On sign-changing solutions for (p,q)-Laplace equations with two parameters
-
Vladimir Bobkov
and
Mieko Tanaka
Abstract
We investigate the existence of nodal (sign-changing) solutions to the Dirichlet problem for a two-parametric family of partially homogeneous
1 Introduction
In this article, we study the existence and nonexistence of sign-changing solutions for the problem
where
Boundary value problems with a combination of several differential operators of different nature (in particular, as in ((${\mathrm{GEV};\alpha,\beta}$))) arise mainly as mathematical models of physical processes and phenomena, and have been extensively studied in the last two decades; see, e.g., [15, 30, 19, 13] and the references below. Among the historically first examples one can mention the Cahn–Hilliard equation [12] describing the process of separation of binary alloys, and the Zakharov equation [33, (1.8)] which describes the behavior of plasma oscillations.
Elliptic equations with the
The considered problem ((${\mathrm{GEV};\alpha,\beta}$)) attracts special attention due to its symmetric and partially homogeneous structure; cf. [31, 32, 28, 20, 34, 10, 4].
By developing the results of [31, 28, 20], the authors of the present article obtained in [10] a reasonably complete description of the subsets of the
The aim of the present article is to allocate and characterize the sets of parameters α and β for which problem ((${\mathrm{GEV};\alpha,\beta}$)) possesses or does not possess sign-changing solutions (see Figure 1). In this sense, this work can be seen as the second part of the article [10].
1.1 Notations and preliminaries
Before formulating the main results, we introduce several notations.
In what follows,
respectively, and
By a (weak) solution of ((${\mathrm{GEV};\alpha,\beta}$)) we mean function
for all
where
Notice that the supports of
Remark 1.
Any solution
Next, we recall several facts related to the eigenvalue problem for the Dirichlet r-Laplacian,
has a nontrivial (weak) solution.
Analogously to the linear case, the set of all eigenvalues of ((${\mathrm{EV};r,\lambda}$)) will be denoted as
The eigenvalue
The following lemma directly follows from the definition of
Lemma 2.
Assume that
Let
Let
Although the structure of
where
It is known [18] that
and it can be alternatively characterized as in [16]:
where the first eigenfunction
Finally, we introduce the notation for the eigenspace of
It is clear that
1.2 Main results
Let us state the main results of this article. We begin with the nonexistence of nodal solutions for ((${\mathrm{GEV};\alpha,\beta}$)).
Theorem 3.
Assume that
Then ((${\mathrm{GEV};\alpha,\beta}$)) has no nodal solutions.
In the one-dimensional case Theorem 3 can be refined as follows.
Theorem 4.
Let
In the case of general dimensions an additional information on hypothetical nodal solutions to ((${\mathrm{GEV};\alpha,\beta}$)) for
The case
Now we formulate the existence result for nodal solutions with a positive energy.
Let us define the following “lower” critical value depending on
where
and put
Theorem 5.
Let
Several main properties of the function
Next, we state the existence of negative energy nodal solutions for ((${\mathrm{GEV};\alpha,\beta}$)). Consider the “upper” critical value
where
Theorem 6.
Let
Evidently, if
One of the main ingredients for the proof of Theorem 6 is the result on the existence of three nontrivial solutions (positive, negative and sign-changing) to the problem with the
Theorem 6 can be refined as follows.
Theorem 7.
Assume that
Then ((${\mathrm{GEV};\alpha,\beta}$)) has a nodal solution u
satisfying
Remark 8.
In the one-dimensional case we have
Let us note that unlike the case of positive solutions, the structure of the set of nodal solutions for problem ((${\mathrm{GEV};\alpha,\beta}$)) is more complicated, and we are not aware of the maximality of the regions obtained in Theorems 5 and 6.
The article is organized as follows: In Section 2, we apply the method of the Nehari manifold in order to prove Theorem 5. In Section 3, by means of linking arguments and the descending flow method, we provide two general existence results which yield, in particular, Theorems 6 and 7. For the convenience of the reader we collect the proofs of the main theorems in Section 4. In Appendix A, we prove several additional facts on the relation between eigenvalues and eigenfunctions of the p- and q-Laplacians in the one-dimensional case. Finally, in Appendix B, we give a sketch of the proof of Theorem 13.
2 Nodal solutions with positive energy
The classical Nehari manifold for problem ((${\mathrm{GEV};\alpha,\beta}$)) is defined by
It can be readily seen that
These equalities bring us to the definition of the so-called nodal Nehari set for ((${\mathrm{GEV};\alpha,\beta}$)):
By construction,
Let us divide
Evidently,
2.1 Preliminary analysis
In this subsection, we mainly study the properties of the sets
Lemma 1.
Let
then
We start our consideration of the sets
Lemma 2.
Let
If
If
Proof.
Let us first prove assertion (i). Assume that
Let us introduce the following sets:
Obviously,
Lemma 3.
Let
If
If
Proof.
We give the proof of assertion (i). The second part can be proved analogously. Suppose, by contradiction, that
On the other hand, it is shown in [9, Proposition 4.2]
that the second eigenvalue
Comparing (2.4) and (2.5) (with
Lemmas 2 and 3 readily entail the following information about the emptiness of
Lemma 4.
If
Lemma 5.
Let
Let us now subsequently treat the emptiness and nonemptiness of
for each
We assume that
where
The main properties of
Lemma 6.
The following assertions hold:
Proof.
(i) If
Consider the set
It is known that for any
which gives the boundedness of
(ii) If
(iii) Let us fix an arbitrary
Recalling that
Combining (2.8) and (2.9), we obtain the inequality
(iv) Let
Therefore,
provided
(v) If
This means that
Suppose now that there exists
Consider now the set
where the admissible set
The main properties of
Lemma 7.
The following assertions hold:
Proof.
(i) It is easy to see that for any
(ii) If
where the equality follows from
(2.5) with
Assertions (iii) and (iv) can be proved in much the same way as in Lemma 6. ∎
For the further proof of the existence of nodal solutions to ((${\mathrm{GEV};\alpha,\beta}$)) in
where the admissible set
We put
First we give two auxiliary results.
Lemma 8.
Let
Then the following assertions hold:
Proof.
Obviously,
Proposition 9.
For any
Proof.
If
which means that
Remark 10.
The definition (1.7) of
This can be seen by testing
Consider now the critical value
The following lemma contains the main properties of
Lemma 11.
The following assertions hold:
- (2.14)
Proof.
(i) As stated in the proof of Lemma 3, we easily see that
Assertion (ii) can be proved as in Lemma 6.
(iii) Due to assertion (ii), it is sufficient to show that
According to Proposition 9,
for each
(iv)
Assume that
Suppose now that there exists
which completes the proof. ∎
In the sequel, it will be convenient to use the notation
Remark 12.
Due to Lemmas 6 and 11,
the definitions of
Remark 13.
In the one-dimensional case we have
Indeed, if
and, consequently, (2.16) holds.
2.2 Existence of positive energy nodal solutions
In this subsection, we prove the existence of nodal solutions in the set
First, we prepare the following auxiliary lemma.
Lemma 14.
Let
Proof.
Assume that
hold for all
Consequently, the weak lower semicontinuity and the assumption that
Combining this inequality with the fact that
From Remark 12 we know that
Theorem 15.
Let
Proof.
Assume
Let us now show that
since
Now, choosing an appropriate subsequence, we get
Let us show that
Indeed, since
This means that
According to (2.20),
Lemma 1 implies the existence of unique maximum points
The last inequality in this formula is due to the fact that
Lemma 16.
Let
2.3 Qualitative properties
In this subsection, we show that
any minimizer u of
Lemma 17.
Let
Proof.
Suppose, contrary to our claim, that there exists a minimizer
Assume, without loss of generality, that
Moreover, we may assume that
Now we will subsequently show a contradiction for each case.
(i) It is easy to see that
(ii) Since
(iii) Recall
Since
On the other hand, since
Consequently,
(iv) Consider a function
that is, w is also a minimizer of
On the other hand, since
for all
for each
3 Nodal solutions with negative energy
In this section, we provide the main ingredients for the proofs of Theorems 6 and 7.
3.1 Auxiliary results
Consider the set
where
Lemma 1.
Let
Proof.
Fix any
It is not hard to see that
Lemma 2.
Let
Proof.
Assume that
which implies the desired conclusion since
Lemma 3.
Assume
Proof.
Since
as
where the last inequality is obtained by Hölder’s inequality.
Hence,
On the other hand, for any
Therefore, passing to the limit as
for all
Lemma 4.
If
Proof.
Let
as
Remark 5.
For the reader’s convenience we show that
the operator
converges strongly in
which implies that
Recall the definition (1.9):
Lemma 6.
If
Proof.
Let
for any
In the one-dimensional case we can clarify the bounds for
Lemma 7.
Let
Proof.
Let
Note that the eigenspace
The following lemma ensues readily from the definition (3.2).
Lemma 8.
Let
Lemma 9.
Let
Proof.
Let
By the definition of
Note that by taking
Since
By the continuity of
Considering
Due to the compactness of
where
Note that
Since
for all
Evidently,
Let us show that
where we used that
since
for all
since
In the sequel, we will also need the following variant of the deformation lemma. We refer the reader to [14, Theorem 3.2] for the proof.
Lemma 10.
Let Ψ be a
If Ψ is even, then
That is,
3.2 General existence result via minimax arguments
In this subsection, we prove a result on the existence of an abstract nontrivial solution to ((${\mathrm{GEV};\alpha,\beta}$)). Let us emphasize that this result does not guarantee that the obtained solution is sign-changing. (However, it is shown in [10] that for sufficiently large α and β problem ((${\mathrm{GEV};\alpha,\beta}$)) has no sign-constant solutions).
Recall that we denote
Theorem 11.
Assume that
Proof.
Since
Case (i):
Let
Moreover, by the definition of
Since
On the other hand, noting that
whence
Case (ii):
Let
Recalling that
for all
Let
as
To obtain a converse estimate, we show that
as
see Lemma 8. Hence, we obtain a contradiction since
Remark 12.
Note that the proof of case (ii) gives more. Namely, if
3.3 General existence result via the descending flow
In the last part of this section, we use the descending flow method to provide an existence result for
Suppose that
Under (3.19), we define a
For simplicity, we denote the positive cone in
The following result can be proved by the same arguments as [27, Theorem 11]. For the reader’s convenience, we give a sketch of the proof in Appendix B.
Theorem 13.
Assume that the following conditions hold:
There exists
There exists
If, moreover, J is coercive on
and ν denotes the unit outer normal vector to
We say that
Applying Theorem 13 to a truncated functional corresponding to
Proposition 14.
Let
Proof.
Let
It is easy to see that
f is the Carathéodory function and
Define a corresponding truncated
Note that the boundedness of v in Ω implies the boundedness of f, and therefore I is coercive on
Using the density arguments (as in the proof of Lemma 9), we can obtain a path
for every
and hence
Therefore,
for sufficiently small
Thus, for such a small
As a result, according to Theorem 13,
we obtain a sign-changing critical point
which implies that
4 Proofs of the main results
In this section, we collect the proofs of our main results stated in Section 1.2.
Proof of Theorem 3.
Recall that any sign-changing solution of ((${\mathrm{GEV};\alpha,\beta}$)) belongs to
the nodal Nehari set
Proof of Theorem 5.
The desired conclusion follows directly from the combination of Theorem 15 and Lemmas 16 and 17. ∎
Proof of Theorem 6.
Note that problem ((${\mathrm{GEV};\alpha,\beta}$)) possesses an abstract nontrivial solution
by Theorem 11.
If u is a nodal solution, then we are done.
If u is a nontrivial nonnegative solution, then
Proof of Theorem 7.
If
Finally, we will prove the nonexistence result in the one-dimensional case.
Proof of Theorem 4.
Let
for any
Taking now v as a test function for (1.1), we arrive at
and hence we have equalities in
(4.1).
On the other hand, the simplicity of
Funding source: Japan Society for the Promotion of Science
Award Identifier / Grant number: 15K17577
Funding source: Ministerstvo Školství, Mládeže a Tělovýchovy
Award Identifier / Grant number: LO1506
Funding statement: This work was supported by JSPS KAKENHI grant number 15K17577. The work of the first author was also supported by the project LO1506 of the Czech Ministry of Education, Youth and Sports.
A Appendix A
In this section, we show some relations between eigenvalues and eigenfunctions of the p- and q-Laplacians in the one-dimensional case. Consider the eigenvalue problem
where
(It is not hard to see that
For the convenience of the reader we briefly prove that the first eigenfunctions
Lemma 1.
Let
Proof.
Suppose, by contradiction, that
for all
Using a Taylor series, we get
for sufficiently small
Next, we prove the main result of the section.
Lemma 2.
Let
Proof.
The first inequality is trivial because the first eigenvalue
Note that
Using the formulas
from [11, Proposition 3.1],
where
We will subsequently simplify (A.2), to obtain an easier sufficient condition. Note that, by definition,
Note that
where
Applying (A.3) to
Therefore, using the estimate
we arrive at
Thus, comparing the right-hand sides of (A.2) and (A.4), we get the following sufficient condition for the assertion of the lemma:
To prove this inequality, we first obtain an appropriate upper bound for its left-hand side. From [3, p. 8] we know that
since for all
Next, we will get a suitable lower bound for the right-hand side of (A.5).
Since
since
Let us consider three cases. Assume first that
of (A.6) is strictly increasing with respect to
Combining these facts, we prove that (A.5) holds for
Secondly, assume that
for all
Finally, we assume that
It is not hard to check that
which again implies (A.5).
Therefore, (A.5) holds for all
If we swap p and q in Lemma 2, then an opposite situation occurs.
Lemma 3.
Let
Proof.
The case
Note first that
and for
Therefore, (A.8) can be simplified as
Note that
we arrive at the following sufficient inequality:
At the same time,
However, (A.9) is obviously satisfied for any
B Appendix B: Sketch of the proof of Theorem 13
Let us consider a map
for
Lemma 1.
Let us define
for
By the standard calculations, we have the following facts (cf. [5, Lemmas 3.7 and 3.8] for details).
Lemma 2.
Let
Then similar arguments as in [27, Lemma 17] (see also [5, Lemma 4.1]) can be applied to prove the following result on the existence of a locally Lipschitz continuous pseudo-gradient vector field in order to produce an invariant descending flow with respect to the positive and negative cones
Lemma 3.
Let
For any
Here
Let
Then for any
If
There exists a constant
For every
Now, we will give the proof of Theorem 13.
Proof of Theorem 13.
Note first that the boundary of
Take
Denote by
It is known that
Choose a constant c satisfying
which implies that
Hence, the coercivity of J
guarantees the existence of
Therefore, if
by assertion (i) of Lemma 3 and (B.1).
Thus,
Assume that
Note that
there exists a sequence
and
for all
The cases
Combining now (B.2) with (B.1), we conclude that
Acknowledgements
The first author wishes to thank the Tokyo University of Science, where the main constructions and results of the article were obtained, for the invitation and hospitality.
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- Multiple solutions for an elliptic system with indefinite Robin boundary conditions
- New solutions for critical Neumann problems in ℝ2
- A fractional Kirchhoff problem involving a singular term and a critical nonlinearity
- Existence and non-existence of solutions to a Hamiltonian strongly degenerate elliptic system
- Characterizing the strange term in critical size homogenization: Quasilinear equations with a general microscopic boundary condition
- Nonlocal perturbations of the fractional Choquard equation
- A pathological example in nonlinear spectral theory
- Infinitely many solutions for cubic nonlinear Schrödinger equations in dimension four
- On Cauchy–Liouville-type theorems
- Maximal Lp -Lq regularity to the Stokes problem with Navier boundary conditions
- Besov regularity for solutions of p-harmonic equations
- The classical theory of calculus of variations for generalized functions
- On the Cauchy problem of a degenerate parabolic-hyperbolic PDE with Lévy noise
- Hölder gradient estimates for a class of singular or degenerate parabolic equations
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Critical and subcritical fractional Trudinger–Moser-type inequalities on
- Multiple nonradial solutions for a nonlinear elliptic problem with singular and decaying radial potential
- Quantization of energy and weakly turbulent profiles of solutions to some damped second-order evolution equations
- An elliptic system with logarithmic nonlinearity
- The Caccioppoli ultrafunctions
- Equilibrium of a production economy with non-compact attainable allocations set
- Exact behavior around isolated singularity for semilinear elliptic equations with a log-type nonlinearity
- The higher integrability of weak solutions of porous medium systems
- Classification of stable solutions for boundary value problems with nonlinear boundary conditions on Riemannian manifolds with nonnegative Ricci curvature
- Regularity results for p-Laplacians in pre-fractal domains
- Carleman estimates and null controllability of a class of singular parabolic equations
- Limit profiles and uniqueness of ground states to the nonlinear Choquard equations
- On a measure of noncompactness in the space of regulated functions and its applications
- p-fractional Hardy–Schrödinger–Kirchhoff systems with critical nonlinearities
- On the well-posedness of a multiscale mathematical model for Lithium-ion batteries
- Global existence of a radiative Euler system coupled to an electromagnetic field
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On the existence of a weak solution for some singular
- Choquard-type equations with Hardy–Littlewood–Sobolev upper-critical growth
- Clustered solutions for supercritical elliptic equations on Riemannian manifolds
- Ground state solutions for the Hénon prescribed mean curvature equation
- Quasilinear equations with indefinite nonlinearity
- Concentrating solutions for a planar elliptic problem with large nonlinear exponent and Robin boundary condition
- Retraction of: Concentrating solutions for a planar elliptic problem with large nonlinear exponent and Robin boundary condition
