Positive radial solutions for Dirichlet problems via a Harnack-type inequality

Abstract

We deal with the existence and localization of positive radial solutions for Dirichlet problems involving \(\phi\)-Laplacian operators in a ball. In particular, \(p\)-Laplacian and Minkowski-curvature equations are considered.

Our approach relies on fixed point index techniques, which work thanks to a Harnack-type inequality in terms of a seminorm. As a consequence of the localization result, it is also derived the existence of several (even infinitely many) positive solutions.

Authors

Radu Precup
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania

Jorge Rodríguez-López
CITMAga and Departamento de Estatística, Análise Matemática e Optimización, Facultade de Matemáticas, Universidade de Santiago de Compostela, Santiago, Spain

Keywords

compression–expansion, Dirichlet problem, fixed point index, Harnack-type inequality, mean cur-vature operator, Positive radial solution

Paper coordinates

R. Precup, J. Rodríguez-López, Positive radial solutions for Dirichlet problems via a Harnack-type inequality, Mathematical Methods in the Applied Sciences, 46 (2023) no. 2, pp. 2972-2985, https://doi.org/10.1002/mma.8682

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Mathematical Methods in the Applied Sciences

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Wiley

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0170-4214

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1099-1476

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Positive radial solutions for Dirichlet problems via a Harnack type inequality

Positive radial solutions for Dirichlet problems via a Harnack type inequality

Radu Precup Jorge Rodríguez-López Institute of Advanced Studies in Science and Technology STAR-UBB, Babeş-Bolyai University & Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania, E-mail: r.precup@math.ubbcluj.roDepartamento de Estatística, Análise Matemática e Optimización, Instituto de Matemáticas, Universidade de Santiago de Compostela, 15782, Facultade de Matemáticas, Campus Vida, Santiago, Spain, E-mail: jorgerodriguez.lopez@usc.es
Abstract

We deal with the existence and localization of positive radial solutions for Dirichlet problems involving ϕ-Laplacian operators in a ball. In particular, p-Laplacian and Minkowski-curvature equations are considered. Our approach relies on fixed point index techniques, which work thanks to a Harnack type inequality in terms of a seminorm. As a consequence of the localization result, it is also derived the existence of several (even infinitely many) positive solutions.

MSC: 35J25, 35J60, 34B18, 35J92, 35J93.

Keywords and phrases: Positive radial solution, Dirichlet problem, mean curvature operator, Harnack type inequality, fixed point index, compression-expansion.

1 Introduction

In this paper, we deal with the existence, localization and multiplicity of positive radial solutions to the Dirichlet problem involving ϕ-Laplacian operators:

div(ψ(|v|)v)=f(|x|,v) in ,v=0 on , (1.1)

where is the unit open ball in n (n3) centered at the origin, the function f:[0,1]×++ is continuous and ψ:(a,a) is C1, such that ϕ(s):=sψ(s) is an increasing homeomorphism between two intervals (a,a) and (b,b) (0<a,b+).

The following particular cases are of much interest due to their corresponding models arising from physics:

(a) ϕ:,ϕ(s)=|s|p2s, where p>1(here a=b=+), when the left side Lv in (1.1) is

Lv=div(|v|p2v)(p-Laplace operator),

involved in a nonlinear Darcy law for flows through porous media [3];

(b) (singular homeomorphism) ϕ:(a,a),ϕ(s)=sa2s2 (here 0<a<+ and b=+), when

Lv=div(va2|v|2)(Minkowski mean curvature operator),

involved in the relativistic mechanics [2, 10];

(c) (bounded homeomorphism) ϕ:(b,b),ϕ(s)=bs1+s2 (here a=+ and 0<b<+), when

Lv=bdiv(v1+|v|2)(Euclidian mean curvature operator),

associated to capillarity problems [1, 14].

Looking for radial solutions of (1.1), that is, functions of the form v(x)=u(r) with r=|x|, the Dirichlet problem (1.1) reduces to the mixed boundary value problem

(rn1ϕ(u))=rn1f(r,u),u(0)=u(1)=0. (1.2)

Radial and nonradial solutions for the Dirichlet problem involving ϕ-Laplace operators have been intensively investigated in the literature, both by means of topological and variational methods. We refer the interested reader to the papers [4, 5, 6, 7, 8, 9, 11, 12, 15] and the references therein.

Our approach here is based on fixed point index theory, namely on compression-expansion type homotopy arguments. The most known are those from Krasnosel’skiĭ’s compression-expansion theorem in a conical annulus defined by using the max-norm of the space. Applications to one-dimensional ϕ-Laplace equations are given in [13]. In the radial case considered in the present paper, the absence of a Harnack type inequality in terms of the max-norm, makes Krasnosel’skiĭ’s theorem inoperative and forces us to use instead, some other homotopy conditions and properties of the fixed point index.

The first paper in radial solutions that uses the compression-expansion technique, but in a variational form and only for p-Laplacian equations, is [17]. As explained there, the difficulty in applying the compression-expansion method consists in the necessity that, for the considered differential operator, a Harnack type inequality be available. In the present paper, such a key inequality is established for problem (1.2) with a general homeomorphism ϕ satisfying some additional conditions. With its help, a precise localization of positive solutions is possible, allowing in a natural way to obtain multiple solutions. The results apply in particular for the p-Laplacian and the Minkowski mean curvature operator.

Our basic assumptions are as follow:

(Hϕ)

ϕ:(a,a)(b,b) (0<a,b+) is an odd increasing homeomorphism such that

λϕ(x)ϕ(λx) for all λ[0,1],x[0,a)(ϕ is convex on [0,a)). (1.3)
(Hf)

f:[0,1]×+[0,b) is continuous, with f(,s) nonincreasing in [0,1] for every s+ and f(r,) nondecreasing in + for every r[0,1].

Note that the homeomorphisms related to the p-Laplacian for p2 and the Minkowski mean curvature operator both satisfy condition (Hϕ). Contrarily, the bounded homeomorphisms with a=+, for example the one involved by the Euclidian mean curvature operator, are not convex on [0,+) and thus they do not satisfy our assumption (Hϕ).

2 A Harnack type inequality

In the space of functions uC1[0,1] satisfying u(0)=u(1)=0 we consider the following norms:

up = (01(rn1|u(r)|)p𝑑r)1/pfor 1p<;
u = supr[0,1]rn1|u(r)|for p=+.

Let ϕ satisfy (Hϕ) and denote

h0(r) = rn1ϕ(u(r)),
J(u) = h0(r)=(rn1ϕ(u(r))).

First we prove a Harnack type inequality for problem (1.2), given in terms of the norm p, with 1p+.

Theorem 2.1.

Let uC1[0,1] be such that u(r)(a,a) for all r[0,1], h0C1[0,1] andJ(u)0 on [0,1]. Then u0 on [0,1].

If in addition r1nJ(u) is nonincreasing on (0,1], then

u(r)(1r)rnup,r[0,1],

for every 1p+.

Proof.

By assumption,

h0=J(u)0.

Hence h0 is nondecreasing in [0,1]. Since h0(0)=0, one has h00 and so u0 on [0,1]. Thus

h0(r)=rn1ϕ(|u(r)|). (2.1)

Next

ϕ1(h0(r))=ϕ1(rn1ϕ(|u(r)|))ϕ1(ϕ(|u(r)|))=|u(r)|,r[0,1]. (2.2)

Also

ϕ1(h0(1))=|u(1)|.

Since both ϕ1 and h0 are nondecreasing, using (2.2) we have

u(r)=r1|u(s)|𝑑sr1ϕ1(h0(s))𝑑s(1r)ϕ1(h0(r)).

Therefore

u(r)(1r)ϕ1(h0(r)),r[0,1].

Next we prove the inequality

Φ(r):=h0(r)rnh0(1)0on [0,1]. (2.3)

One has

Φ(r)=J(u)(r)nrn1h0(1)=rn1(r1nJ(u)(r)nh0(1)).

By assumption, Ψ(r):=r1nJ(u)(r)nh0(1) is nonincreasing. As in the proof of [17, Theorem 2.1], since Φ(0)=Φ(1)=0, we deduce (2.3).

Then

ϕ1(h0(r))ϕ1(rnh0(1))=ϕ1(rnϕ(|u(1)|))rn|u(1)|,

where the last inequality is based on (1.3) . Hence

u(r)(1r)rn|u(1)|,r[0,1].

Finally, by (2.1) and using again (1.3), one has ϕ1(h0(r))rn1|u(r)|. Hence, we have

upp :=01(rn1|u(r)|)p𝑑r01ϕ1(h0(r))p𝑑r
ϕ1(h0(1))p=|u(1)|p,(1p<).

Therefore,

u(r)(1r)rnup(1p), (2.4)

for all r[0,1]. ∎

A similar result has been established in [17] for the particular case of the p-Laplacian with p>n, for which ϕ(s)=|s|p2s and a=b=+. More exactly, it has been proved that

u(r)(pnp1)1p(1r)rnp1u1,p(r[0,1]), (2.5)

where u1,p=(01rn1|u(r)|p𝑑r)1p. In this case, since by using Hölder’s inequality one has

|u(r)| r1|u(s)|𝑑s=r1sn1p|u(s)|sn1p𝑑s
u1,p(01sn1p1𝑑s)p1p=(p1pn)p1pu1,p,

a Harnack type inequality in terms of the max-norm |u|=maxr[0,1]|u(r)| can be immediately derived from (2.5), namely

u(r)pnp1(1r)rnp1|u|for r[0,1].

It is an open problem to obtain an analogue result for more general homeomorphisms ϕ satisfying (Hϕ).   At this moment we are only able to establish such an inequality in terms of a max-seminorm on C[0,1]. For example, taking p=+ in (2.4), we have the following Harnack type inequality related to a seminorm on C[0,1].

Corollary 2.2.

Under the assumptions of Theorem 2.1, if for a fixed subinterval [η,ν] with 0<η<ν<1, one define in C[0,1] the semi-norm [u]=maxr[η,ν]|u(r)|, then

u(r)(1ν)η2n2(n2)[u]for r[η,ν]. (2.6)
Proof.

Clearly

|u(r)| r1|u(s)|𝑑s=r1sn1|u(s)|s(n1)𝑑s
ur1s(n1)𝑑sr2nn2u,

which implies

[u]η2nn2u.

This combined with (2.4) yields (2.6). ∎

In the sequel, inequality (2.6) is a key ingredient for the localization and multiplicity of positive radial solutions. We will use the main ideas in [16] in order to localize the solutions in terms of a norm and a semi-norm.

3 Positive radial solutions

Recall that by a (nonnegative) solution of (1.2) we mean a function uC1([0,1],+) with u(0)=u(1)=0, |u(r)|<a for all r[0,1], such that rn1ϕ(u)C1[0,1] and (1.2) is satisfied. We will say that a nonnegative solution is positive if it is distinct from the identically zero function. Let X be the Banach space of continuous functions X=C[0,1] and K0 its positive cone K0={uX:u0 on [0,1]}. It is not difficult to see that a nonnegative function u is a solution of problem (1.2) if and only if u is a fixed point of the operator T:K0K0 given by

T(u)(r)=r1ϕ1(τ1n0τsn1f(s,u(s))𝑑s)𝑑τ. (3.1)

As proved in [4, 6], the operator T is completely continuous.

Let us now consider a subcone of K0 related to the Harnack inequality (2.6), namely

K={uK0:u is nonincreasing on [0,1] and minr[η,ν]u(r)c[u]}, (3.2)

where c:=(1ν)η2n2(n2).

Lemma 3.1.

The operator T maps the cone K into itself.

Proof.

Indeed, take uK and let us show that v:=Tu belongs to K. Since f is nonnegative, v0 and, moreover, J(v)0 and so v0 (see Theorem 2.1), that is v is nonincreasing on [0,1]. Furthermore, by the monotonicity properties of f imposed in (H)f and the fact that u is nonincreasing in [0,1], the composed function rf(r,u(r)) is nonincreasing in [0,1]. Hence,

r1nJ(v)=f(r,u)

is nonincreasing in [0,1]. Then Corollary 2.2 ensures that

v(r)(1ν)η2n2(n2)[v] for r[η,ν].

Therefore, vK, as claimed. ∎

Now, for any number α>0 consider the set

Uα:={uK:|u|<α}.

The operator T being completely continuous, the set T(U¯α) is bounded, so there is a number α~α such that T(U¯α)U¯α~. Define the operator T~:U¯α~U¯α~ by

T~(u)=T(min{α|u|,1}u).
Lemma 3.2.

If

T(u)λufor uK with |u|=α and λ1, (3.3)

then the fixed point index  i(T~,Uα,U¯α~)=1.

Proof.

Clearly U¯α~ is a convex closed set and T~ is a compact map. Consider the homotopy H:[0,1]×U¯α~U¯α~ given by

H(τ,u)=τT~(u).

By (3.3), this homotopy is admissible and so

i(T~,Uα,U¯α~)=i(H(1,),Uα,U¯α~)=i(H(0,),Uα,U¯α~)=1,

where the last equality is due to the normalization property of the fixed point index, since 0Uα. ∎

Next, for a number β>0 consider the set

Vβ:={uU¯α~:[u]<β}.

It is clear that Vβ is open in U¯α~.

Lemma 3.3.

Assume that there exists a function hK such that |h|=α, [h]>β and

(1λ)T~(u)+λhufor uK with |u|α~,[u]=β and λ[0,1]. (3.4)

Then i(T~,Vβ,U¯α~)=0.

Proof.

Observe that Vβ={uK:[u]=β,|u|α~}. Thus (3.4) implies that

(1λ)T~(u)+λhufor uVβ.

By the homotopy property of the fixed point index, one has

i(T~,Vβ,U¯α~)=i(h,Vβ,U¯α~).

Finally, i(h,Vβ,U¯α~)=0, since hU¯α~V¯β. ∎

Remark 3.1.

If the operator T maps U¯α into itself, then α~=α and condition (3.4) reduces to

(1λ)T(u)+λhufor uK with |u|α,[u]=β and λ[0,1].

By using the previous fixed point index computations, we deduce the following existence result.

Lemma 3.4.

Under the assumptions of Lemmas 3.2 and 3.3, the operator T has a fixed point u in UαV¯β, that is problem (1.2) has a solution such that

β<[u]and |u|<α.
Proof.

One has

1 = i(T~,Uα,U¯α~)=i(T~,UαV¯β,U¯α~)+i(T~,UαVβ,U¯α~),
0 = i(T~,Vβ,U¯α~)=i(T~,VβU¯α,U¯α~)+i(T~,UαVβ,U¯α~).

As a result

i(T~,UαV¯β,U¯α~)i(T~,VβU¯α,U¯α~)=1.

In addition i(T~,VβU¯α,U¯α~)=0 since otherwise there would exist vVβU¯α with T~(v)=v, that is

T(α|v|v)=v,

or equivalently T(w)=λw, where w=α|v|v and λ=|v|α. Since |w|=α and λ>1 we arrived to a contradiction with (3.3). Therefore i(T~,UαV¯β,U¯α~)=1, which implies our conclusion. ∎

Now we give sufficient conditions in order to guarantee the assumptions of the previous lemmas hold.

We will use the following notation. If b<+, denote

A:=101ϕ1(bτ)𝑑τandB:=η1ϕ1(bτ)𝑑τ.

If b=+, denote

A:=101ϕ1(τ)𝑑τandB:=η1ϕ1(τ)𝑑τ.
Theorem 3.5.

Assume that n3 and conditions (Hϕ) and (Hf) are fulfilled. If there exist α,β>0 with β<ABα, such that

ϕ1(f(0,α)) <α, (3.5)
(1ν)ϕ1((νη)ηn1f(ν,cβ)) >β, (3.6)

then problem (1.2) has at least one solution uK such that β<[u] and |u|<α.

Proof.

We shall apply Lemma 3.4. First, we show that (3.3) holds. Indeed, for uK with |u|α, by the monotonicity assumptions on f we have that

f(s,u(s))f(0,α),

and thus from (3.5),

|T(u)(r)| 01ϕ1(τ1n0τsn1f(s,u(s))𝑑s)𝑑τ
01ϕ1(0τf(s,u(s))𝑑s)𝑑τϕ1(f(0,α))<α.

Hence, |T(u)|<α for all uK with |u|α, which implies (3.3). In addition, on the basis of Remark 3.1, we can take α~=α.

Next, we prove that (3.4) holds for the following choice of h: if b<+,

h(r)=Aαr1ϕ1(τ1n0τbnsn1𝑑s)𝑑τ=Aαr1ϕ1(bτ)𝑑τ,

and, otherwise, for b=+,

h(r)=Aαr1ϕ1(τ1n0τnsn1𝑑s)𝑑τ=Aαr1ϕ1(τ)𝑑τ.

Note that |h|=h(0)=α and [h]=h(η)=ABα>β. Assume that (3.4) does not hold. Then there exist uK with |u|α, [u]=β and λ[0,1] such that

(1λ)T(u)+λh=u.

In particular, since [u]=maxr[η,ν]u(r)=β, one has

βu(η) =(1λ)T(u)(η)+λh(η)
=(1λ)η1ϕ1(τ1n0τsn1f(s,u(s))𝑑s)𝑑τ+λ[h]
(1λ)ν1ϕ1(τ1nηνsn1f(s,u(s))𝑑s)𝑑τ+λβ.

Since uK with [u]=β, we have u(r)cβ for all r[η,ν]. Thus, by (Hf),

β (1λ)ν1ϕ1(τ1nηνsn1f(ν,cβ)𝑑s)𝑑τ+λβ
(1λ)(1ν)ϕ1((νη)ηn1f(ν,cβ))+λβ,

that is,

(1λ)β(1λ)(1ν)ϕ1((νη)ηn1f(ν,cβ)),

which contradicts (3.6) for any λ[0,1). Note that, in case λ=1, one has the contradiction

βu(η)=h(η)=[h]>β.

Finally, the conclusion follows from Lemma 3.4. ∎

Remark 3.2 (Asymptotic conditions).

Existence of both positive numbers α and β satisfying inequalities (3.5) and (3.6) is guaranteed if the following asymptotic conditions at zero and infinity hold:

lim supx0+ϕ1((νη)ηn1f(ν,x))x>1c(1ν),lim infx+ϕ1(f(0,x))x<1.

Obviously, if ϕ is a classical homeomorphism (a=b=+), conditions (3.5) and (3.6) can be rewritten as

f(0,α)<ϕ(α),f(ν,cβ)>1(νη)ηn1ϕ(β1ν).

Hence, if we assume in addition that ϕ satisfies:

lim supx0+ϕ(τx)ϕ(x)<+ for all τ>0, (3.7)

then the existence of both positive numbers α and β is guaranteed under suitable asymptotic conditions about f at zero and at infinity.

Note that assumption (3.7) holds in the case of the p-Laplacian operator and so it is commonly employed in the literature, see for instance [6, 11].

Theorem 3.6.

Assume that n3, conditions (Hϕ) and (Hf) are fulfilled and ϕ is a classical homeomorphim. If (3.7) and

lim supx0+f(ν,x)ϕ(x)=+,lim infx+f(0,x)ϕ(x)<1 (3.8)

hold, then problem (1.2) has at least one positive solution.

Proof.

By (3.7), with τ=1/(c(1ν)), there exists L>0 so that

L>lim supx0+ϕ(x/(c(1ν)))ϕ(x),

and thus there exists ρ>0 such that

Lϕ(x)ϕ(xc(1ν))for all x(0,ρ).

Now, by (3.8), there exists r>0 (we may suppose r<ρ) such that

f(ν,r)>1(νη)ηn1Lϕ(r),

which implies that

f(ν,r)>1(νη)ηn1ϕ(rc(1ν)).

Finally, taking β=r/c, condition (3.6) is obtained.

On the other hand, condition

lim infx+f(0,x)ϕ(x)<1

clearly implies the existence of a positive number α satisfying (3.5) and such that β<ABα.

Therefore, Theorem 3.5 ensures the existence of at least one positive solution for problem (1.2). ∎

Corollary 3.7.

Assume that n3, p2 and (Hf) holds. If

lim supx0+f(ν,x)xp1=+andlim infx+f(0,x)xp1<1, (3.9)

then problem

div(|v|p2v)=f(|x|,v) in ,v=0 on , (3.10)

has at least one positive radial solution.

Proof.

It suffices to show that problem (1.2) has at least one positive solution with ϕ(x)=|x|p2x, p2. Since ϕ is a classical homeomorphism which satisfies (Hϕ) and (3.7), the conclusion follows from Theorem 3.6. ∎

We show the applicability of our theory with an example involving radial solutions of p-Laplacian equations.

Example 3.8.

Consider the function f given by

f(s,x)=f(x)=xq+x,

with 0q<p1 and p2, which clearly satisfies condition (Hf).

It is immediate to check that

limx0+f(x)xp1=limx0+xq+xxp1=+ and limx+f(x)xp1=limx+xq+xxp1=0.

Therefore, problem (3.10) associated to this function f has at least one positive radial solution, as a consequence of Corollary 3.7.

Finally, we highlight that due to the asymptotic behavior of f at zero and at infinity, this problem falls outside the scope of the results in [11, 17].

On the other hand, it is worth to mention that in the case of a singular homeomorphism ϕ (i.e., with a<+, b=+), condition (3.5) is trivially satisfied for α large enough. Hence, in that case, we only need to ensure the existence of the number β in order to obtain positive solutions for problem (1.2).

Let us assume in the rest of this section that ϕ is singular. We present an existence result inspired by those in [6].

Theorem 3.9.

Assume that n3, conditions (Hϕ) and (Hf) are fulfilled and ϕ is a singular homeomorphim. If (3.7) and

lim supx0+f(ν,x)ϕ(x)=+ (3.11)

hold, then problem (1.2) has at least one positive solution.

Proof.

Arguing as in the proof of Theorem 3.6, conditions (3.7) and (3.11) imply the existence of a positive number β satisfying (3.6). Therefore, Theorem 3.5 ensures the existence of at least one positive solution for problem (1.2). ∎

As a consequence, we derive a simple existence result concerning positive radial solutions for Dirichlet problems involving the Minkowski mean curvature operator.

Corollary 3.10.

Assume that n3 and condition (Hf) holds. If

lim supx0+f(ν,x)x=+, (3.12)

then problem

div(v1|v|2)=f(|x|,v) in ,v=0 on , (3.13)

has at least one positive radial solution.

Proof.

If suffices to show that problem

(rn1u1u2)=rn1f(r,u),u(0)=u(1)=0,

has at least one positive solution. The conclusion follows from Theorem 3.9 with ϕ(x)=x/1x2, 1<x<1. Note that

limx0+ϕ(τx)ϕ(x)=τ for all τ>0,

and that, for this homeomorphism ϕ, condition (3.12) implies (3.11). ∎

Example 3.11.

Consider problem (3.13) with a function f of the form

f(s,x)=g(s)xq,s[0,1],x0,

where 0q<1 and g is a nonincreasing positive and continuous function. Clearly, f satisfies (Hf) and the asymptotic condition (3.12), so Corollary 3.10 ensures the existence of a positive radial solution.

Example 3.12.

If 0q<1p, λ>0 and g is a nonincreasing nonnegative continuous function, then problem (3.13) with

f(s,x)=λxq+g(s)xp,s[0,1],x0,

has at least one positive radial solution. Clearly, f satisfies (Hf) and the asymptotic condition (3.12), so the claim is again a consequence of Corollary 3.10.

4 Multiplicity results

Obviously, the localization of positive solutions provided by Theorem 3.5 makes possible to obtain multiplicity results for problem (1.2) if there exist several (perhaps infinitely many) well–ordered pairs of numbers (α,β) satisfying (3.5)–(3.6). Nevertheless, a suitable computation of the fixed point index related to the operator T allows us to establish a three solutions type result under less stringent assumptions.

First, we present the three solutions type fixed point theorem concerning the operator T defined in (3.1).

Lemma 4.1.

Under the assumptions of Lemma 3.4, if in addition there exists a positive number α0 with α0<β and

T(u)λufor uK with |u|=α0 and λ1, (4.1)

then T has at least three fixed points u1, u2 and u3 such that

β<[u1],|u1|<α;|u2|<α0;α0<|u3|<α,[u3]<β.
Proof.

By Lemma 3.4, T has a fixed point u1 such that

β<[u1],|u1|<α.

Moreover, assumption (4.1) ensures that i(T~,Uα0,U¯α~)=1 and thus the operator T has a fixed point u2 in Uα0, that is, |u2|<α0. Since α0<β, one has U¯α0Vβ, and so the properties of the fixed point index together with its computation in Lemma 3.3 imply

i(T~,VβU¯α0,U¯α~)=i(T~,Vβ,U¯α~)i(T~,Uα0,U¯α~)=01=1.

Therefore, the existence property of the fixed point index ensures that the operator T has a third fixed point u3 located in VβU¯α0. ∎

As a consequence, we obtain a three solutions type result for problem (1.2).

Theorem 4.2.

Assume that n3 and conditions (Hϕ) and (Hf) are fulfilled. If there exist α0,α1,β>0 with α0<β<ABα1, such that

ϕ1(f(0,αi)) <αi,i=0,1,
(1ν)ϕ1((νη)ηn1f(ν,cβ)) >β,

then problem (1.2) has at least three solutions u1, u2 and u3 such that

β<[u1],|u1|<α1;|u2|<α0;α0<|u3|<α1,[u3]<β.
Remark 4.1.

Theorem 4.2 ensures the existence of at least two positive solutions, namely, u1 and u3 with the localizations above. Furthermore, if f(,0)0, then u2 is also a positive solution.

A multiplicity result can be also obtained under a suitable behavior of the nonlinearity at zero and infinity.

Corollary 4.3.

Assume that n3 and conditions (Hϕ) and (Hf) are fulfilled. In addition, suppose that there exists β>0 satisfying condition (3.6) and

lim infx0+ϕ1(f(0,x))x<1,lim infx+ϕ1(f(0,x))x<1. (4.2)

Then problem (1.2) has at least two positive solutions v1 and v2 such that [v1]>β and [v2]<β.

Proof.

By the asymptotic behavior of f at zero and at infinity given by (4.2), there exist 0<α0<β (sufficiently small) and α1>β/(AB) (sufficiently large) such that

ϕ1(f(0,αi))<αi,i=0,1.

Therefore, the conclusion follows from Theorem 4.2. ∎

Next we emphasize the multiplicity result in the remarkable particular cases of Dirichlet problems involving the p-Laplacian and Minkowski mean curvature operators.

Corollary 4.4.

Assume that n3, p2 and condition (Hf) holds. In addition, suppose that there exists β>0 satisfying condition (3.6) and

lim infx0+f(0,x)xp1<1,lim infx+f(0,x)xp1<1.

Then problem (3.10) has at least two positive radial solutions.

Corollary 4.5.

Assume that n3 and condition (Hf) holds. In addition, suppose that there exists β>0 satisfying condition (3.6) and

lim infx0+f(0,x)x<1.

Then problem (3.13) has at least two positive radial solutions.

Proof.

The conclusion follows from Corollary 4.3 with ϕ(x)=x/1x2, 1<x<1. Since ϕ is singular, condition

lim infx+ϕ1(f(0,x))x<1

is trivially satisfied. ∎

We illustrate the applicability of the previous multiplicity results with the following example.

Example 4.6.

Consider problem (3.13) with a function f of the form

f(s,x)=f(x)=λxq,s[0,1],x0,

where q>1 and λ>0.

We shall study the existence of two positive solutions for problem (1.2) with f as above and ϕ being the singular homeomorphism given by ϕ(x)=x/1x2, 1<x<1.

Clearly, f satisfies (Hf) and it is immediate to check that

limx0+f(x)x=0.

Finally, taking η=1/4 and ν=3/4, it is a simple matter to see that β=1/16 satisfies condition (3.6) for any λ large enough (for instance, with λ>4(2n+1)q+n1/2/(n2)q15).

Therefore, Corollary 4.5 guarantees that problem

div(v1|v|2)=λvq in ,v=0 on ,

has at least two positive radial solutions for any q>1 provided that λ>0 is sufficiently large.

Finally, the existence of infinitely many positive solutions for (1.2) is obtained if the nonlinearity f has an oscillating behavior at zero or at infinity.

Corollary 4.7.

Assume that n3 and conditions (Hϕ) and (Hf) are fulfilled.

  1. (a)

    If

    lim supx0+ϕ1((νη)ηn1f(ν,x))x>1c(1ν),lim infx0+ϕ1(f(0,x))x<1, (4.3)

    then (1.2) has a sequence of positive solutions uk such that |uk|0 as k.

  2. (b)

    If

    lim supx+ϕ1((νη)ηn1f(ν,x))x>1c(1ν),lim infx+ϕ1(f(0,x))x<1, (4.4)

    then (1.2) has a sequence of positive solutions uk such that |uk|+ as k.

Proof.

Let us prove the claim (a) (the case (b) is analogous). By (4.3), there exist two decreasing sequences {αi}i and {βi}i tending to zero such that αi+1<βi<ABαi and

ϕ1(f(0,αi))<αi,(1ν)ϕ1((νη)ηn1f(ν,cβi))>βi.

Therefore, Theorem 3.5 can be applied to each pair (βi,αi) and so the conclusion is immediately obtained. ∎

Remark 4.2.

Note that case (b) in Corollary 4.7 is not possible if ϕ is a singular homeomorphism. Indeed, in that case, ϕ1 is bounded and thus

limx+ϕ1((νη)ηn1f(ν,x))x=0,

which makes impossible that (4.4) holds.

To finish, we provide an example concerning the existence of infinitely many positive radial solutions for a Dirichlet problem involving the relativistic operator.

Example 4.8.

Consider the problem (3.13) associated to the function

f(s,x)=f(x)=x[λ+ρsin(γln1x)], for x>0,f(0)=0,

where λ,ρ,γ>0. Observe that f is continuous and, moreover, it is nondecreasing if λρ(γ+1).

Furthermore, since ϕ1(x)=x/1+x2 and f(0)=0, one has

lim supx0+ϕ1((νη)ηn1f(x))x=lim supx0+(νη)ηn1f(x)x=(νη)ηn1(λ+ρ)

and

lim infx0+ϕ1(f(x))x=lim infx0+f(x)x=λρ.

Then the asymptotic condition (4.3) holds if

(νη)ηn1(λ+ρ)>1c(1ν) and λρ<1.

Therefore, Corollary 4.7 ensures that the corresponding problem (3.13) associated to this nonlinearity f has a sequence of positive radial solutions uk such that |uk|0 as k provided that

λρ(γ+1),(νη)ηn1(λ+ρ)>1c(1ν) and λρ<1.

In particular, taking η=1/3 and ν=2/3, the previous inequalities reduce to

λρ(γ+1),λ+ρ>33nn2 and λρ<1.

Acknowledgements

Jorge Rodríguez-López was partially supported by Institute of Advanced Studies in Science and Technology of Babeş-Bolyai University of Cluj-Napoca (Romania) under a Postdoctoral Advanced Fellowship, project CNFIS-FDI-2021-0061; by Xunta de Galicia (Spain), project ED431C 2019/02 and AIE, Spain and FEDER, grant PID2020-113275GB-I00.

References

  • [1] M. Athanassenas and J. Clutterbuck, A capillarity problem for compressible liquids, Pacific J. Math., 243 (2009), 213–232.
  • [2] R. Bartnik and L. Simon, Spacelike hypersurfaces with prescribed boundary values and mean curvature, Commun. Math. Phys., 87 (1982), 131–152.
  • [3] J. Benedikt, P. Girg, L. Kotrla, P. Takac, Origin of the p-Laplacian and A. Missbach, Electron. J. Differential Equations, 2018 (2018), No. 16, 1–17.
  • [4] C. Bereanu, P. Jebelean and J. Mawhin, Radial solutions for some nonlinear problems involving mean curvature operators in Euclidean and Minkowski spaces, Proc. Amer. Math. Soc., 137 1 (2009), 161–169.
  • [5] C. Bereanu, P. Jebelean and C. Şerban, Dirichlet problems with mean curvature operator in Minkowski space. In: NewTrends in Differential Equations, Control Theory and Optimization: Proceedings of the 8th Congress of Romanian Mathematicians, (2016), 1–20.
  • [6] C. Bereanu, P. Jebelean and P. J. Torres, Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space, J. Funct. Anal., 264 (2013), 270–287.
  • [7] C. Bereanu, P. Jebelean and P. J. Torres, Multiple positive radial solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space, J. Funct. Anal., 265 (2013), 644–659.
  • [8] I. Coelho, C. Corsato and S. Rivetti, Positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball, Topol. Methods Nonlinear Anal., 44 1 (2014), 23–39.
  • [9] C. Corsato, F. Obersnel, P. Omari and S. Rivetti, Positive solutions of the Dirichlet problem for the prescribed mean curvature equation in Minkowski space, J. Math. Anal. Appl., 405 1 (2013), 227–239.
  • [10] K. Ecker, Mean curvature evolution of spacelike hypersurfaces, Proceedings of the Centre for Mathematics and its Applications, Australian National University, 37 (1999), 119–132.
  • [11] M. García–Huidobro, R. Manásevich and K. Schmitt, Positive radial solutions of quasilinear elliptic partial differential equations on a ball, Nonlinear Anal., 35 (1999), 175–190.
  • [12] X. He, Multiple radial solutions for a class of quasilinear elliptic problems, Appl. Math. Lett., 23 1 (2010), 110–114.
  • [13] D.-R. Herlea and R. Precup, Existence, localization and multiplicity of positive solutions to ϕ-Laplace equations and systems. Taiwan. J. Math., 20 (2016), 77–89.
  • [14] J. López-Gómez, P. Omari and S. Rivetti, Positive solutions of a one-dimensional indefinite capillarity-type problem: a variational approach, J. Differential Equations, 262 (2017), 2335–2392.
  • [15] R. Ma, H. Gao and Y. Lu, Global structure of radial positive solutions for a prescribed mean curvature problem in a ball, J. Funct. Anal., 270 (2016), 2430–2455.
  • [16] R. Precup, Moser–Harnack inequality, Krasnosel’skiĭ type fixed point theorems in cones and elliptic problems, Topol. Methods Nonlinear Anal., 40 (2012), 301–313.
  • [17] R. Precup, P. Pucci and C. Varga, Energy-based localization and multiplicity of radially symmetric states for the stationary p-Laplace diffusion, Complex Var. Elliptic Equ., 65 (2020), 1198–1209.
  • [18] R. Precup and J. Rodríguez-López, Positive solutions for discontinuous problems with applications to ϕ-Laplacian equations, J. Fixed Point Theory Appl., 20:156 (2018), 1–17.

[1] Benedikt J, Girg P, Kotrla L, Takac P. Origin of thep-Laplacian and A. Missbach.Electron J Differ Equ. 2018;2018(16):1-17.
[2] Bartnik R, Simon L. Spacelike hypersurfaces with prescribed boundary values and mean curvature.Commun Math Phys. 1982;87:131-152.
[3] Ecker K. Mean curvature evolution of spacelike hypersurfaces, Proceedings of the Centre for Mathematics and its Applications.AustralianNat Univ. 1999;37:119-132.
[4] Athanassenas M, Clutterbuck J. A capillarity problem for compressible liquids.Pacific J Math. 2009;243:213-232.
[5] López-Gómez J, Omari P, Rivetti S. Positive solutions of a one-dimensional indefinite capillarity-type problem:A variational approach.J Differ Equ. 2017;262:2335-2392.
[6]  Bereanu C, Jebelean P, Mawhin J. Radial solutions for some nonlinear problems involving mean curvature operators in Euclidean and Minkowski spaces.Proc Amer Math Soc. 2009;137(1):161-169.
[7] Bereanu C, Jebelean P, ̧Serban C. Dirichlet problems with mean curvature operator in Minkowski space. In: New Trends in Differential Equations, Control Theory and Optimization: Proceedings of the 8th Congress of Romanian Mathematicians; 2016:1-20.
[8] Bereanu C, Jebelean P, Torres PJ. Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space.J FunctAnal. 2013;264:270-287.
[9] Bereanu C, Jebelean P, Torres PJ. Multiple positive radial solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space.J Funct Anal. 2013;265:644-659.
[10] Coelho I, Corsato C, Rivetti S. Positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball, Topol.Methods Nonlinear Anal. 2014;44(1):23-39.
[11] Corsato C, Obersnel F, Omari P, Rivetti S. Positive solutions of the Dirichlet problem for the prescribed mean curvature equation in Minkowski space.J Math Anal Appl. 2013;405(1):227-239.
[12] García–Huidobro M, Manásevich R, Schmitt K. Positive radial solutions of quasilinear elliptic partial differential equations on a ball. Nonlinear Anal. 1999;35:175-190.
[13] He X. Multiple radial solutions for a class of quasilinear elliptic problems. Appl Math Lett. 2010;23(1):110-114.
[14] Ma R, Gao H, Lu Y. Global structure of radial positive solutions for a prescribed mean curvature problem in a ball. J Funct Anal.2016;270:2430-2455.
[15] Herlea D-R, Precup R. Existence, localization and multiplicity of positive solutions to -Laplace equations and systems. Taiwan J Math.2016;20:77-89.
[16] Precup R, Rodríguez-López J. Positive solutions for discontinuous problems with applications to -Laplacian equations. J Fixed PointTheory Appl. 2018;20:1-17.
[17] Precup R, Pucci P, Varga C. Energy-based localization and multiplicity of radially symmetric states for the stationary p-Laplace diffusion.Complex Var Elliptic Equ. 2020;65:1198-1209.
[18] Precup R. Moser–Harnack inequality, Krasnosel’skiı type fixed point theorems in cones and elliptic problems. Topol Methods Nonlin Anal.2012;40:301-313.

2023

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