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The estimation and profile of the critical value for a Schrödinger equation
Boundary Value Problems volume 2014, Article number: 202 (2014)
In this paper, we are concerned with the following Schrödinger problem: , , in , where is of class . The estimation and profile of the critical value of the corresponding functional is proved, which entails the relationship between the critical value on the balls and the least-energy value on the whole space. Our results are also true for three cases of the potential function .
The main subject of this paper is the following problem:
where is the Laplace operator. Compactness and noncompactness assumptions posed on the potential function are also discussed.
The nonlinear Schrödinger equation (1.1) serves as a model for various problems in physics. For the last 20 years, (1.1) has received considerable attention as its solutions seem both mathematically intriguing and scientifically useful. We would like to mention earlier results on the existence of entire solutions of Schrödinger type equations with or without potentials, which was studied in – (see references therein).
A more general form of nonlinearity, i.e.
is also studied by many authors. Kryszewski-Szulkin  considered the existence of a nontrivial solution of (1.2) in a situation when , are periodic in the x-variables, is superlinear at and , and 0 lies in a spectral gap of . In addition, if is odd in u, they proved that (1.2) has infinitely many solutions. The result from Bartsch-Wang  suggested that (1.2) should have one sign changing solution. Bartsch-Liu-Weth  further proved the existence of sign changing solutions of (1.2) in with superlinear and subcritical nonlinearity , and the number of nodal domains can be controlled. If is odd, they obtained an unbounded sequence of sign changing solutions (), and they have at most nodal domains.
For the power nonlinearity (), since , it is well known that this problem has a positive solution which goes to zero at infinity. This solution is, besides, radially symmetric around some point and unique up to translations; see  and . Moreover, the linearized equation around w is nondegenerate in the sense that the equation
has linear combinations of the functions as its only solutions which go to zero at infinity , . These facts are crucial in the formulation of a Lyapunov-Schmidt type procedure, which was first introduced by Floer-Weinstein  for the one-dimensional case, and then was extended by Oh ,  to higher dimensions. Ni-Wei  studied the critical value of the energy functional
of the classical singular perturbation equation () on Ω, where Ω is a bounded smooth domain in .
Throughout this paper the following hypotheses on and will be assumed.
(): , , as , uniformly in x.
(): There exists a number , with if , such that , where .
(): There exists a number such that , for all .
(): For any x, is nonincreasing in t.
(): and .
Under assumptions () ∼ () and (), there exists a least-energy ground state solution of (1.1).
Under assumptions () ∼ () and (), there exists a least-energy ground state of (1.1), which is radially uniqueness solution, with the corresponding least-energy value.
The assumptions () and () are adopted in , which means that the potential functions possess certain compactness conditions. () indicates that the x dependence is radially symmetric. For this case, is compactly embedded in ().
By standard variational arguments, the assumptions () ∼ () and () guarantee the results of Theorem 1.1, which can be proved by the traditional Minmax Theory.
In fact, the critical value of the energy functional
can be characterized as
The associated critical point actually solves (1.1) and is called a least-energy solution. It decays exponentially at infinity.
As far as we know, the most general result of uniqueness of (1.1) type is obtained by Serrin-Tang , which would guarantee radial uniqueness in (1.1) if additionally one assumes (). The proof of Theorem 1.2 is similar to the steps in ; we omit the details.
Now we define the energy functional of (1.1) on :
The main result of this paper is the following.
Under assumptions () ∼ () and (), the critical valueof the functionalsatisfies
whereis the least-energy value in Theorem 1.2or Remark 1.4, γ is defined as
where w is the unique solution in Theorem 1.2.
The assumptions () ∼ () and () can guarantee the existence of on the bounded domain . In fact, it can be proved by the Minmax Theory as in Remark 1.4.
Assuming the conditions () or () on , we can get a similar equality as (1.5).
(): There exists such that, for any ,
where m denotes the Lebesgue measure on .
(): , , and as .
The assumptions () and () are certain compactness conditions, listed in . () is a more general condition, which gives a compact embedding. For (), we have a compact embedding from in for .
(): , . is 1-periodic in each of .
Assumption () is periodic, i.e., the x-dependence is periodic. () means that has a bounded potential well in the sense that exists and is equal to .
A particularly interesting case is whether one can come to the same conclusion as Theorem 1.6 under the noncompact assumptions () and ().
Our theorems generalize the results in  to three cases of compactness potential function entailing a type of nonlinear Schrödinger equation. The existence of the least-energy ground state solution of (1.1) is essential. Our results show the relationship between the critical value on the balls and the least-energy value on the whole space. The estimation of the critical value can be used to locate the geometrical shape of the solution.
In this section, we give some preliminary lemmas, which will be adopted in the proof of the theorems.
Assume w is a solution of (1.1), andis a solution of
where γ is defined in (1.6).
The proof of the two equalities is similar, we only prove the latter. Let w be the unique positive solution of (1.1), then the function
satisfies the equation
Next we consider the solution of the equation
Note , ; then the solution of (2.3) is
We have , as . For ρ big enough,
where γ is defined in (1.6); then .
For the lower boundary estimation, given , consider the equation
Similarly to the computation of (2.3), we get, for ρ big enough, , and is a subsolution of the above equation. So . Therefore, for ρ big enough, , so . We conclude that . □
Let u be a solution of
Let v be a solution of
whereis a function of ρ. Then.
By computation is a solution of (2.5), and . Similarly, it can be checked that
is a solution of (2.6), and
So , i.e.. □
3 The estimation of the critical value
This section is devoted to the proof of Theorem 1.6.
Proof of Theorem 1.6
is the critical value of the functional , is the least-energy value of .
First we find the upper bound of . Let be the solution of the equation
where w is the solution of (1.1) in Theorem 1.2, then
Moreover, by the definition of ,
where (3.1) is used in the second inequality.
Similar to the computation in Lemma 2.1, we have
Next we find the lower bound of . Let be the solution of
For the second part in (3.8), by , for ,
Choose such that , and let in , then as , . Moreover, , as .
Next we consider , which gives the solution of
By Lemma 2.2, . And by Lemma 2.1 and (3.11),
So we conclude
The work was carried out by the author.
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The author is supported by China Postdoctoral Science Foundation (2014M551830) and the Nonlinear Analysis Innovation Team (IRTL1206) funded by Fujian Normal University.
The author declares to have no competing interests.
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Zeng, J. The estimation and profile of the critical value for a Schrödinger equation. Bound Value Probl 2014, 202 (2014). https://doi.org/10.1186/s13661-014-0202-7
- critical value
- Schrödinger equation
- potential function
- compactness and noncompactness condition