Existence and concentration behavior of solutions for a class of quasilinear elliptic equations with critical growth
-
Kaimin Teng
and
Xiaofeng Yang
Abstract
In this paper, we study a class of quasilinear elliptic equations involving the Sobolev critical exponent
where
1 Introduction
In this paper, we are concerned with the existence, multiplicity and concentration behavior of solutions of the following quasilinear elliptic equations involving the Sobolev critical exponent:
where
where
When κ=1 and
where
When
The existence and concentration behavior of positive solutions of the above equation have been extensively investigated under various hypotheses on the potential
In recent years, many scholars have been interested in the study of the existence and multiplicity of solutions for equation (1.4). For example: the existence of a positive ground state solution has been obtained in [33] by using a constrained minimization argument which gives a solution of (1.4) with an unknown Lagrange multiplier λ in front of the nonlinear term. In [30], the existence of both one-sign and nodal ground states of soliton type solutions were established by the Nehari manifold method. In [29], Liu, Wang and Wang developed the methods of change of variables such that the quasilinear problem reduces to a semilinear one. They used an Orlicz space framework to prove the existence of positive solutions of (1.4) by the mountain pass theorem. Meanwhile, Colin and Jeanjean [14] developed the dual methods to treat the quasilinear problem (1.5), and the usual Sobolev space
Regarding critical problems, there are also some important results appearing in the literature. For example, in [24], the authors established the existence, multiplicity and concentration behavior of ground states for quasilinear Schrödinger equation with critical growth
by using the variational methods and combining them with the theory of the Ljusternik–Schnirelman category which was used by Alves, Figueiredo and Severo [2] to establish the existence and multiplicity of nontrivial weak solutions for quasilinear elliptic equations of the form
Yang and Ding [40] applied the perturbed methods to consider the following critical quasilinear Schrödinger equation:
and showed the existence of positive solutions as
Unlike [24, 2, 40], where the minimum of
where the potential
Motivated by the above-cited papers, the main purpose of this paper is to establish the existence, multiplicity and concentration behavior of the ground states for the quasilinear elliptic equation with critical growth
where
Assume that the nonlinearity
there exists
the function
there exist
As far as we know, little work has been done for the existence and concentration behavior of positive solutions for the quasilinear problem (Pe) where the nonlinearity has a critical growth. Our main result complements the corresponding conclusion of [2] and extends the main result of [24]. Alves, Figueiredo and Severo [2], He, Qian and Zou [24] and Wang and Zou [38] chose the Sobolev space E which is defined by
equipped with the norm
The Orlicz–Sobolev space E may not be reflexible for
For stating our main result, we set
and
In view of (V), the set M is compact. We recall that, if Y is a closed subset of a topological space X, the Ljusternik–Schnirelman category
Theorem 1.1.
Suppose that conditions (V) and (H
The paper is organized as follows: In Section 2, we present the abstract framework of the problem as well as some preliminary results. In Section 3, we show the existence of ground state solution for autonomous problem. Section 4 is devoted to the proof of Theorem 1.1.
2 Variational framework and preliminary results
Formally, the energy functional associated to (Pe) is defined by
where
and
The following properties were proved in [35].
Lemma 2.1.
The following properties involving f and its derivative hold:
f is a uniquely defined
the function
there exists a positive constant C such that
Proposition 2.2.
The following properties involving f and h hold:
Proof.
(1) By computation, we have
Since
(2) By Lemma 2.1 (4) and (7), we have that
(3) Since
by using conclusion (2) and (H
(4) By Lemma 2.1 (4) and (7), we have that
(5) From (H
as desired. ∎
Under the change of variables, we can rewrite the functional I defined by (2.1) in the following form:
which is well defined on the Banach space
endowed with the norm
In view of conditions (H
for all
We observe that if
for all
3 Autonomous problem
In this section, we will study the existence of a positive ground state solution for the following equation:
where μ is an arbitrary positive constant and
which is well defined on the Banach space
From the hypotheses (H
for all
Let us denote the Nehari manifold associated to (Qm) by
3.1 Mountain pass geometry
Theorem 3.1 ([36]).
Let E be a real Banach space and
there is
Then J possesses a
where
Consider the set
Since
Lemma 3.2.
Suppose that conditions (V) and (H
Proof.
By the hypotheses (H
By (3.1), Lemma 2.1 (10) and the Sobolev inequality, we have that
where
Similar to the proofs of [2, Lemma 3.2 and 3.3], we can deduce the following two Lemmas.
Lemma 3.3.
Suppose that (V), (H
if
if
Lemma 3.4.
For every
From Lemmas 3.2–3.4 it follows that
where
and
Therefore, by Theorem 3.1, there exists a
Now, we will give the detailed properties of the above
Lemma 3.5.
Let
Proof.
(1) By (3.3), (H
which implies that
On the other hand, using Lemma 2.1 (9) and the Sobolev inequality, we have that
Therefore,
Moreover, using [27, Theorem 1.6], we can get
Indeed, by translation, equation (Qm) is reduced to
Let
where
(2) Since
thus we need to show that the following limits hold:
for all
By (3.5), it is easy to show that (3.6) holds by the weak convergence argument.
Let
where
By the Hölder inequality, we have
From the Lebesgue dominated convergence theorem it follows that
By (3.4), we deduce that
Take
Since
Hence (3.9) is proven.
Consequently, we obtain
Meanwhile, if
for any
(3) Since
Thus
for some
Lemma 3.6.
Let
there exist
Proof.
Assume that (2) dose not occur, that is, for all
By the vanishing Lemma in [28], we can assume
Next, since
and thus
that is,
Denote by
Assume that
thus,
which yields a contradiction because
For the least energy level
Lemma 3.7.
For any
where S denotes the best constant for the embedding
Proof.
Define a functional
By the equivalent characteristic of
From Lemma 3.3 we know that
Fix
where
and
where
By computations,
and
Obviously,
It is clear that
We have
so
Since
which attains its unique global maximum at
Thus, by (3.12) and (3.13), using the fact that
for
3.2 The existence of the ground state solution for (Qm)
Now, we are able to prove the existence of positive ground state solution for problem (Qm).
Theorem 3.8.
Suppose that conditions (H
Proof.
From Section 3.1, we know that
From Lemma 3.5, up to a subsequence, there is a
From
Under the assumptions (H
Therefore, we conclude that
which is a contradiction with
Define
Then
so we can assume that
and
From (H
Combining (3.14)–(3.15) with Fatou’s Lemma, we obtain
Then
Now we show that v is nonnegative since
for all
implies that
Next, we will prove the
For any
and
with
By (H
Choose τ sufficiently small; by Lemma 2.1 (10), we have that
For every
where we have chosen
On the other hand, by the Sobolev inequality and the Hölder inequality, we have
Combining (3.17) with (3.18), we obtain
Let
From the definition of
Since
Therefore,
Using Fatou’s Lemma in the variable l, we get
Next, we note that if
then
By (3.19) and the Hölder inequality, we have that
where we have used
Using Fatou’s Lemma, we obtain
If we take
Setting
Note that
Letting
Therefore, for any
4 The nonautonomous problem
In this section, we will study the following problem (which is equivalent to (Pe)), which can be obtained under the change of variable
The functional
which is well defined on the Banach space
endowed with the norm
Obviously,
for all
Set
We first show that the Nehari manifold
Lemma 4.1.
There exists a constant
Proof.
By Lemma 2.1 (10), the Hölder inequality and the Sobolev inequality, we have that
where
By Lemma 2.1 (7) and the Sobolev inequality, we have
Thus, for
Since
for all
It is easy to check, by arguing as in Section 3, that
and thus
Similar to the proof Lemma 3.4, there is a unique
Similar to the proof of Lemma 3.6, we can characterize the
Lemma 4.2.
Let
there exist
Lemma 4.3.
Suppose that
Proof.
Let
Also since
Given
This together with Proposition 2.2 (1) and (2) and the boundedness of
If
Define
Let
for any
Next, we distinguish the following two cases.
Case 1:
By the boundedness of
Using the mean value theorem and (H
where τ is between 1 and
and
Combining (4.8) with (4.9), we obtain the following inequality:
Letting
Case 2:
for
Letting
4.1 Compactness condition
Lemma 4.4.
Let
where
Proof.
Firstly, we show that the following equalities hold:
for all
From this and Young’s inequality, for each
Define
which verifies that
Hence, by Lebesgue’s theorem, we have
By the definition of
Thus, we get
which implies that
and (4.14) follows.
Secondly, similar to the proof of (3.6) and from [10, Brezis–Lieb Lemma], we can deduce that
Finally, by (4.10)–(4.16) and (3.6)–(3.9), we obtain
and
for all
Lemma 4.5.
The functional
Proof.
Let
By the boundedness of
From (H
Since
In order to apply the Ljusternik–Schnirelman category theory, we need the functional
Lemma 4.6.
The Nehari manifold
Proof.
Observe that
By
Let
for
From Proposition 2.2 (3) and
Thus
Therefore, by (4.17), (4.18) and Lemma 2.1 (4) and (7), we have that
∎
Lemma 4.7.
The functional
Proof.
Let
For
and then we have that
By (4.19), we see that
as
and so
By a similar argument, or using Lemma 4.6, we get the following corollary.
Corollary 4.8.
The critical points of
4.2 The existence of the ground state solution for (Pe*)
Theorem 4.9.
Suppose that conditions (V) and (H
Proof.
From the above statement, we know that
Without loss of generalization, we may assume that
This achieves a contradiction. Since
Therefore, for all
which implies that
By Lemma 4.5, there exists a
That is,
4.3 Multiplicity of solutions to (Pe*)
In this subsection, we will study the multiplicity of solutions and study the behavior of its maximum points concentrating on the set M of global minima of V given in Section 1. The main result of this section is equivalent to Theorem 1.1 and it can be restated as follows.
Theorem 4.10.
Suppose that conditions (V) and (H
To prove Theorem 4.10, we fix some notation and give some preliminary lemmas. Fix
and
By construction,
For any
Lemma 4.11.
The function
and
Proof.
The proof of this lemma can be found in [2]; we omit its proof. ∎
Lemma 4.12 (A compactness lemma).
Let
there exists a sequence
Proof.
Since
Using the boundedness of
where
So without loss of generality, we may suppose that
Case 1:
We claim that equality holds in the last inequality. Otherwise, using the fact that
are nonnegative functions for
which leads to a contradiction. So we obtain
Combining (4.20) with
By an argument similar to (4.20), we also have that
Hence, up to a subsequence still denoted by
where we have used Lemma 2.1 (4) and the fact that
which implies that
Hence, up to a subsequence, there exists
Using the convexity of
Since
Therefore, by Lemma 2.1 (9), the Sobolev inequality, (4.20), and (4.21), we deduce that
Consequently, we conclude that
Case 2:
Let
Lemma 4.13.
Let
Proof.
In view of
Let
Hence,
It remains to show that
which gives a contradiction. Thus,
Let
Lemma 4.14.
Let
Proof.
Let
Thus it suffices to find a sequence
To obtain this sequence, we note that
This implies that
Recalling that
The following Lemma plays a fundamental role in the study of the behavior of the maximum points of the solutions.
Lemma 4.15.
Suppose that conditions (V) and (H
where
Proof.
We only replace v by
Lemma 4.16.
Under the hypotheses of Lemma 4.15, there exists
Proof.
Suppose by contradiction that
Thus, by Lemma 2.1 (4) and (9), for n large enough, we have
which implies that
4.4 Proof of Theorem 4.10
Proof.
For a fixed
Since
The remaining proof of concentration behavior can be deduced by using Lemma 4.15 and Lemma 4.16 and its proof is a standard argument; we refer the interested readers to [1, 2, 24]. We omit its details here. ∎
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11501403
Funding source: Natural Science Foundation of Shanxi Province
Award Identifier / Grant number: 2013021001-3
Funding statement: This work is supported by the NSFC (No. 11501403) and the Natural Science Foundation of Shanxi Province for Youths (No. 2013021001-3).
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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
