5. Every (PS)c-sequence (uk) for I on N , with c ̸= 0, contains a subsequence which
is a (PS)c-sequence for I.

Proof. We evoke the proof of [6, Lemma 2.5]. For the functional J we have |J ′(uk)uk| ≤
∥∇J(uk)∥∥uk∥. We claim that |J ′(uk)uk| → ρ ≥ 0 and additionally that ρ > 0. Indeed, since
J(v) = 0 for v ∈ N , then

J ′(v) · v = 2

∫
(|∇v|2 − λv2)dx− (ν + 1)

∫
b(x)|v|ν+1

= [2− (ν + 1)]

∫
b(x)|v|ν+1 ≥ 0.

Also, for ∥uk∥ ≤M with uk ⇀ u and uk ∈ S+, we have uk → u in Lν+1 and uk(x) → u(x).
Therefore,

lim
k→∞

|J ′(uk)uk| = |2− (ν + 1)| lim
k→∞

∣∣ ∫ b(x)|uk|ν+1dx
∣∣

= (ν − 1)
∣∣ ∫ b(x)|u|ν+1

∣∣ = ρ

We then divide into two cases. First, suppose u ≡ 0. From (2.7) it holds that I(uk) → 0, which
is also not possible, since I(uk) → c ̸= 0.

For the second case u ̸≡ 0, the convergences uk ⇀ u and uk → u in Lν+1, and if ρ = 0, would
lead to u

∥u∥ ∈ L− ∩B0 = ∅. The contradiction on both cases u ≡ 0 and u ̸≡ 0 gives us ρ > 0. □

Proof of Theorem 3.4. The Nehari manifold may be written as N = S0 ∪ S+ = {J−1(0)}, which
is closed in H1

0 (Ω). That might allow us to apply Ekeland Variational Principle on N , which
is a closed metric space. In fact, since I : N → R ∪ ∞ is continuous and bounded from below
by Theorem 3.1, 0 > I(u) ≥ d > −∞. As a consequence, item (ii) can be proved as follows.
We suppose by contradiction that for all fixed ρ with 0 < ρ < 2∥φ∥, there exists a sequence
(un) ⊂ N ∩ ∂Bρ(φ) such that I(un) → d. That is, (un) is a minimizing sequence, and then
from Ekeland Variational Principle we would get the existence of (vn) ∈ N with I(vn) → d,

∥vn − un∥ → 0 and I|′N (vn) → 0. Hence, by Lemma 3.5 and since I satisfies (PS)c (see [12,

Lemma 2.1]), vn → v, up to a subsequence. Then I(v) = d, Ĩ ′(v) = 0 and ∥v − φ∥ = ρ. Since ρ is
arbitrary, we can find critical points of I in any ball centered in φ, which contradicts Lemma 3.3.

Therefore, the Mountain Pass geometry on N is verified, and knowing that the functional I
satisfies (PS)c, there exists a critical point u∗ of the functional I constrained to N , and d < c ≤ 0.
Recalling that N is a natural constraint, then u∗ is a solution. □

Note that the solution just found may be the trivial one. In this case, knowing it is a Mountain
Pass solution constrained to N , with Morse index at least one, we would be able to conclude that
zero has at least one unstable direction in the parabolic setting.



10 J. FERNANDES, L. MAIA EJDE-2025/60

Next, we give a sufficient condition for such min-max solution u∗ not to be trivial.

Theorem 3.6. Assume the hypotheses of Theorem 3.4, and λ1(Ω) < λ2(Ω) < λ < λ1(Ω0), then
d < I(u∗) < 0 and u∗ is a sign-changing solution of problem (2.1).

Proof. First we want to show that I(u∗) = c < 0, which gives u∗ ̸= 0. To do so, we consider
the positive (normalized in L2(Ω)) first eigenfunction of −∆ in Ω, denoted by ϕ1, associated
with the eigenvalue λ1(Ω), a normalized eigenfunction ϕ2, associated with the second eigenvalue
λ2(Ω), ϕ

0
1 the positive (normalized) eigenfunction associated with the first eigenvalue λ1(Ω0),

and a normalized eigenfunction ϕ02, associated with the second eigenvalue λ2(Ω0). Note that the
supports of ϕ0i , i = 1, 2, are subsets of Ω0. Moreover, ⟨ϕ1, ϕ2⟩ = 0, ⟨ϕ01, ϕ02⟩ = 0. To construct a
convenient path in Γ not passing through zero we define w = t1(ϕ1 + εϕ01) + t2(ϕ2 + εϕ02), with
constants t1, t2 > 0, and for some ε > 0 to be chosen sufficiently small. Using that b(x) ≤ 0, there
is a positive constant C such that

I(w) =
t21
2

∫
Ω

|∇ϕ1|2 − λϕ21 + ε2
t21
2

∫
Ω

|∇ϕ01|2 − λ(ϕ01)
2

+
t22
2

∫
Ω

|∇ϕ2|2 − λϕ22 + ε2
t22
2

∫
Ω

|∇ϕ02|2 − λ(ϕ02)
2

+
1

2

2∑
i=1

2∑
j=1

titj

{∫
Ω

∇(ϕi + εϕ0i )∇(εϕ0j )− λ(ϕi + εϕ0i )(εϕ
0
j )
}

− 1

ν + 1

∫
Ω

b(x)|w|ν+1

≤ t21
2
(λ1(Ω)− λ) +

t22
2
(λ2(Ω)− λ) + ε2

t21
2
(λ01(Ω)− λ)

+ ε2
t22
2
(λ02(Ω)− λ) + ϵ

2∑
i=1

2∑
j=1

titj
2

{∫
Ω

∇ϕi∇ϕ0j − λϕiϕ
0
j

}
+ C

{
ε∥w∥2 + ∥w∥ν+1

}
.

Since ∥w∥ ≤
√
t21 + t22 + Cε2, taking t1, t2 > 0 and ε sufficiently small, recalling ν + 1 > 2, and

using the hypothesis λ1(Ω) < λ2(Ω) < λ, we obtain

I(w) ≤ (t21 + t22)

2
max{λ1(Ω)− λ, λ2(Ω)− λ}+O(ε(t21 + t22)) ≤ −δ1 < 0,

for some constant δ1 > 0.
Now, let w1 := t1(ϕ1 + εϕ01) and w2 := t2(ϕ2 + εϕ02), and wθ := cos(θ)w1 + sin(θ)w2, so that

wπ/4 =
√
2
2 w and ∥wπ/4∥ =

√
2
2 ∥w∥ and for some constant δ2 > 0 and for all θ ∈ [0, π],

I(wθ) ≤
t21
2
cos2(θ)(λ1(Ω)− λ) +

t22
2
sin2(θ)(λ2(Ω)− λ)

+ ε2
t21
2
cos2(θ)(λ01(Ω)− λ) + ε2

t21
2
sin2(θ)(λ02(Ω)− λ)

+ C
{
ε∥wθ∥2 + ∥wθ∥ν+1

}
≤ −δ2 < 0.

(3.1)

Finally, define the path in H1
0 (Ω) by

γ(s) :=


[(1− 3s)φ+ 3s(w1)], s ∈ [0, 1/3]

wθ(s), s ∈ [1/3, 2/3] and θ(s) = 3(s− 1/3)π,

[3(1− s)(−w1) + 3(s− 2/3)(−φ)], s ∈ [2/3, 1],

(3.2)

which can be projected on N by the multiplication τ(s)γ(s), with

τ(s) =
[∫

Ω
|∇γ(s)|2 − λ(γ(s))2∫
Ω
b(x)|γ(s)|ν+1

]
.



EJDE-2025/6- DEGENERATE LOGISTIC PARABOLIC EQUATIONS 11

Indeed, using that φ ∈ S+, φ is a solution of (2.1), and the definition of w1 which involves the
eigenfunction ϕ1, simple calculations for s ∈ [0, 1/3] yield∫

Ω

|∇γ(s)|2 − λ(γ(s))2 = (1− 3s)2
∫
Ω

|∇φ|2 − λφ2 + (3s)2
∫
Ω

|∇w1|2 − λw2
1

+ 2(1− 3s)(3s)

∫
Ω

∇φ∇w1 − λφw1 < 0.

Also, since supp{φ} ∩ (Ω \ Ω0) ̸= ∅, and supp{w1} ∩ (Ω \ Ω0) ̸= ∅, it follows that∫
Ω

b(x)|γ(s)|ν+1 =

∫
Ω\Ω0

b(x)|γ(s)|ν+1 < 0,

for all s ∈ [0, 1/3]. Henceforth, γ(s) ∈ S+, for all s ∈ [0, 1/3], yielding max0≤s≤1/3 I(γ(s)) < 0.
Analogously for s ∈ [2/3, 1].

The second segment of the path γ, for each s ∈ [1/3, 2/3], by (3.1) also satisfies both∫
Ω

|∇γ(s)|2 − λ(γ(s))2 =

∫
Ω

|∇wθ|2 − λw2
θ ≤ −δ2 < 0,∫

Ω

b(x)|γ(s)|ν+1 =

∫
Ω\Ω0

b(x)|γ(s)|ν+1 < 0.

The second inequality uses the fact that λ2(Ω) < λ, which implies that supp{w2} ∩ (Ω \ Ω0) ̸= ∅.
This shows that on the continuous path τ(s)γ ∈ S+ it holds by Remark 2.3 that there is a negative
upper bound I(τ(s)γ(s)) ≤ max0≤t≤1 I(τ(s)γ(s)) < 0, for all s ∈ [0, 1]. By the definition of the
min-max level c, it follows that I(u∗) = c < 0. To conclude, suppose by contradiction that u∗ is

w.l.o.g. non-negative and nontrivial. If the subset Ω̃ ⊂ Ω, on which u∗ = 0 is non-empty, then
its boundary points ∂Ω̃ ⊂ Ω. Let x0 be a point in ∂Ω̃ ⊂ Ω, u∗(x0) = 0. Moreover, u∗ ∈ C1(Ω)

(see Remark 3.2), hence ∂Ω̃ is smooth enough, and compact. Because of the higher power of the

nonlinear term in f(x, u), it holds that near the points of ∂Ω̃ we have −∆u = λu+o(|u|) > 0. Hopf
Lemma gives that Du∗(x0) ̸= 0, which is impossible in an interior minimum point. Therefore,
u∗ > 0, which is impossible by the uniqueness of the positive solution. This leads to the conclusion
that u∗ changes sign. □

4. Parabolic problem

The local existence in time for equation (1) follows directly from the fact that f(λ, x, u) is
locally Lipschitz in u, see [16]. Then we have a locally defined semigroup u(t) := S(t, u0), for
0 ≤ t < Tu0

and Tu0
being the maximum time of existence.

In addition, note that for u0 ∈ H1
0 (Ω), if we differentiate the map t 7→ I(u(t)) with respect

to t, we obtain d
dtI(u(t)) = −

∫
Ω
u2t (t) for all t > 0, which implies that I is decreasing along

non-stationary solutions. In this case, I is referred to as Lyapunov functional and the dynamical
system generated by the semigroup is said to have a gradient structure.

To analyze the parabolic problem, we begin by proving global existence in time t.

Theorem 4.1. Let λ1 < λ < λ1(Ω0). Then the solutions of (1) exist for all forward time.
Additionally, no solution may blow-up in infinite-time (i.e. grow-up).

Proof. For any u0 ∈ H1
0 (Ω) we claim that the corresponding solution S(t, u0) is uniformly bounded

in time. Indeed, suppose there exists u0 such that S(t, u0) blows-up in finite or infinite-time. Since
S+ and N− are bounded sets in H1

0 (Ω), it should exist t̄ ≥ 0 such that S(t, u0) ∈ N+ for all t > t̄.
We also obtain, from the gradient structure of the system that S(t, u0) ∈ Ik for all t > t̄. with
k = I(S(t̄, u0)). By applying Lemma 2.6, we conclude that {S(t, u0) : t ≥ t̄}is contained in a
bounded subset of H1

0 (Ω), which gives a contradiction. We conclude that any solution of (1)
remains uniformly bounded in time and, in particular, it is defined for all t ≥ 0 (see [16, Theorem
3.3.4]). □



12 J. FERNANDES, L. MAIA EJDE-2025/60

It is known that all nonnegative solutions of (1.1) converge to the unique positive equilibrium φ,
see [3]. In what follows we address the local evolutionary dynamics close to a stationary Mountain
Pass solution, inspired by the ideas in [14].

Let ϕ be a nontrivial stationary solution of (1.1). Then the linearized operator at ϕ Lu = −∆u−
fu(λ, x, ϕ)u is self-adjoint in L2(Ω) with domain H1

0 (Ω)∩H2(Ω) and spectrum entirely composed

of eigenvalues. We denote by {µϕ
i }∞i=1 the nondecreasing sequence of eigenvalues of L, repeated

according to their (finite) multiplicities, and let {ψϕ
i }∞i=1 the corresponding eigenfunctions.

We know that µϕ
i → +∞ as i → ∞ and, if we take ϕ = u∗, then µu∗

1 < 0, by Theorem 3.4.
Thus we may define

q := max{i ∈ N : µu∗

i ≤ 0}. (4.1)

Since {ψu∗

i }∞i=1 form a Hilbert basis of L2(Ω), we can decompose any v ∈ H1
0 (Ω),

v =

q∑
i=1

aiψ
u∗

i +

∞∑
i=q+1

aiψ
u∗

i .

Theorem 4.2. Let u∗ be Mountain Pass solution obtained in Theorem 3.4. Then there exist
initial data u0, v0 ∈ H1

0 (Ω) with u0 ∈ N+ and v0 ∈ N− with both converging to u∗ as time goes to
infinity, i.e. u0, v0 ∈W s(u∗).

To prove Theorem 4.2, we need the following lemma. For simplicity we denote µi = µu∗

i and

ψi = ψu∗

i .

Lemma 4.3. For u∗ there exists i > q, where q is given in (4.1), such that

ai :=

∫
Ω

ϕψi ̸= 0. (4.2)

Proof. We know that u∗ is a Mountain Pass critical point constrained to N , by Theorem 3.4,
hence its Morse index is at least equal to 1 and, by definition of q, µi > 0 for i > q.

Suppose by contradiction that ai = 0 for all i > q. Then u∗ may be written as u∗ =
∑q

i=1 aiψi.
Then, since

L(u∗) = L
( q∑

i=1

aiψi

)
=

q∑
i=1

aiL(ψi) =

q∑
i=1

aiµiψi,

we obtain

⟨L(u∗), u∗⟩ =
〈 q∑

i=1

aiµiψi,

q∑
i=1

aiψi

〉
=

q∑
i=1

a2iµi ≤ 0.

On the other hand ⟨L(u∗), u∗⟩ = I ′′(u∗)u∗2 > 0, since u∗ ∈ S+. □

Proof of Theorem 4.2. We denote by X1 the finite dimensional subspace of H1
0 (Ω) spanned by

{ψi : 1 ≤ i ≤ q} and by X2 the infinite dimensional subspace of H1
0 (Ω) spanned by {ψi : i > q}.

It is known that the local stable manifold of u∗, denoted by W s
loc(u

∗), is tangent to X2 at u∗. In
addition, there exists a neighborhood V of 0 in X2 and a C1 map h : V → X1 such that

W s
loc(u

∗) = {u∗ + η + h(η) : η ∈ V }. (4.3)

By Lemma 4.3, there exists at least one i > q such that ai ̸= 0. Notice that, for any ψ ∈ H1
0 , it

holds that

⟨Lu∗, ψ⟩L2(Ω) = −
∫
Ω

∆u∗ψ − fu(λ, x, u
∗)u∗ψ =

∫
Ω

∇u∗∇ψ − fu(λ, x, u
∗)u∗ψ

and

⟨J ′(u∗), ψ⟩L2(Ω) = 2

∫
Ω

∇u∗∇ψ −
∫
Ω

fu(λ, x, u
∗)u∗ψ −

∫
Ω

f(λ, x, u∗)ψ

=

∫
Ω

∇u∗∇ψ −
∫
Ω

fu(λ, x, u
∗)u∗ψ.



EJDE-2025/6- DEGENERATE LOGISTIC PARABOLIC EQUATIONS 13

Therefore, for ψ = ψi, the normalized eigenfunction with i > q,

⟨J ′(u∗), ψi⟩L2(Ω) = ⟨Lu∗, ψ⟩L2(Ω) = aiµi ̸= 0.

It follows from (4.3) that u0 := u∗ + ϵψi + h(ϵψi) ∈W s
loc(u

∗) if |ϵ| ≠ 0 is small enough. Therefore,
the corresponding solution u(x, t) converges to u∗ as t → ∞. Now we claim that J(u0) has
the same sign as ϵ, for small |ϵ|. Indeed, by Taylor’s expansion at u∗, since J(u∗) = 0 then
J(u0) = J(u∗) + J ′(u∗)(ϵψi) + Rϵ = ϵµiai + Rϵ. The statement follows since we can assume,
without loss of generality that ai > 0. □

Acknowledgements. J. Fernandes was supported by ARC/FAPERJ #010.002597/2019. L.
Maia was supported by FAPDF, CAPES, and CNPq grant 309866/2020-0. The authors want
to thank the referee for the careful reading of the manuscript and for several suggestions that
helped to improve the manuscript. Moreover, they want to express their deepest gratitude to
Filomena Pacella for her support.

References

[1] S. Alama, G. Tarantello; On semilinear elliptic equations with indefinite nonlinearities. Calc. Var. Partial
Differential Equations, 1 (1993), 439–475.

[2] A. Ambrosetti, P. H. Rabinowitz; Dual variational methods in critical point theory and applications. J. Funct.
Anal., 14 (1973), 349–381.

[3] J. M. Arrieta, R. Pardo, A. Rodŕıguez-Bernal; Asymptotic behavior of degenerate logistic equations. J. Diff.

Eq., 259 (2015), 6368–6398.
[4] K.J. Brown, Y. Zhang; The Nehari manifold for a semilinear elliptic equation with a sign-changing weight

function. J. Diff. Eq., 193 (2003), 481–499.

[5] M. Cardoso, F. Furtado, L. Maia; Positive and sign-changing stationary solutions of degenerate logistics type
equations. Nonlinear Anal. TMA, 245 (2024), 113575.

[6] M. Clapp, L. A. Maia; A positive bound state for an asymptotically linear or superlinear Schrödinger equation.

J. Diff. Eq., 260 (2016) 3173–3192.
[7] P. Cubillos, J. López-Gómez, A. Tellini; Multiplicity of nodal solutions in classical non-degenerate logistic

equations. Electron. Res. Arch., 30 (2022), 898–928.

[8] P. Cubillos, J. López-Gómez, A. Tellini; Multiplicity of nodal solutions in classical non-degenerate logistic
equations. Global structure of the set of 1-node solutions in a class of degenerate diffusive logistic equations.

Communications in Nonlinear Science and Numerical Simulation, 125 (2023), 1–24.

[9] E. N. Dancer; Competing species systems with diffusion and large interaction. Rend. Sem. Mat. Fis. Milano,
65 (1995) 23–33.

[10] E. N. Dancer, Y. H. Du; On sign-changing solutions of certain semilinear elliptic problems. Appl. Anal., 56
(1995) 193–206.

[11] E. N. Dancer, Y. H. Du; Competing species equations with diffusion, large interactions, and jumping nonlin-

earities. J. Diff. Eq., 114 (1994) 434–475.
[12] M. Del Pino, P. Felmer; Multiple solutions for a semilinear elliptic equation. Trans. Amer. Math. Soc., 347

(1995) 4839–4853.

[13] F. Dickstein, N. Mizoguchi, P. Souplet, F. Weissler; Transversality of stable and Nehari manifolds for a semi-
linear heat equation, Calc. Var. Partial Differential Equations, 42 (2011), 547–562.

[14] J. Fernandes, L. A. Maia; Blow-up and bounded solutions for a semilinear parabolic problem in a saturable

medium. Discrete Contin. Dyn. Syst. Ser. A, 41 (2021) 1297–1318.
[15] F. Gazzola, T. Weth; Finite time blow-up and global solutions for semilinear parabolic equations with initial

data at high energy level Diff. Int. Eq., 18, 9 (2005), 961–990.

[16] D. Henry; Geometric theory of semilinear parabolic equations. Lecture Notes in Mathematics, 840. Springer-
Verlag, Berlin-New York, 1981.

[17] R. Kajikiya; Stability and instability of stationary solutions for sublinear parabolic equations. J. Diff. Eq.,
264 (2018), 786–834.

[18] J. López-Gómez; Metasolutions: Malthus versus Verhulst in population dynamics. A dream of Volterra, in

Stationary Partial Differential Equations, vol II, in: Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam.
(2005), 211–309.

[19] J. López-Gómez, P. Rabinowitz. Nodal solutions for a class of degenerate boundary value problems. Advanced

Nonlinear Studies, 15 (2015), 253–288.
[20] J. López-Gómez, P. Rabinowitz. The effects of spatial heterogeneities on some multiplicity results. Discrete

Contin. Dyn. Syst. Ser. A, 36 (2016), 941–952.

[21] J. López-Gómez, P. Rabinowitz; Nodal solutions for a class of degenerate one-dimensional bvp’s. Top Methods
Nonlinear Anal., 49 (2017), 359–376.

[22] J. López-Gómez, P. Rabinowitz; The structure of the set of 1-node solutions of a class of degenerate bvp’s. J.

Diff. Eq., 268 (2020), 4691–4732.



14 J. FERNANDES, L. MAIA EJDE-2025/60

[23] J. López-Gómez, M. Molina-Meyer, P. Rabinowitz; Global bifurcation diagrams of one-node solutions on a

class of degenerate boundary value problems. Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 923–946.

[24] T. Ouyang. On the positive solutions of semilinear equations ∆u+λu−hup = 0 on compact manifolds. Trans.
Amer. Math. Soc., 331 (1992), 503–527.

[25] A. Rodŕıguez-Bernal, A. Vidal-López; Extremal equilibria for reaction-diffusion equations in bounded domains

and applications. J. Diff. Eq., 244 (2008), 2983–3030.

Juliana Fernandes
Universidade Federal do Rio de Janeiro, Instituto de Matemática, 21941-909 Rio de Janeiro - RJ, Brazil

Email address: jfernandes@im.ufrj.br

Liliane de A. Maia

Universidade de Brasilia, Departamento de Matemática, 70.910-900, Braśılia, DF, Brazil

Email address: lilimaia@unb.br


	1. Introduction
	2. Stationary problem
	3. Existence and nonexistence results
	4. Parabolic problem
	Acknowledgements

	References