Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 30, pp. 1–9.

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

POSITIVE SOLUTIONS TO A DIRICHLET PROBLEM WITH

NON-LIPSCHITZ NONLINEARITIES

GIOVANNI ANELLO

Abstract. Let Ω be a bounded smooth domain in RN . We study the exis-
tence of positive solutions to the Dirichlet problem

−∆u = (1 − u)us−1 − λur−1, in Ω,

u = 0, on ∂Ω,

where 1 < r < s ≤ 2, and λ > 0. In particular, we answer to some questions

posed in the recent paper [3] where this problem was considered.

1. Introduction

In the recent paper [3], the authors investigate the existence and multiplicity of
nonzero nonnegative solutions to the problem

−∆u = (1− u)us−1 − λur−1, in Ω,

u = 0, on ∂Ω,
(1.1)

where Ω is a bounded smooth domain in RN (N ≥ 1), 1 < s ≤ 2, r > 1, and λ
is a real parameter. The equation −∆u = (1 − u)us−1 − λur−1, with λ > 0, is
the stationary version of a reaction diffusion equation employed to model certain
isothermal autocatalytic chemical reactions (see [6]). In [3], various situations,
which correspond to different values of the parameters s, r, λ, are analyzed and, for
each of them, the authors prove some existence and multiplicity results of nonzero
nonnegative solutions. Moreover, the cases in which a nonzero nonnegative solution
is actually a positive solution (via the Strong Maximum Principle) are pointed out
and, for each of these cases, the uniqueness of positive solution is also investigated.

In particular, one can observe that the Strong Maximum Principle applies in the
following cases (see Lemma 2.2 of [3])

• λ ≤ 0;
• λ > 0, r ≥ 2;
• λ > 0, r > s;
• 0 < λ < 1, r = s.

It is worth noting to recall that positive solutions are often the only ones that have a
physical meaning. If the function t ∈ (0,+∞)→ (1− t)ts−1−λtr−1 is negative and
non-Lipschitz near 0 (for instance, when 1 < r < s ≤ 2), the standard conditions

2010 Mathematics Subject Classification. 35J20, 35J25.

Key words and phrases. Positive solution; non-Lipschitz nonlinearity; variational methods.
c©2021 Texas State University.

Submitted October 1, 2020. Published April 20, 2021.

1



2 G. ANELLO EJDE-2021/30

that allow to apply the Strong Maximum Principle are not satisfied any longer,
and in this case the detection of positive solutions to problem (1.1) becomes a more
delicate question.

In particular, the possible existence of positive solutions (as well as compact
support solutions) in the case λ > 0 and 1 < r < s ≤ 2 is an open question posed
in [3].

In this article, we will prove that a positive solution actually exists for λ > 0
small enough. Moreover, we will also show that the existence of a positive solution
entails the existence of a second nonnegative solution with positive energy. This
means that the technical condition involving the exponents r, s imposed in [3] to
guarantee the existence of this second solution can be removed.

In what follows, by a nonnegative solution of problem (1.1) we mean a nonneg-

ative function u ∈ L∞(Ω) ∩W 1,2
0 (Ω) such that∫

Ω

∇u∇vdx =

∫
Ω

[(1− u)us−1 − ur−1]vdx, for each v ∈W 1,2
0 (Ω).

By the standard regularity theory of elliptic equations, if Ω is of class C2, any
nonnegative solution u to problem (1.1) (in the sense given above) is classical (see [8,
Appendix B]). More precisely, one has u ∈ C1,α(Ω)∩C2(Ω), for some α ∈ (0, 1), and
u satisfies the equation and the boundary condition of (1.1) pointwise. Hereafter,
we will always assume that Ω is of class C2.

Throughout this article, we denote by(
‖ · ‖ :=

∫
Ω

|∇(·)|2dx
)1/2

the Poincaré norm of W 1,2
0 (Ω) and, for p ∈ [1,∞), we denote by

‖ · ‖p =
(∫

Ω

| · |pdx
) 1
p

the standard norm of Lp(Ω). Moreover, we put

cp = sup
u∈W 1,2

0 (Ω), ‖u‖=1

‖u‖p and λ1 = c−2
2

By the Sobolev embeddings, we know that cp < +∞, if p ≤ 2N
N−2 , when N ≥ 3.

The number λ1 is the first eigenvalue of the Laplacian in Ω and it is well known
that λ1 is simple.

We denote by φ1 ∈ C1(Ω) the unique (positive) eigenfunction associated with
λ1 and normalized with respect to sup-norm ‖u‖∞ := supΩ |u|. The function φ1

satisfies

−∆φ1 = λ1φ1, in Ω,

0 ≤ φ1 ≤ 1, in Ω,

φ1 = 0, on ∂Ω,

and, in particular,

‖φ1‖2 = λ1‖φ1‖22 (1.2)

Finally, given any u : Ω → R, we denote by u+, u− : Ω → R the functions defined
by

u+(x) = max{u(x), 0}, u−(x) = max{−u(x), 0}, all x ∈ Ω.



EJDE-2021/30 POSITIVE SOLUTIONS TO A DIRICHLET PROBLEM 3

2. Main result

In what follows, we assume that r, s ∈]1, 2] are real numbers such that 1 < r < s.
Moreover, for each λ ≥ 0, we consider the continuous function fλ : R→ R defined
by

fλ(t) =


0, if t ∈ (−∞, 0),

(1− t)ts−1 − λtr−1, if t ∈ [0, 1),

−λtr−1, if t ∈ [1,∞),

and the functional Iλ : W 1,2
0 (Ω)→ R defined by

Iλ(u) =
1

2
‖u‖2 −

∫
Ω

Fλ(u(x))dx, for each u ∈W 1,2
0 (Ω),

where

Fλ(ξ) =

∫ ξ

0

fλ(t)dt, for each ξ ∈ R.

A routine argument shows that Iλ is sequentially weakly lower semicontinuous and
of class C1 in W 1,2

0 (Ω), with

I ′λ(u)(v) =

∫
Ω

∇u∇vdx−
∫

Ω

fλ(u)vdx, for each u, v ∈W 1,2
0 (Ω).

A further property of Iλ is given by the next lemma.

Lemma 2.1. Let u ∈W 1,2
0 (Ω). Then

(1) u is a nonnegative solution of (1.1) if and only if I ′λ(u) = 0;
(2) I ′λ(u) = 0 implies 0 ≤ u ≤ 1 in Ω.

Proof. The proof is standard. We give it for completeness. Let u ∈ W 1,2
0 (Ω) be a

nonnegative solution to (1.1). Then∫
Ω

∇u∇vdx =

∫
Ω

[(1− u)us−1 − λur−1]vdx, for each v ∈W 1,2
0 (Ω). (2.1)

Testing this equation with v(x) = max{u(x)− 1, 0}, x ∈ Ω, we obtain∫
u≥1

|∇u(x)|2dx =

∫
u≥1

[(1− u)us−1(x)− λur−1(x)](u(x)− 1)dx ≤ 0.

This clearly implies 0 ≤ u(x) ≤ 1, for all x ∈ Ω, which in turn, recalling the
definition of fλ, implies the equivalence of (2.1) and Iλ(u) = 0. Suppose now that

u ∈ W 1,2
0 (Ω) satisfies I ′λ(u) = 0. Then, if we take again v(x) = max{u(x) − 1, 0},

x ∈ Ω, as a test function, we obtain

0 = I ′λ(u)(v) =

∫
u≥1

|∇u|2dx+

∫
u≥1

λur−1(x)dx ≥
∫
u≥1

|∇u|2dx

which implies u(x) ≤ 1, for all x ∈ Ω. While, testing with v = u−, we obtain

0 = I ′λ(u)(v) =

∫
u≤0

|∇u|2dx+

∫
u≤0

f(u)udx =

∫
u≤0

|∇u|2dx

and so u(x) ≥ 0, for all x ∈ Ω. �



4 G. ANELLO EJDE-2021/30

The previous lemma says that the nonnegative solutions of (1.1) are exactly

the critical points of Iλ. In particular, u ∈ W 1,2
0 (Ω) is a positive solution to (1.1)

if and only if u is a positive critical point of Iλ. We will see, via the Strong
Maximum Principle, that for λ = 0 the non-zero critical points of Iλ are positive
in Ω. To be more precise, let P be the interior of the positive cone of the space
C1

0 (Ω) := {u ∈ C1(Ω) : u(x) = 0, for each x ∈ ∂Ω} equipped with its standard

norm ‖u‖C1(Ω) := supΩ |u| +
∑N
i=1 supΩ | ∂u∂xi |. As it is well known, the set P is

given by

P =
{
u ∈ C1

0 (Ω) : u > 0 in Ω and
∂u

∂ν
< 0 on ∂Ω

}
,

where ν denotes the outer unit normal to ∂Ω.

Lemma 2.2. For λ = 0, any nonnegative and nonzero critical point of Iλ = I0
belongs to P.

Proof. Assume λ = 0 and let u ∈W 1,2
0 (Ω) be a nonzero critical point of I0. Then,

in view of Lemma 2.1, u is a nonzero nonnegative solution of the problem

−∆u = f0(u), in Ω,

u = 0, on ∂Ω,
(2.2)

where fλ(t) = f0(t) = max{0, (1−t) max{0, t}s−1}, t ∈ R, is a nonnegative function.
Hence, conclusion follows by the Strong Maximum Principle (see [5, Lemma 3.4 and
Theorem 3.5]). �

For λ > 0, fλ is negative and is not Lipschitz continuous in a right-neighborhood
of 0, so we cannot use, as for the case λ = 0, the Strong Maximum Principle to
deduce that any nonnegative and nonzero critical point of Iλ is positive in Ω.
Nevertheless, using the regularity theory for elliptic equations, we will see that for
λ small, positive critical points of Iλ actually exist. More precisely, we have the
following lemma.

Lemma 2.3. If λ1 < 1 when s = 2, there exists Λ > 0 such that, for each λ ∈
[0,Λ), Iλ admits a global nonzero minimum point uλ in W 1,2

0 (Ω), with Iλ(uλ) < 0.
Moreover, there also exists Λ0 ∈ (0,Λ) such that, for each λ ∈ [0,Λ0), any global
minimum point of Iλ belongs to P.

Proof. Consider the positive eigenfunction φ1 associated to λ1 and normalized with
respect to the sup-norm. For each τ ∈ (0, 1) and λ ≥ 0, taking in mind (1.2) and
that 0 ≤ φ1 ≤ 1 in Ω, one has

Iλ(τ1φ1) =
τ2

2
‖φ1‖2 −

∫
Ω

(∫ τφ1(x)

0

fλ(t)dt
)
dx

=
τ2

2
τ2‖φ1‖2 −

∫
Ω

(∫ τφ1(x)

0

[(1− t)ts−1 − λtr−1]dt
)
dx

=
λ1τ

2

2
‖φ1‖22 −

τs

s
‖φ1‖ss +

τs+1

s+ 1
‖φ1‖s+1

s+1 +
λτ r

r
‖φ1‖rr.

Since 1 < s ≤ 2, and λ1 < 1 when s = 2, we can find τ0 > 0 such that

C0 :=
λ1τ

2
0

2
‖φ1‖22 −

τs0
s
‖φ1‖ss +

τs+1
0

s+ 1
‖φ1‖s+1

s+1 < 0.



EJDE-2021/30 POSITIVE SOLUTIONS TO A DIRICHLET PROBLEM 5

Consequently, if Λ = − rC0

τr0 ‖φ1‖rr
, then

inf
W 1,2

0 (Ω)
Iλ ≤ Iλ(τ0φ1) < 0, for each λ ∈ [0,Λ) (2.3)

In addiction, since fλ(t) ≤ 1 for each t ∈ [0,∞), and fλ(t) = 0 for each t ∈ (−∞, 0),
one has

Iλ(u) ≥ 1

2
‖u‖2 − ‖u‖1 ≥

1

2
‖u‖2 − c1‖u‖, for each u ∈W 1,2

0 (Ω).

From this inequality, it follows that

lim
‖u‖→+∞

Iλ(u) = +∞. (2.4)

Therefore, since Iλ is sequentially lower semicontinuous, it admits at least a global
minimum point in W 1,2

0 (Ω). Moreover, if λ ∈ [0,Λ) and uλ is any global minimum
point of Iλ, in view of (2.3) one has Iλ(uλ) < 0. In particular, uλ is nonzero.

Now, let us to show that the global minimum points of Iλ belong to P, provided
that λ > 0 is small. Arguing by contradiction, assume that there exists a sequence
{λn}n∈N in (0,Λ), with λn → 0, and a sequence {un}n∈N in W 1,2

0 (Ω) such that, for
each n ∈ N, un is a global minimum point of Iλn and un /∈ P.

For each n ∈ N, one has I ′λn(un) = 0. Therefore, by Lemma 2.1, it turns out
0 ≤ un(x) ≤ 1, for each x ∈ Ω. Moreover, if we fix q > N/2, by a classical
regularity result (see [1, Theorem 8.2’]), we have u ∈ W 2,q(Ω) and there exists a
constant C > 0, independent of n, such that

‖un‖W 2,q(Ω) ≤ C(‖fλn(un)‖q + ‖un‖q) ≤ C(2 + λn)|Ω|1/q, (2.5)

where the last inequality follows by 0 ≤ un(x) ≤ 1 and 0 ≤ |f(un(x))| ≤ (1 + λn),
for each x ∈ Ω. In particular, in view of the embedding W 2,q(Ω) ↪→ C1,α(Ω), for
some α ∈ (0, 1), from (2.5) we infer that

‖un‖C1,α(Ω) ≤ C1(2 + λn) (2.6)

for a suitable constant C1 > 0 independent of n ∈ N. As a consequence, we obtain
the boundedness of {un}n∈N in C1,α(Ω). Thus, by the Ascoli-Arzelá Theorem, we
can assume that, up to a subsequence, {un}n∈N converges in C1(Ω) to a function
u0 ∈ C1(Ω). We claim that u0 is a global minimum point of I0. Indeed, consider
the function g : [0,Λ)→ R defined by

g(λ) = inf
u∈W 1,2

0 (Ω)
Iλ(u)

= inf
u∈W 1,2

0 (Ω)

(1

2
‖u‖2 −

∫
0≤u≤1

∫ u(x)

0

(1− t)ts−1dx+
λ

r

∫
u≥0

u(x)rdx
)
,

for each λ ∈ [0,Λ). The function g is convex and non-decreasing in [0,Λ) (as a
lower envelope of a family of affine non-decreasing functions). In particular, g is
continuous in [0,Λ). Consequently, recalling (2.3) and taking in mind that un → u0

in C1(Ω) and λn → 0, we infer that

0 > inf
u∈W 1,2

0 (Ω)
I0(u) = g(0)

= lim
n→+∞

g(λn) = lim
n→+∞

inf
u∈W 1,2

0 (Ω)
Iλn(u)

= lim
n→+∞

Iλn(un) = I0(u0)



6 G. ANELLO EJDE-2021/30

which proves our claim.
Therefore, the function u0 is in particular a nonzero critical point of I0. So, by

Lemma 2.2, one has u0 ∈ P. Finally, being P an open set of C1(Ω), from un → u0

in C1(Ω) we infer that un ∈ P, for n ∈ N large. This is a contradiction with un /∈ P,
for each n ∈ N. �

Lemma 2.3 ensures that the set

S := {λ > 0 : (1.1) admits a solution u ∈ P} (2.7)

is nonempty since it contains an interval of the type [0,Λ0). In the next Lemma we
show that the set S is an interval. This means that we can take Λ0 = supS.

Lemma 2.4. Let S be as in (2.7). Then, for each λ ∈ (0, supS), there exists a
solution uλ ∈ P of problem (1.1) which is also a local minimum point of Iλ.

Proof. Let λ ∈ (0, supS) and fix λ′ ∈ (λ, supS]∩S. Moreover, fix a solution uλ′ of

(Pλ′) belonging to P. Consider the function f̃λ : Ω× R→ R defined by

f̃λ(x, t) =

{
fλ(uλ′(x)), if (x, t) ∈ Ω× (−∞, uλ′(x)),

fλ(t) if (x, t) ∈ Ω× [uλ′(x),∞).

and the functional Ĩλ : W 1,2
0 (Ω)→ R, defined by

Ĩλ(u) =
1

2
‖u‖2 −

∫
Ω

(∫ u(x)

0

f̃λ(x, t)dt
)
dx, for each u ∈W 1,2

0 (Ω).

Clearly, as for Iλ, we have that Ĩλ is sequentially weakly lower continuous and of
class C1 in W 1,2

0 (Ω). Moreover, since f̃λ(x, t) ≤ 1, for each x ∈ Ω and t ∈ R, with

f̃λ(x, t) = 0 if t ≤ 0, one has

Ĩλ(u) ≥ 1

2
‖u‖2 − ‖u‖1 ≥

1

2
‖u‖2 − c1‖u‖, for each u ∈W 1,2

0 (Ω),

which implies lim‖u‖→+∞ Ĩλ(u) = +∞. Therefore, Ĩλ admits a global minimum

point uλ in W 1,2
0 (Ω). We claim that uλ ∈ P and that uλ is also a local minimum

point of Iλ. First of all, note that uλ is a solution of the problem

−∆u = f̃λ(x, u), in Ω,

u = 0, on ∂Ω,
(2.8)

Next, let us consider the open set A = {x ∈ Ω : uλ(x) < uλ′(x)} and assume A
non-empty. Then, the function uλ − uλ′ , which is negative in A, satisfies

−∆(uλ − uλ′)(x) = f̃λ(x, uλ(x))− fλ′(uλ′(x))

= (λ′ − λ)uλ′(x)r−1 > 0, if x ∈ A,
(uλ − uλ′)(x) = 0, if x ∈ ∂A.

As a consequence, by the Maximum Principle, we should also have that uλ−uλ′ is
positive in A, a contradiction. This means that A is empty and thus

uλ(x) ≥ uλ′(x), for each x ∈ Ω.

In addiction, note that there can be no point x0 ∈ Ω such that uλ(x0) = uλ′(x0)
for, otherwise, x0 should be a global minimum point for uλ − uλ′ , and then

0 ≥ −∆(uλ − uλ′)(x0) = (λ′ − λ)uλ′(x0)r−1 > 0,



EJDE-2021/30 POSITIVE SOLUTIONS TO A DIRICHLET PROBLEM 7

a contradiction. Thus, the function uλ − uλ′ is actually positive in Ω. Moreover, it
satisfies

−∆(uλ − uλ′)(x)

= uλ(x)s−1 − uλ′(x)s−1 − [uλ(x)s − usλ′(x) + λuλ(x)r−1 − λ′ur−1
λ′ (x)]

≥ −
[
uλ(x)s−1

(uλ(x)
uλ(x)

)s − 1

uλ(x)
uλ(x) − 1

+ λuλ′(x)r−2

( uλ(x)
uλ′ (x)

)r−1 − 1

uλ(x)
uλ′ (x) − 1

]
(uλ − uλ′)(x)

≥ −c(x)(uλ − uλ′)(x)

(2.9)

for all x ∈ Ω, where

c(x) = uλ(x)s−1 sup
t>1

ts − 1

t− 1
+ λuλ′(x)r−2 sup

0<t<1

tr−1 − 1

t− 1
.

Now, observe that, since u′λ ∈ P, one has

0 < c(x) ≤ kd(x, ∂Ω)r−2, for all x ∈ Ω,

for some constant k > 0, where d(·, ∂Ω) denotes the distance from ∂Ω. Thus, we can
apply Lemma 1 of [2] and obtain uλ− uλ′ ∈ P or, equivalently, uλ ∈ U := uλ′ +P,
where U ⊂ P is an open set in C1(Ω).

Finally, observe that, for each u ∈ U , one has u > uλ′ in Ω, and

Ĩλ(u)

=
1

2
‖u‖2 −

∫
Ω

(∫ uλ′ (x)

0

f̃λ(x, t)dt+

∫ u(x)

0

fλ(x, t)dt−
∫ uλ′ (x)

0

fλ(x, t)dt
)
dx

= Iλ(u)−M
where

M =

∫
Ω

(∫ uλ′ (x)

0

(f̃λ(x, t)dt− fλ(x, t))dt
)
dx.

Recalling that uλ is a global minimum point of Ĩλ, from the previous identity we
infer that uλ is a local minimum point of Iλ with respect to the C1(Ω) topology.
Then, by a classical result (see [4]), uλ turns out to be a local minimum point of

Iλ with respect to the W 1,2
0 (Ω)-topology as well. �

We are now in position to prove our main result.

Theorem 2.5. For each λ ∈ (0, supS), problem (1.1) admits at least two nonzero
nonnegative solutions, one of which belongs to P and is a local minimum point of
Iλ.

Proof. Let λ ∈ (0, supS) and fix q ∈ (2,∞), with q < 2N
N−2 , if N ≥ 3. Moreover,

we put

M = sup
t>0

Fλ(t)

tq
.

It is easy to check that M ∈ (0,+∞) and

Iλ(u) ≥ 1

2
‖u‖2 −M‖u‖qq ≥

1

2
‖u‖2 −Mcqq‖u‖q > 0,

for each u ∈ W 1,2
0 (Ω), with 0 < ‖u‖ <

(
2Mcqq

) 1
2−q . In particular, 0 is a (strict)

local minimum point for Iλ. By Lemma 2.4 we know that there exists another



8 G. ANELLO EJDE-2021/30

local minimum point uλ ∈ P. In addiction, in view of (2.4) one has that any

Palais-Smale sequence for Iλ (that is any sequence {un}n∈N in W 1,2
0 (Ω) such that

{Iλ(un)}n∈N converges in R and I ′λ(un) → 0 in W 1,2
0 (Ω)) is bounded. Therefore,

by a standard result (see Proposition 2.2 of [8], for instance), we see that functional
Iλ satisfies the Palais-Smale condition. At this point, by applying a well known
Mountain Pass Theorem (see [7]), we can deduce the existence of a second critical

point vλ ∈ W 1,2
0 (Ω) (besides uλ) for Iλ, with Iλ(vλ) > 0. This completes the

proof. �

Remark 2.6. Theorem 2.5 gives a positive answer to [3, Conjectures 3.5 and 7.4]
about the existence of positive solutions to problem (1.1). However, the existence of
possible compact-support solutions remains an open problem. Moreover, Theorem
2.5 confirms that, as anticipated in [3], condition (3.3) in [3] is technical and can
be removed.

Remark 2.7. Consider the set

S̃ = {λ > 0 : (1.1) admits a nonnegative solution u ∈W 1,2
0 (Ω) \ {0}}.

Clearly, S̃ ⊇ S and it is easy to see that S̃ is an interval. Indeed, if λ ∈ (0, sup S̃),

choosing λ′ ∈ (λ, sup S̃)∩ S̃, and fixing a nonzero nonnegative solution uλ′ to (1.1)
with λ′ instead of λ, we can see that u′λ is a sub-solution of (1.1). Moreover, if
ū is the unique positive solution of −∆u = 1 in Ω, u = 0 on ∂Ω, choosing k > 0
large enough, kū turns out to be a super-solution of (1.1), with kū > uλ′ in Ω.
Then, by the sub-supersolution method, we easily derive the existence of a nonzero
nonnegative solution to (1.1). Again in view of [3, Conjectures 3.5 and 7.4], it would
be interesting to give an answer to the following questions:

(1) Does Problem (1.1) admit a nonzero nonnegative solution for λ = sup S̃?

(2) Is it true that sup S̃ = supS? In other words, is it true that problem (1.1)
admits no nonzero solution for λ > supS?

The next result answers positively to the first question.

Theorem 2.8. Let S̃ be as in Remark 2.7 and put λ∗ = sup S̃. Then Problem
(1.1), with λ∗ instead of λ, admits a nonzero and nonnegative solution.

Proof. Let {λn}n∈N be a sequence in (0, λ∗) such that λn ↑ λ∗. Moreover, let

{un}n∈N be a sequence in W 1,2
0 (Ω) such that un is a nonzero and nonnegative

solution to Problem (1.1), with λn instead of λ, for each n ∈ N. Since I ′λn(un) = 0,
arguing as in Lemma 2.3, we obtain an inequality as (2.6). Therefore, up to a
subsequence, un → u∗ in the C1(Ω)-topology, for some u∗ ∈ C1(Ω). Consequently,

I ′λ∗(u
∗) = lim

n→+∞
I ′λn(un) = 0.

To conclude, it remains to show that u∗ is nonzero. To this end, fix q ∈ (2,∞),
with q ≤ 2N

N−2 if N ≥ 3, and put

M = sup
t>0

fλ1(t)

tq−1
.

Then, M ∈ (0,∞) and, for each n ∈ N, one has

0 = I ′λn(un)(un) = ‖un‖2 −
∫

Ω

fλn(un(x))un(x)dx



EJDE-2021/30 POSITIVE SOLUTIONS TO A DIRICHLET PROBLEM 9

≥ ‖un‖2 −
∫

Ω

fλ1
(un(x))un(x)dx

≥ ‖un‖2 −M‖un‖qq ≥ ‖un‖2 −Mcqq‖un‖q,
which implies

‖un‖ ≥ (Mcqq)
1

2−q > 0.

Since un → u∗ in C1(Ω), passing to the limit in the above inequality, we finally

obtain ‖u∗‖ ≥ (Mcqq)
1

2−q , that is u∗ 6= 0. �

References

[1] S. Agmon; The Lp approach to the Dirichlet problem, Ann. Scuola Norm. Sup. Pisa, 13
(1959), 405–448.

[2] G. Anello; Existence and multiplicity of nonnegative solutions for semilinear elliptic problems

involving nonlinearities indefinite in sign, Nonlinear Anal., 75 (2012), 6103–6107.
[3] V. Bobkov, P. Drábek, J. Hernández; Existence and multiplicity results for a class of semi-

linear elliptic equations, Nonlinear Analysis, 200, (2020), Article ID 112017, 24 p.

[4] H. Brezis, L. Nirenberg; H1 versus C1 local minimizers, C. R. Acad. Sci. Paris Sér. I Math.,
317(5) (1993), 465-472.

[5] D. Gilbarg, N. S. Trudinger; Elliptic partial differential equations of second order, Springer,

Berlin Heidelberg, 1998.
[6] J. Leach & D. Needham; Matched asymptotic expansions in reaction-diffusion theory,

Springer, London, 2004.
[7] P. Pucci, J. Serrin; A mountain pass theorem, J. Differ. Equations, 60 (1985), 14–149.

[8] M. Struwe; Variational Methods, Springer Verlag, Berlin, 1996.

Giovanni Anello

Dept. of Mathematics and Computer Science, Physical Science and Earth Science,
University of Messina, Viale F. Stagno d’Alcontres 31, Messina, Italy

Email address: ganello@unime.it


	1. Introduction
	2. Main result
	References