Abstract
It is proved that if the bounded function of coefficient in the following equation is positive in a region contained in Ω and negative outside the region, the sets shrink to a point as , and then the sequence generated by the nontrivial solution of the same equation, corresponding to , will concentrate at with respect to and certain -norms. In addition, if the sets shrink to finite points, the corresponding ground states only concentrate at one of these points. These conclusions extend the results proved in the work of Ackermann and Szulkin (2013) for case .
1. Introduction
We study a new concentration phenomenon for the following p-Laplacian equations: where is a smooth domain and , and , where if and if . If is unbounded, we assume additionally that .
And an assumption of is as follows.
The set contained in the neighborhood of zero has positive measure, and with the constant is independent of . Moreover, for each there exist constants and such that whenever and .
As it is known, is the only solution to (1) if for all . In addition, if is based on a bounded set of positive measures, it is clear that there exists a solution (see Theorem 1). Hence, without loss of generality, we assume that and let be such that on the ball and on and are the solutions to (1) associated with . Accordingly, the question is what happens to as . Furthermore, this phenomenon can be found in physics. For instance, considering the materials separately from positive or negative (see [1]), it corresponds to investigating the existence of bright or dark solitons.
Equations of these types have been studied extensively in many monographs and lectures (e.g., [2–10] for , [11–18] for general ). In [2], Byeon and Wang considered the standing wave solutions for the nonlinear Schrödinger equation: Thus, they needed only to discuss the function which satisfies and rewrote it in the following form: By a rescaling, it is transformed to
Let the zero set and be an isolated component of , and they distinguished three cases of to prove the concentration as . And then, in [3] by replacing with a fairly general class nonlinearity , they also obtained the concentration. Furthermore, in [4], Byeon and Jeanjean gave the almost optimal condition on for the concentration. Recently, in [19], different from above with the linearity term , Ackermann and Szulkin considered the concentration phenomenon in the nonlinearity; that is, . In contrast, by following the similar strategy as in [19], we first show that the concentration phenomenon also occurs in the general p-Laplacian equation. It seems that this concentration phenomenon was unknown earlier, but to some extent, it answers the question mentioned above.
This paper is organized as follows. In Section 2, we prove that the solutions to (1) concentrate at the origin in the and the -norm; in Section 3, concentration in the -norms for different is considered and Section 4 shows that the ground states only concentrate at one of these points when is positive in a neighbourhood of a finite number of points.
2. Concentration in the and
We begin with some notations.
Let and is an equivalent norm in (due to ). Set , and we abbreviate to sometimes. Moreover, denotes a ball.
Here we offer the existence result for (1).
Theorem 1. Suppose that satisfies the assumption above and ; then for all sufficiently large , there is a positive ground state solution to problem (1). Moreover, there exists a constant independent of , such that .
Proof. As in [19], let and
Suppose that is a minimizing sequence for , normalized by ; then is bounded. Hence, in and a.e. in (by choosing a subsequence). Note that on for large. The Rellich-Kondrachov Theorem and Fatous’s Lemma say that
Thus is a minimizer.
And then, the lagrange multiple rule implies that is a solution to (1) for some appropriate constant . Moreover, since may be replaced by , (and hence ). To show that , we note that satisfies
where . Since , it follows from the strong maximum principle (see [20, 21]) that .
If is a solution to (1), then, via multiplying the equation by , integrating by parts, and using the Sobolev inequality, one deduces that
hence, for some and all large .
The next step is to consider the property of the nontrivial solution to (1) and .
Lemma 2. Consider
Proof. We present an abridged version of the proof highlighting the main differences to that in [19]. It will be proved by contradiction. Assume in and in after passing to a subsequence. Multiplying (1) (with ) by , integrating by parts, and recalling that for each and , it holds that
If , in . It is a contradiction to given in Theorem 1.
Lemma 3. Consider
Proof. We prove it by contradiction as well. We may assume that in . Multiplying (1) (with ) by yields that
Due to Lemma 2 with , .
On the other hand, we have for
We may choose small such that the second integral on the right-hand side above is positive as . Then we get the contradiction as .
In the sequel, we study concentration of as . Let be given and be such that for and for .
Multiplying (1) (with ) by we obtain
namely,
Given , we have on supp, provided that is large enough. Hence for all such ,
where is a constant independent of . Since in according to Lemma 3, it follows from Hölder inequality that
So (20) implies
Theorem 4. Suppose that satisfies the assumption and . Let be a nontrivial solution to (1) and put . Then for every they hold that Moreover,
Proof. (23) and (24) can be easily obtained by (22). Note that From (23), one concludes that According to (16), we get This and (24) imply
3. Concentration in the -Norm
The next is to consider the concentration in other norms.
Theorem 5. Let denote a nontrivial solution to (1) for each . Suppose that the assumption holds and there exists , such that whenever , and there exists such that ; then one can get that(a), for all , , ;(b)if in can be chosen independently of , then , for every ;(c)for all , one has and (d)if , then for it holds that If the hypotheses in (b) are satisfied, then (30) also holds for this .
Proof. There is clearly a positive classical solution to the equation
In fact, by [22, 23], the radial solution satisfies the ordinary differential equation
Set and
and hence from
Note that in when . Due to the continuity of and the fact that as , there is such that on . Moreover, on . If is bounded, the maximum principle says that in (see [20, 21]). If is unbounded, by virtue of tending to as by construction, thus for any , we may pick such that in . Moreover, applying regularity theory to , we can get as . Now the same maximum principle is applied on , which implies that in all of . Letting , we obtain again. By analogy we obtain (take ); hence
Hence (a) follows from above arguments with the fact that is continuous in .
Next, the hypotheses in (b) imply that there is such that on for each large enough. Let be a positive solution to
Then the sequence is monotone decreasing, by using the maximum principle to on for every . Therefore, converges locally and uniformly to a nonnegative solution to (37) on . It follows from our hypotheses on and that is an entire solution to (37) by applying the argument as in [24]. And then, due to [25], . For another, the function dominates the solution on for some , as seen in the proof of (a). Thus, also converges to locally and uniformly in ; that is, .
For (c), we first consider the case . By interpolation inequality, we have the following estimate for solution :
Here , are independent of , and satisfies that
According to Lemma 2, it suffices to impose that or equivalent . This and (a) prove the case . And then, (38) and (a) yield ; hence for every as . Using (a) again we get (30).
Note that (38) implies (30) for , so case (d) is easily followed.
4. Concentration at Several Points
Now we assume that the function is positive in a neighbourhood of two distinct points (indeed, the following argument is also valid for any finite number of points in ). More precisely, we assume.
in a neighbourhood of , and there exists a constant such that for all . Moreover, for each there exist constants and such that whenever and .
As in Section 2, we put :
Theorem 6. Suppose satisfies and , and is a ground state solution to (1). Then, for large, concentrates at or . More precisely, for each we have by passing to a subsequence for or (but not for and ).
Remark 7. Note that, in view of the obvious modification of Theorem 4, the limits in (41) are if is replaced by . So if in (41), then concentration occurs at and if , it occurs at .
Proof. As in [19], we may assume that by renormalizing ( may not be a solution to (1), but we still have ). Let be a function such that on and on , , where is so small that and . Set , , and . Since supp and the conclusion of Theorem 4 remains valid after a modification, we have
First, we assume that and . By passing to a subsequence, we may assume that and . If , recalling that , we get a contradiction from the following inequality:
So or . If (say), then the second limit in (41) is for because supp. The first limit is as well, since is otherwise bounded away from for large , and we obtain a contradiction again from
Finally, suppose (the case is of course analogous); it passes to a subsequence for some when is large enough. Then a contradiction (44) holds for such because .
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The author is greatly indebted to Professor A. Szulkin for providing the report about the concentration for the case , which inspires them to consider concentration for the p-Laplacian equation. This paper is partially supported by the Science Foundation of Fujian Province (2012J05002), Post-Doctor Foundation of China Grant (2011M501074), and the Innovation Foundation of Fujian Normal University (IRTL1206).