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
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
About this paper
Journal
Mathematical Methods in the Applied Sciences
Publisher Name
Wiley
Print ISSN
0170-4214
Online ISSN
1099-1476
google scholar link
Paper in HTML form (preprint)
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 -Laplacian operators in a ball. In particular, -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 -Laplacian operators:
(1.1) |
where is the unit open ball in centered at the origin, the function is continuous and is such that is an increasing homeomorphism between two intervals and
The following particular cases are of much interest due to their corresponding models arising from physics:
(a) where (here ), when the left side in (1.1) is
involved in a nonlinear Darcy law for flows through porous media [3];
Looking for radial solutions of (1.1), that is, functions of the form with , the Dirichlet problem (1.1) reduces to the mixed boundary value problem
(1.2) |
Radial and nonradial solutions for the Dirichlet problem involving -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 -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 -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 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 -Laplacian and the Minkowski mean curvature operator.
Our basic assumptions are as follow:
- (Hϕ)
-
is an odd increasing homeomorphism such that
(1.3) - (Hf)
-
is continuous, with nonincreasing in for every and nondecreasing in for every .
Note that the homeomorphisms related to the -Laplacian for and the Minkowski mean curvature operator both satisfy condition (Hϕ). Contrarily, the bounded homeomorphisms with for example the one involved by the Euclidian mean curvature operator, are not convex on and thus they do not satisfy our assumption (Hϕ).
2 A Harnack type inequality
In the space of functions satisfying we consider the following norms:
Let satisfy (Hϕ) and denote
First we prove a Harnack type inequality for problem (1.2), given in terms of the norm , with .
Theorem 2.1.
Let be such that for all and on Then on
If in addition is nonincreasing on then
for every
Proof.
By assumption,
Hence is nondecreasing in Since one has and so on Thus
(2.1) |
Next
(2.2) |
Also
Since both and are nondecreasing, using (2.2) we have
Therefore
A similar result has been established in [17] for the particular case of the -Laplacian with , for which and . More exactly, it has been proved that
(2.5) |
where In this case, since by using Hölder’s inequality one has
a Harnack type inequality in terms of the max-norm can be immediately derived from (2.5), namely
It is an open problem to obtain an analogue result for more general homeomorphisms satisfying (Hϕ). At this moment we are only able to establish such an inequality in terms of a max-seminorm on For example, taking in (2.4), we have the following Harnack type inequality related to a seminorm on
Corollary 2.2.
Under the assumptions of Theorem 2.1, if for a fixed subinterval with one define in the semi-norm then
(2.6) |
3 Positive radial solutions
Recall that by a (nonnegative) solution of (1.2) we mean a function with for all , such that and (1.2) is satisfied. We will say that a nonnegative solution is positive if it is distinct from the identically zero function. Let be the Banach space of continuous functions and its positive cone It is not difficult to see that a nonnegative function is a solution of problem (1.2) if and only if is a fixed point of the operator given by
(3.1) |
Lemma 3.1.
The operator maps the cone into itself.
Proof.
Indeed, take and let us show that belongs to . Since is nonnegative, and, moreover, and so (see Theorem 2.1), that is is nonincreasing on Furthermore, by the monotonicity properties of imposed in H and the fact that is nonincreasing in , the composed function is nonincreasing in . Hence,
is nonincreasing in . Then Corollary 2.2 ensures that
Therefore, as claimed. ∎
Now, for any number consider the set
The operator being completely continuous, the set is bounded, so there is a number such that Define the operator by
Lemma 3.2.
If
(3.3) |
then the fixed point index
Proof.
Clearly is a convex closed set and is a compact map. Consider the homotopy given by
By (3.3), this homotopy is admissible and so
where the last equality is due to the normalization property of the fixed point index, since . ∎
Next, for a number consider the set
It is clear that is open in
Lemma 3.3.
Assume that there exists a function such that and
(3.4) |
Then
Proof.
Observe that . Thus (3.4) implies that
By the homotopy property of the fixed point index, one has
Finally, , since . ∎
Remark 3.1.
If the operator maps into itself, then and condition (3.4) reduces to
By using the previous fixed point index computations, we deduce the following existence result.
Lemma 3.4.
Proof.
One has
As a result
In addition since otherwise there would exist with that is
or equivalently where and Since and we arrived to a contradiction with (3.3). Therefore 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 , denote
If , denote
Theorem 3.5.
Assume that and conditions and are fulfilled. If there exist with , such that
(3.5) | ||||
(3.6) |
then problem (1.2) has at least one solution such that and .
Proof.
We shall apply Lemma 3.4. First, we show that (3.3) holds. Indeed, for with , by the monotonicity assumptions on we have that
and thus from (3.5),
Hence, for all with , which implies (3.3). In addition, on the basis of Remark 3.1, we can take .
Next, we prove that (3.4) holds for the following choice of : if ,
and, otherwise, for ,
Note that and . Assume that (3.4) does not hold. Then there exist with , and such that
In particular, since , one has
Since with , we have for all . Thus, by ,
that is,
which contradicts (3.6) for any . Note that, in case , one has the contradiction
Finally, the conclusion follows from Lemma 3.4. ∎
Remark 3.2 (Asymptotic conditions).
Obviously, if is a classical homeomorphism (), conditions (3.5) and (3.6) can be rewritten as
Hence, if we assume in addition that satisfies:
(3.7) |
then the existence of both positive numbers and is guaranteed under suitable asymptotic conditions about at zero and at infinity.
Note that assumption (3.7) holds in the case of the -Laplacian operator and so it is commonly employed in the literature, see for instance [6, 11].
Theorem 3.6.
Proof.
By (3.7), with , there exists so that
and thus there exists such that
Now, by (3.8), there exists (we may suppose ) such that
which implies that
Finally, taking , condition (3.6) is obtained.
On the other hand, condition
clearly implies the existence of a positive number satisfying (3.5) and such that .
Corollary 3.7.
Assume that , and holds. If
(3.9) |
then problem
(3.10) |
has at least one positive radial solution.
Proof.
We show the applicability of our theory with an example involving radial solutions of -Laplacian equations.
Example 3.8.
Consider the function given by
with and , which clearly satisfies condition .
On the other hand, it is worth to mention that in the case of a singular homeomorphism (i.e., with , ), condition (3.5) is trivially satisfied for large enough. Hence, in that case, we only need to ensure the existence of the number in order to obtain positive solutions for problem (1.2).
Let us assume in the rest of this section that is singular. We present an existence result inspired by those in [6].
Theorem 3.9.
Proof.
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 and condition holds. If
(3.12) |
then problem
(3.13) |
has at least one positive radial solution.
Proof.
Example 3.11.
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 satisfying (3.5)–(3.6). Nevertheless, a suitable computation of the fixed point index related to the operator 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 defined in (3.1).
Lemma 4.1.
Under the assumptions of Lemma 3.4, if in addition there exists a positive number with and
(4.1) |
then has at least three fixed points , and such that
Proof.
By Lemma 3.4, has a fixed point such that
Moreover, assumption (4.1) ensures that and thus the operator has a fixed point in , that is, . Since , one has , and so the properties of the fixed point index together with its computation in Lemma 3.3 imply
Therefore, the existence property of the fixed point index ensures that the operator has a third fixed point located in . ∎
As a consequence, we obtain a three solutions type result for problem (1.2).
Theorem 4.2.
Assume that and conditions and are fulfilled. If there exist with , such that
then problem (1.2) has at least three solutions , and such that
Remark 4.1.
Theorem 4.2 ensures the existence of at least two positive solutions, namely, and with the localizations above. Furthermore, if , then 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 and conditions and are fulfilled. In addition, suppose that there exists satisfying condition (3.6) and
(4.2) |
Then problem (1.2) has at least two positive solutions and such that and .
Proof.
Next we emphasize the multiplicity result in the remarkable particular cases of Dirichlet problems involving the -Laplacian and Minkowski mean curvature operators.
Corollary 4.4.
Assume that , and condition holds. In addition, suppose that there exists satisfying condition (3.6) and
Then problem (3.10) has at least two positive radial solutions.
Corollary 4.5.
Assume that and condition holds. In addition, suppose that there exists satisfying condition (3.6) and
Then problem (3.13) has at least two positive radial solutions.
Proof.
The conclusion follows from Corollary 4.3 with , . Since is singular, condition
is trivially satisfied. ∎
We illustrate the applicability of the previous multiplicity results with the following example.
Example 4.6.
We shall study the existence of two positive solutions for problem (1.2) with as above and being the singular homeomorphism given by , .
Clearly, satisfies and it is immediate to check that
Finally, taking and , it is a simple matter to see that satisfies condition (3.6) for any large enough (for instance, with ).
Therefore, Corollary 4.5 guarantees that problem
has at least two positive radial solutions for any provided that is sufficiently large.
Finally, the existence of infinitely many positive solutions for (1.2) is obtained if the nonlinearity has an oscillating behavior at zero or at infinity.
Corollary 4.7.
Proof.
Remark 4.2.
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
where . Observe that is continuous and, moreover, it is nondecreasing if .
Furthermore, since and , one has
and
Then the asymptotic condition (4.3) holds if
Therefore, Corollary 4.7 ensures that the corresponding problem (3.13) associated to this nonlinearity has a sequence of positive radial solutions such that as provided that
In particular, taking and , the previous inequalities reduce to
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.
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[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.
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[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.