Elliptic Equations with Hardy Potential and Gradient-Dependent Nonlinearity
-
Konstantinos T. Gkikas
Abstract
Let
1 Introduction and Main Results
Let Ω be a
where
1.1 Background and Main Contributions
The boundary value problem for (E$\pm$) without Hardy potential, i.e.
admits a unique solution. Afterwards, equation (1.1) in measure framework was first considered by Gmira and Véron in [18] where they showed that boundary value problem for (1.1) is not always solvable for every measure boundary datum. Because of its applications in many areas, equation (1.1) with
We refer to [32] for the case when g depends only on
Equation
has been studied in various directions. Necessary and sufficient criteria in terms of capacities for the existence of a solution with measure boundary data were obtained in [9]. Singular solutions of (1.2) with
The case with Hardy potential has been intensively studied over the last decade. See e.g. Bandle, Moroz and Reichel [5], Bandle, Marcus and Moroz [4], Marcus and Nguyen [25], Gkikas and Véron [17], Marcus and Moroz [23], Nguyen [31], Gkikas and Nguyen [16]. In the aforementioned papers, the best constant in the Hardy inequality
is deeply involved in the analysis. It is well known that
Moreover, Brezis and Marcus [11, Remark 3.2] proved that, for any
admits a minimizer
We see from (1.3) and (1.4) that
Throughout the present paper, we assume that
This assumption implies the validity of the representation theorem which states that every positive
In order to investigate the boundary behavior of
Let
admits a unique solution which allows to define the
A sequence of domains
Definition 1.1.
Let
The boundary trace of u is denoted by
In [17, Proposition 2.34], Gkikas and Véron proved that if
A complete description of the structure of positive solutions to (1.6) with “plus sign” was established in [25, 17], and various existence results for (1.6) with “minus sign” were given in [15, 31] in connection to the critical exponent
In particular, it was proved that, when
Very recently, a thorough study of the boundary value problem
was carried out in [16], revealing that the value
is a critical exponent for the solvability of (1.8). This means that if
Motivated by the aforementioned works, in the present paper, we aim to investigate related issues for (E$\pm$). Main features of a boundary value problem for (E$\pm$) with measure are
the presence of the Hardy potential which blowups strongly at the boundary,
the dependence of the nonlinearity on both solution and its gradient,
rough data which cause the invalidity of some classical results.
The interplay between the features leads to new essential difficulties, hence complicates drastically the analysis and produces interestingly new phenomena.
Our contributions are the following.
We establish the existence of weak solutions of (E$\pm$) with prescribed boundary trace ν under sharp assumption on g. In particular, by using the standard approximation method, combined with the estimates of the Green kernel and the Martin kernel as well as their gradient [15], the sub- and supersolutions theorem and the Vitali convergence theorem, we show that, for every measure
We show sharp a priori estimates for singular solutions of (E$\pm$). This allows to study solutions with an isolated singularity. As a matter of fact, we show that there are two types of solutions with an isolated singularity of
Our results cover and refine most of the aforementioned works in the literature and provide a full understanding of equations with Hardy potential and gradient-dependent nonlinearity.
1.2 Main Results
First we are concerned with a boundary value problem for equations with absorption term of the form
Before stating the main results, let us give the definition of weak solutions of (P+ν).
Definition 1.2.
Let
and
where the space of test function
We notice that this definition is inspired by the definition in [17, Section 3.2]. For more properties of the space of test functions
Our first result is the existence of a weak solution of (P+ν) under an integral condition on g.
Theorem 1.3 (Existence).
Assume g satisfies
Then, for any
Remark 1.4.
We remark the following.
It is worth noticing that this theorem is established by developing the sub- and supersolutions method in [16], in combination with the Schauder fixed point theorem and the Vitali convergence theorem.
It seems infeasible to obtain the uniqueness in case of general nonlinearity; however, when
we are able to prove the comparison principle, which in turn implies the uniqueness. The method is delicate, relying on a regularity result (see Proposition 4.1), maximum principle (see Lemma 4.2), estimates on the gradient of subsolutions of a nonhomogeneous linear equation (see Lemma 4.4). We emphasize that this result is new even in the case without Hardy potential, i.e.
Theorem 1.5 (Comparison Principle).
Assume
Assume
Theorem 1.6.
Assume
Weak singularity. For any
(1.13)Then there exists a constant
Moreover,
(1.14)Furthermore, the mapping
Strong singularity. Put
(1.15)There exists a constant
Moreover,
(1.16)locally uniformly on the upper hemisphere
Let us discuss briefly the proof of Theorem 1.6. The main ingredients in the proof of convergence (1.14) are the estimates on the Green kernel (2.1) and the Martin kernel (2.2) and condition (1.11). From the monotonicity of the sequence
When
Theorem 1.7 (Absolute Continuity).
Assume
Let
If
If
Put
We note that, by [17, Propositions 2.17, 2.18], for any
In addition,
The following result provides a removability result for solutions with “zero boundary condition” on
Theorem 1.8 (Removability).
Assume
Then any nonnegative solution
such that
is identically zero.
Next we deal with the boundary value problem for equations with source term of the form
Weak solutions of (P$-$ν) are defined similarly to Definition 1.2.
Phenomena occurring in this case are different from those in the case of absorption nonlinearity. This is reflected in Theorem 1.9 which ensures the existence of a weak solution under a smallness assumption on the total mass of the boundary data.
In order to make the statement clear and lucid, we rewrite equation (P$-$ν) as
where ϱ is a positive parameter and
Theorem 1.9 (Existence Result for (P$-$ρν) in Subcritical Case).
Let
for some
This result is established by combining an idea in [34] and the Schauder fixed point theorem.
Remark 1.10.
It is easy to see that if
The next result provides sufficient conditions for the existence of a solution to (P$-$ρν) with
Theorem 1.11 (Existence Result for (P$-$ρν)).
Assume that
There exists a constant
Here
There exists a positive constant
(1.22)
Then there exists
where
Organization of the paper
In Section 2, we recall main properties of the first eigenvalue and the corresponding eigenfunction of
1.3 Notations
We list below some notations that we use frequently in the present paper.
For
Denote
We denote by
For
Throughout the paper,
The notation
Denote by
For
2 Preliminaries
We recall that, throughout the paper, we assume that
2.1 Eigenvalue and Eigenfunction
We recall important facts of the eigenvalue
2.2 Green Kernel and Martin Kernel
Denote by
When there is no ambiguity, we will drop the superscript Ω, i.e. we write
By [14, Theorem 4.11], it holds
Since (1.5) holds, by [3] and [17, Proposition 2.29], the Martin kernel
For estimates on the Green kernel and the Martin kernel of a more general Schrödinger operator, we refer to [21].
Next we recall estimates of Green kernel and Martin kernel in weak
and
Moreover, for any
Proposition 2.1 ([15, Proposition 2.4]).
The following statements hold.
Let
(2.5)Let
(2.6)
Proposition 2.2 ([16, Proposition A]).
The following statements hold.
Let
(2.7)where
Let
where
2.3 Linear Equations
The Green kernel and the Martin kernel play an important role in the study of the boundary value problem for the linear equation
Definition 2.3.
Assume
where
Proposition 2.4 ([15, Proposition 2.11]).
Assume that
For
A similar result has been established for (2.9).
Proposition 2.5 ([17]).
For any
where
and there exists a constant
Finally, we will need the following classical properties of
Proposition 2.6 ([29]).
There exists a positive constant
3 Nonlinear Equations with Subcritical Absorption
In this section, we establish the existence of a positive solution of (P+ν). The approach is based on a combination of the idea in [20], estimates on the Green kernel, the Martin kernel, their gradient and the Vitali convergence theorem.
Proof of Theorem 1.3.
We divide the proof into three steps.
Step 1. In this step, we assume that
Let
First we note that
In this step, we use the idea in [20] in order to construct a solution
which satisfies
Let
Recall that
We define an operator
by standard elliptic estimates, there exists a constant
Also, by (3.3) and standard elliptic estimates, there exists a positive constant
By using an argument similar to the proof of [16, Theorem B, Step 1], we can show that
is continuous and compact. Now set
Now we allege that
which is clearly a contradiction. Therefore,
As a consequence,
Step 2. In this step, we still assume that (3.1) holds. Let
This implies that there exists a subsequence, still denoted by
Furthermore,
Step 3. In this step, we drop condition (3.1). Set
satisfying
Then
Choosing
Now, by (3.9) and Proposition 2.2, we obtain
This, together with (3.10) and (2.6), implies
Similarly, we can show that
Next we prove that
For
(3.12)
Then, for
For any Borel set
Note that the first term on the right-hand side of (3.13) tends to 0 as
Then, by (3.13),
Therefore, the sequence
From (3.7), we deduce that
4 Absorption g ( u , | ∇ u | ) = | u | p | ∇ u | q
In this section, we assume
admits a positive weak solution.
Next we prove the following regularity result.
Proposition 4.1.
Assume
then
Proof.
Let
4.1 Comparison Principle
Lemma 4.2.
Let
Proof.
By Young’s inequality,
Now set
Next we state the comparison principle for (4.2).
Lemma 4.3.
Let
Then
Proof.
Suppose by contradiction that
By (4.4), we deduce that there exists
Let
Now note that
where
and
Since
Combining (4.5) and (4.6), we have
Hence, by the maximum principle,
In order to prove the comparison principle for (4.1), we need the following result.
Lemma 4.4.
[
16
, Lemma 4.5]
Let
Then, for any
Proof of Theorem 1.5.
Since
for any
Without loss of generality, we assume that
Let
which implies
Note that
Without loss of generality, we assume that
Put
By (4.7), we derive that
Let
Note that, for this choice of κ, we have
Using (4.8), Lemma 4.4 and Hölder’s inequality, we get
Since
By the above inequality and (4.9), we obtain
for some constant
This implies the existence of
Next we take
Lemma 4.5.
Let
where
Proof.
The proof is similar to that of [16, Lemma 4.6], and hence we omit it. ∎
4.2 Isolated Singularities
In this section, we assume the origin
Proposition 4.6.
Assume
locally uniformly in
Proof.
We split the proof into two steps.
Step 1. Let
Notice that there exists a constant
Let
As a consequence, there is a constant
Set
where
Gathering estimates (4.16)–(4.18) leads to, for
Moreover, we can choose
Step 2. For
Let
4.3 Weak Singularities
Proof of Theorem 1.6.
We use the same idea as in the proof of [16, Theorem F]. Let
This and (2.2) imply that
By proceeding as in the proof of (4.13), we obtain
It follows from (2.1), (4.21) and (4.22) that
Case 1:
Since p and q satisfy (1.11), it follows that
Combining (4.24), (4.25) and (2.2) yields
As a consequence,
Case 2:
where
Let
We first deal with the first term on the right-hand side of (4.28). By the definition of ϕ and the inequality (which follows from (4.27))
we deduce that there exists
Now we deal with the second term on the right-hand side of (4.28). Let
and let
Note that, by the choice of
Combining the above estimates, we deduce
Now note that
On the other hand,
By collecting the above estimates, we obtain
It follows from integration by parts, (4.30) and (4.31) that
Since
Combining (4.23), (4.28), (4.29), (4.32) and (4.33) implies that there exists a positive constant
such that
Since p and q satisfy (1.11), we deduce (4.26) from (4.34).
Thus, from (4.26) and (4.20), we obtain (1.14). Finally, the monotonicity comes from the comparison principle. ∎
4.4 Strong Singularities
Let
the spherical coordinates in
where
We look for a particular positive solution of
under the separable form
It follows from a straightforward computation that
where
Let
Notice that the eigenvalue
and the corresponding eigenfunction
Notice that equation (4.36) admits a unique positive solution with supremum 1, and if
We could have defined the first eigenvalue
By [12, Theorem 6.1], the infimum exists since
By (4.36), the following expression holds:
Indeed, since
Taking into account that
Denote
It is asserted below that condition (1.11) is sharp for the existence of a positive solution of (4.35).
Theorem 4.7.
Assume
If ( 1.11 ) holds and
Proof.
(i) By multiplying (4.35) by
Note that the second term on the left-hand side of (4.38) is nonnegative. Thus, if (1.11) does not hold, or equivalently
(ii) The proof is split into two steps.
Step 1: Existence. Set
Then the function
In the above estimates, we note that (1.11) implies
Let
We note that
We allege that there exists a positive constant
provided
For
Therefore, by the standard elliptic theory, there exist a function
Set
Let
Step 2: Uniqueness. Let
Since
Set
In the sequel, we consider
Put
Now let
Since
By applying (4.45) with
Now set
Now, since
By the definition of ψ and (4.43), we have
Here we note that if
By Young’s inequality, we deduce that
where C is the constant in (4.47) and
Gathering (4.47), (4.48) and (4.50) yields
where
which implies
Thus we have proved that
This means that
But
By the above inequality, the fact that
where
The result follows by exchanging the role of
5 Absorption g ( u , | ∇ u | ) = | u | p | ∇ u | q
Let us recall the following result in [25, 16].
Proposition 5.1.
Let
Lemma 5.2.
Assume
If
(5.1)where C depends only on N, μ,
If
(5.2)where C depends only on N, μ, p, Ω, ε and
Proof.
Since u is a nonnegative solution of (4.1) we have
(i) First we assume that
Observe that, for any
Consequently, by Young’s inequality, we can find a constant
By the above estimates, there is a positive constant
By (4.12), Proposition 5.1 and taking into account that
Therefore, by letting
By the dominated convergence theorem, we can send
This implies (5.1).
The proof of (5.2) follows by arguments similar to the proof of (5.1) (with
We recall below some notations concerning the Besov space (see e.g. [1, 33]). For
with norm
Then
with norm
These spaces are fundamental because they are stable under the real interpolation method developed by Lions and Petree. For
with norm
It is extended to open sets and then Borel sets by the fact that it is an outer measure.
Proof of Theorem 1.7.
Let
we denote by
Let
with
Case 1:
where
Following the arguments of the proof of [17, Lemma 3.9, (3.48)], we can obtain
We infer from (5.1) that
where the constant C depends on
Let
then there exists a sequence
This implies that
This implies that
Case 2:
where
Using (5.2) and the ideas of the proof of (5.9), we can obtain the inequality
where the constant C depends on
The rest of the proof follows by using an argument similar to the first case. ∎
Proposition 5.3.
Let
Proof.
If
Proof of Theorem 1.8.
In view of the proof of [17, Proposition A.2], we can obtain the estimates
where C depends on
Case 1. Assume that
Then there exists a sequence
where R is defined by (5.5). Then
Furthermore,
Proceeding as in the proof of [17, Theorem 3.10, (3.65)], we have
Using the expression of
where
By proceeding as in the proof of [17, Theorem 3.10, (3.75)], we can prove
The rest of the proof is similar to the proof of [16, Theorem J], and we omit it.
Case 2. Assume that
for ε as in statement (ii). Then we can obtain the desired result by combining the ideas in Case 1 of this theorem and in Case 2 of Theorem 1.7. ∎
6 Nonlinear Equations with Subcritical Source
In this section, we prove Theorem 1.9. We first establish an existence result for the case when g is smooth and bounded.
Lemma 6.1.
Let
admits a positive weak solution v satisfying
where
Proof.
We shall use the Schauder fixed point theorem to show the existence of a positive weak solution of (6.1). Define the operator
Fix
and
Step 1: Estimate the
Then, by (2.4), we have
With the above notations, we split
First we estimate
Next
We bound
Similarly, we can estimate
By combining (6.2)–(6.6), we obtain (assuming
where
Step 2: Estimate
This and (6.7) imply that
where
which in turn implies
where
Thus,
where
Therefore, if
Since
Step 3. We apply the Schauder fixed point theorem to our setting. By a standard argument, we can show that
Then
Proof of Theorem 1.9.
Let
We observe that
Set
Since
We show that
Next proceeding as in the proof of (3.11), we obtain that
This means u is a nonnegative weak solution of (P$-$ρν). Therefore,
which implies that
7 Nonlinear Equations with Supercritical Source
7.1 Capacities and Existence Results
In this subsection, we introduce the definition of some capacities and provide related results which will be employed to prove Theorem 1.11 in the next subsection.
For
For
Here
Let
Under these conditions, one can define the quasi-metric d by
and denote by
For
For
Proposition 7.1 ([19]).
Let
for any
The equation
For any Borel set
For any Borel set
The inequality
We point out below that
Proposition 7.2 ([9, Lemma 2.2]).
Next we give sufficient conditions for (7.3), (7.4) to hold.
Proof.
If
Indeed, in order to obtain (7.5), we consider four cases.
Case 1:
Case 2:
Case 3:
Set
Then there exist
Now, by (7.6), we get
and hence (7.5) follows.
Case 4:
Then the proof of (7.5) follows due to an argument similar to Case 3.
The rest of the proof can proceed as in the proof of [9, Lemma 2.3], and we omit it. ∎
We recall below the definition of the capacity associated to
Definition 7.4.
Let
Clearly, for any Borel set
Furthermore, by [1, Theorem 2.5.1], we have
for any compact set
Thanks to Propositions 7.2 and 7.3, we can apply Proposition 7.1 to obtain the following result.
Proposition 7.5.
Let
For any Borel set
The inequality
Recall the capacity
Since Ω is a bounded smooth domain in
for any
We then define the
where
The following result is obtained by the same argument as in the proof of [9, Proposition 2.9].
Proposition 7.6.
Let
7.2 Case g ( u , | ∇ u | ) = | u | p | ∇ u | q
Proof of Theorem 1.11.
We see that, under the assumption on p and q, from Proposition 7.5 and Proposition 7.6, conditions (i) and (ii) are equivalent. Therefore, we will prove the existence of a solution by assuming (ii). For
From (2.1), (2.2) and (7.1), we have
(7.7)
and
(7.8)
From (7.7) and (7.8), we obtain
Put
Then, by using (1.22), we deduce that there exists
Define
We can see that
Dedicated to Laurent Véron on his 70th birthday.
Funding source: Grantová Agentura České Republiky
Award Identifier / Grant number: GJ19-14413Y
Funding statement: P.-T. Nguyen was supported by Czech Science Foundation, project GJ19-14413Y.
A Barrier
In this section, we will provide a barrier which plays an important role. This barrier will have the same properties as the barrier in [17, Proposition 6.1]. Let
Proposition A.1.
Let
where b is a constant such that
Proof.
The proof is similar to that of [17, Proposition 6.1] with some minor modifications, and hence we omit it. ∎
B Case g ( u , | ∇ u | ) = | u | p + | ∇ u | q
In this section, we assume that
B.1 Absorption Case
This subsection is devoted to the study of the equation
When
admits a positive weak solution.
Theorem B.1.
Assume p and q satisfy (1.12). Let
Set
Lemma B.2.
Let
Lemma B.3.
Let p and q satisfy (1.12). Assume u is a positive solution of (B.1) in Ω such that (4.14) holds locally uniformly in
Theorem B.4.
Assume
Strong singularity. Put
Moreover,
locally uniformly on upper hemisphere
where
Theorem B.5.
Let
If
If
If
If
Theorem B.6.
Assume
Then any nonnegative solution
B.2 Source Case
Theorem B.7.
Assume
There exists a constant
There exists a positive constant
Then there exists
Acknowledgements
The authors wish to thank Professor L. Véron for useful discussions. The authors are also grateful to the anonymous referee for the valuable comments which help to improve the manuscript.
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Articles in the same Issue
- Frontmatter
- Preface for the Special Issue Honoring Laurent Véron
- A Liouville-Type Theorem for an Elliptic Equation with Superquadratic Growth in the Gradient
- Protection Zones in Periodic-Parabolic Problems
- Improved Interpolation Inequalities and Stability
- Positive Solutions for Systems of Quasilinear Equations with Non-homogeneous Operators and Weights
- Large Time Behavior of Solutions to the Nonlinear Heat Equation with Absorption with Highly Singular Antisymmetric Initial Values
- Fast and Slow Decaying Solutions of Lane–Emden Equations Involving Nonhomogeneous Potential
- A Note on the Sobolev and Gagliardo--Nirenberg Inequality when 𝑝 > 𝑁
- Quasilinear Riccati-Type Equations with Oscillatory and Singular Data
- On Removable Singularities of Solutions of Higher-Order Differential Inequalities
- Elliptic Equations with Hardy Potential and Gradient-Dependent Nonlinearity
- On the Two-phase Fractional Stefan Problem
- The Hopf Lemma for the Schrödinger Operator
- Pointwise Gradient Estimates in Multi-dimensional Slow Diffusion Equations with a Singular Quenching Term
- Strong Maximum Principle for Some Quasilinear Dirichlet Problems Having Natural Growth Terms
Articles in the same Issue
- Frontmatter
- Preface for the Special Issue Honoring Laurent Véron
- A Liouville-Type Theorem for an Elliptic Equation with Superquadratic Growth in the Gradient
- Protection Zones in Periodic-Parabolic Problems
- Improved Interpolation Inequalities and Stability
- Positive Solutions for Systems of Quasilinear Equations with Non-homogeneous Operators and Weights
- Large Time Behavior of Solutions to the Nonlinear Heat Equation with Absorption with Highly Singular Antisymmetric Initial Values
- Fast and Slow Decaying Solutions of Lane–Emden Equations Involving Nonhomogeneous Potential
- A Note on the Sobolev and Gagliardo--Nirenberg Inequality when 𝑝 > 𝑁
- Quasilinear Riccati-Type Equations with Oscillatory and Singular Data
- On Removable Singularities of Solutions of Higher-Order Differential Inequalities
- Elliptic Equations with Hardy Potential and Gradient-Dependent Nonlinearity
- On the Two-phase Fractional Stefan Problem
- The Hopf Lemma for the Schrödinger Operator
- Pointwise Gradient Estimates in Multi-dimensional Slow Diffusion Equations with a Singular Quenching Term
- Strong Maximum Principle for Some Quasilinear Dirichlet Problems Having Natural Growth Terms
