It is proved that if the bounded function of coefficient Qn in the following equation -div{|∇u|p-2∇u}+V(x)|u|p-2u=Qn(x)|u|q-2u,u(x)=0asx∈∂Ω.u(x)⟶0as|x|⟶∞ is positive in a region contained in Ω and negative outside the region, the sets {Qn>0} shrink to a point x0∈Ω as n→∞, and then the sequence un generated by the nontrivial solution of the same equation, corresponding to Qn, will concentrate at x0 with respect to W01,p(Ω) and certain Ls(Ω)-norms. In addition, if the sets {Qn>0} shrink to finite points, the corresponding ground states {un} only concentrate at one of these points. These conclusions extend the results proved in the work of Ackermann and Szulkin (2013) for case p=2.

1. Introduction

We study a new concentration phenomenon for the following p-Laplacian equations:
(1)-div{|∇u|p-2∇u}+V(x)|u|p-2u=Qn(x)|u|q-2u,u(x)=0asx∈∂Ω.u(x)⟶0as|x|⟶∞,
where Ω⊂RN is a smooth domain and V⩾0(∈L∞(Ω)), and p<q<p*, where p*:=Np/(N-p) if N⩾p and p*:=∞ if N<p. If Ω is unbounded, we assume additionally that σ(-div(|∇·|p-2∇·)+V|·|p-2·)⊂(0,∞).

And an assumption of Qn is as follows.

(*) The set {x∣Qn(x)>0} contained in the neighborhood of zero has positive measure, and |Qn|L∞(Ω)⩽C with the constant C is independent of n. Moreover, for each ϵ>0 there exist constants δϵ(>0) and Nϵ such that Qn⩽-δϵ whenever x∉Bϵ(0) and n⩾Nϵ.

As it is known, u≡0 is the only solution to (1) if Qn(x)⩽0 for all x∈Ω. In addition, if Qn(x)>0 is based on a bounded set of positive measures, it is clear that there exists a solution u≢0 (see Theorem 1). Hence, without loss of generality, we assume that 0∈Ω and let Q=Qn be such that Qn>0 on the ball B1/n(0) and Qn<0 on Ω∖B2/n(0) and un≢0 are the solutions to (1) associated with Qn(x). Accordingly, the question is what happens to un as n→∞. Furthermore, this phenomenon can be found in physics. For instance, considering the materials separately from Q positive or negative (see [1]), it corresponds to investigating the existence of bright (Q>0) or dark (Q<0) solitons.

Equations of these types have been studied extensively in many monographs and lectures (e.g., [2–10] for p=2, [11–18] for general p). In [2], Byeon and Wang considered the standing wave solutions ψ(x,t)≡exp (-iEt/ħ)v(x) for the nonlinear Schrödinger equation:
(2)iħ∂ψ∂t+ħ22Δψ-V(x)ψ+|ψ|p-1ψ=0,(t,x)∈R×RN.
Thus, they needed only to discuss the function v which satisfies
(3)ħ22Δv-(V(x)-E)v+|v|p-1v=0,x∈RN,
and rewrote it in the following form:
(4)ϵ2Δv-V(x)v+vp=0,v>0,x∈RNlim|x|→0v(x)=0.
By a rescaling, it is transformed to
(5)Δu-V(ϵx)u+up=0,u>0,x∈RNlim|x|→0u(x)=0.

Let the zero set Z≜{x∈RN∣V(x)=0} and A be an isolated component of Z, and they distinguished three cases of A to prove the concentration as ϵ→0. And then, in [3] by replacing vp with a fairly general class nonlinearity f(v), they also obtained the concentration. Furthermore, in [4], Byeon and Jeanjean gave the almost optimal condition on f for the concentration. Recently, in [19], different from above with the linearity term V(ϵx)u, Ackermann and Szulkin considered the concentration phenomenon in the nonlinearity; that is, -Δu+V(x)u=Qn(x)|u|p-2u. 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 {un} to (1) concentrate at the origin in the W01,p(Ω) and the Lq-norm; in Section 3, concentration in the Ls-norms for different s is considered and Section 4 shows that the ground states only concentrate at one of these points when Qn is positive in a neighbourhood of a finite number of points.

Let E:=W01,p(Ω) and
(6)∥u∥:=(∫Ω(|∇u|p+V|u|p)dx)1/p
is an equivalent norm in E (due to σ(-div(|∇·|p-2∇·)+V|·|p-2·)⊂(0,∞)). Set
(7)|u|s,A:=(∫A|u|sdx)1/s,|u|∞,A=esssupA|u|, and we abbreviate |u|s,A to |u|s sometimes. Moreover,
(8)Br(a):={x∈Rn:|x-a|<r}
denotes a ball.

Here we offer the existence result for (1).

Theorem 1.

Suppose that Qn satisfies the assumption (*) above and q∈(p,p*); then for all sufficiently large n, there is a positive ground state solution un∈E to problem (1). Moreover, there exists a constant α>0 independent of n, such that ∥un∥⩾α.

Proof.

As in [19], let Jn(v)=∫ΩQn|v|qdx and
(9)sn:=infJn(v)>0∥v∥p|Jn(v)|p/q=infJn(v)>0∫Ω|∇v|p+V|v|pdx(∫ΩQn|v|qdx)p/q.
Suppose that (vk) is a minimizing sequence for sn, normalized by Jn(vk)=1; then ∥vk∥ is bounded. Hence, vn⇀v in E and vk(x)→v(x) a.e. in Ω (by choosing a subsequence). Note that Qn<0 on |x|>1 for n large. The Rellich-Kondrachov Theorem and Fatous’s Lemma say that
(10)sn=limk→∞∥vk∥p=limk→∞∥vk∥p(∫|x|<1Qn|vk|qdx+∫|x|>1Qn|vk|qdx)p/q⩾∥v∥pJn(v)p/q⩾sn.
Thus v is a minimizer.

And then, the lagrange multiple rule implies that un=cnvn is a solution to (1) for some appropriate constant cn>0. Moreover, since vn may be replaced by |vn|, vn⩾0 (and hence un⩾0). To show that un>0, we note that un satisfies
(11)-div(|∇v|p-2∇v)+(V(x)unp-2+Qn-(x)un(x)q-2)v=Qn+(x)un(x)q-1⩾0,
where Qn±:=max{±Qn(x),0}. Since V(x)unp-2+Qn-(x)un(x)q-2⩾0, it follows from the strong maximum principle (see [20, 21]) that un>0.

If un≠0 is a solution to (1), then, via multiplying the equation by un, integrating by parts, and using the Sobolev inequality, one deduces that
(12)∥un∥p=∫ΩQn|un|qdx⩽c1|un|qq⩽c2∥un∥q;
hence, ∥un∥⩾α for some α>0 and all large n.

The next step is to consider the property of the nontrivial solution {un} to (1) and wn:=un/∥un∥.

Lemma 2.

Consider
(13)∥un∥⟶∞asn⟶∞.

Proof.

We present an abridged version of the proof highlighting the main differences to that in [19]. It will be proved by contradiction. Assume un⇀u in E and un→u in Llocq(Ω) after passing to a subsequence. Multiplying (1) (with u=un) by un, integrating by parts, and recalling that Qn<0 for each ϵ>0 and n⩾Nϵ, it holds that
(14)limsupn→∞∥un∥p=limsupn→∞∫ΩQn|un|qdx⩽limsupn→∞∫|x|<ϵQn|un|qdx⩽c∫|x|<ϵ|u|qdx.

If ϵ→0, un→0 in E. It is a contradiction to ∥un∥⩾α>0 given in Theorem 1.

Lemma 3.

Consider
(15)wn⇀0inEasn⟶∞

Proof.

We prove it by contradiction as well. We may assume that wn⇀w(≢0) in E. Multiplying (1) (with u=un) by un/∥un∥p yields that
(16)1=∥wn∥p=∥un∥q-p∫ΩQn|wn|qdx.
Due to Lemma 2 with q>p, ∫ΩQn|wn|q→0.

On the other hand, we have for 0<ϵ<ϵ1(17)0=limn→∞∫ΩQn|wn|qdx=limn→∞(∫|x|<ϵQn|wn|qdx+∫|x|>ϵQn|wn|qdx)⩽limn→∞(∫|x|<ϵQn|wn|qdx+∫|x|>ϵ1Qn|wn|qdx)⩽c∫|x|<ϵ|wn|qdx-δϵ1∫|x|>ϵ1|wn|qdx.
We may choose small ϵ1 such that the second integral on the right-hand side above is positive as w≢0. Then we get the contradiction as ϵ→0.

In the sequel, we study concentration of {un} as n→∞. Let ϵ>0 be given and χ∈C∞(Ω,[0,1]) be such that χ(x)=0 for x∈Bϵ/2(0) and χ(x)=1 for x∉Bϵ(0).

Multiplying (1) (with u=un) by χun we obtain
(18)∫Ω(|∇un|p-2∇un·∇(χun)+χVunp)dx=∫ΩχQn|un|qdx,
namely,
(19)∫Ωχ(|∇un|p+Vunp)dx-∫ΩχQn|un|qdx=-∫Ω|∇un|p-2∇un·∇χ·undx.
Given ϵ>0, we have Qn⩽-δϵ on suppχ, provided that n is large enough. Hence for all such n,
(20)0⩽∫Ω∖Bϵ(0)(|∇un|p+Vunp)dx+δϵ∫Ω∖Bϵ(0)|un|qdx⩽∫Ωχ(|∇un|p+Vunp)dx-∫ΩχQn|un|qdx=-∫Ω|∇un|p-2∇un·∇χ·undx⩽dϵ∫Bϵ(0)∖Bϵ/2(0)|un||∇un|p-1dx,
where dϵ is a constant independent of n. Since wn=un/∥un∥→0 in Llocp(Ω) according to Lemma 3, it follows from Hölder inequality that
(21)∫Bϵ(0)∖Bϵ/2(0)|wn||∇wn|p-1dx⟶0.
So (20) implies
(22)∫Ω∖Bϵ(0)(|∇wn|p+Vwnp)dx+δϵ∥un∥q-p∫Ω∖Bϵ(0)|wn|qdx=0.

Theorem 4.

Suppose that Qn satisfies the assumption (*) and q∈(p,p*). Let un be a nontrivial solution to (1) and put wn=un/∥un∥. Then for every ϵ>0 they hold that
(23)limn→∞∫Ω∖Bϵ(0)(|∇wn|p+Vwnp)dx=0,(24)limn→∞∥un∥q-p∫Ω∖Bϵ(0)|wn|qdx=0.
Moreover,
(25)limn→∞∫Ω∖Bϵ(0)(|∇wn|p+Vwnp)dx∫Ω(|∇wn|p+Vwnp)dx=0,limn→∞∫Ω∖Bϵ(0)|wn|qdx∫Ω|wn|qdx=0.

Proof.

(23) and (24) can be easily obtained by (22). Note that
(26)∫Ω(|∇wn|p+Vwnp)dx=∥wn∥p=1.
From (23), one concludes that
(27)limn→∞∫Ω∖Bϵ(0)(|∇un|p+Vunp)dx∫Ω(|∇un|p+Vunp)dx=limn→∞∫Ω∖Bϵ(0)(|∇wn|p+Vwnp)dx∫Ω(|∇wn|p+Vwnp)dx=0.
According to (16), we get
(28)c∥un∥q-p∫Ω|wn|qdx⩾∥un∥q-p∫ΩQn|wn|qdx=∥wn∥p=1.
This and (24) imply
(29)limn→∞∫Ω∖Bϵ(0)|wn|qdx∫Ω|wn|qdx=limn→∞∥un∥q-p∫Ω∖Bϵ(0)|wn|qdx∥un∥q-p∫Ω|wn|qdx=0.

3. Concentration in the <inline-formula>
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The next is to consider the concentration in other norms.

Theorem 5.

Let un denote a nontrivial solution to (1) for each n∈N. Suppose that the assumption (*) holds and there exists R, λ>0 such that V⩾λ whenever x∈Ω∖BR(0), and there exists ϵ>0 such that Bϵ(0)¯⊂Ω; then one can get that

∃C, for all s∈[1,∞], n∈N, |un|s,Ω∖Bϵ(0)⩽C;

if δ=δϵ>0 in (*) can be chosen independently of ϵ(>0), then limn→∞|un|s,Ω∖Bϵ(0)=0, for every s∈[1,∞];

for all s(⩾1)∈(N(q-p)/p,∞], one has limn→∞|un|s=∞ and
(30)limn→∞|un|s,Ω∖Bϵ(0)|un|s=0;

if N(q-p)/p⩾1, then for s=N(q-p)/p it holds that
(31)liminfn→∞|un|s>0.If the hypotheses in (b) are satisfied, then (30) also holds for this s.

Proof.

There is clearly a positive classical solution w to the equation
(32)-div(|∇u|p-2∇u)=-δϵ/2|u|q-2u,x∈Rn∖Bϵ/2(0)¯lim|x|→ϵ/2w(x)=∞,lim|x|→∞w(x)=0.
In fact, by [22, 23], the radial solution up(x)=up(|x|) satisfies the ordinary differential equation
(33)(rn-1|u′|p-2u′)′=-δϵ/2rn-1uqu(r)=∞asr⟶ϵ/2,u(r)⟶0asr⟶∞.
Set zn=w-un and
(34)φn(x):=(q-1)∫01|sw(x)+(1-s)un(x)|q-2ds⩾0,ϕn(x):=(p-1)∫01|sw(x)+(1-s)un(x)|p-2ds⩾0,φn(x)zn=(q-1)∫01|sw(x)+(1-s)un(x)|q-2(w-un)ds=∫01dds(|sw+(1-s)un|p-2(sw+(1-s)un))ds=wq-1-|un|q-2un,ϕn(x)zn=(q-1)∫01|sw(x)+(1-s)un(x)|p-2(w-un)ds=∫01dds(|sw+(1-s)un|p-2(sw+(1-s)un))ds=wp-1-|un|p-2un
and hence from (*)(35)-div(|∇w|p-2∇w)-(-div(|∇un|p-2∇un))+(Vϕn(x)-Qnψn)zn=-div|∇w|p-2+V|w|p-2w-Qnwq-1-[-div|∇u|p-2∇u+V|∇u|p-2∇u-Qn|un|q-2un]=-div|∇w|p-2+V|w|p-2w-Qnwq-1⩾-div|∇w|p-2+δϵ/2wq-1=0.
Note that Vϕn(x)-Qnφn⩾0 in Ω∖Bϵ/2(0)¯ when n⩾Nϵ/2. Due to the continuity of un and the fact that wn(x)→∞ as x→∂Bϵ/2(0), there is r∈(ϵ/2,ϵ) such that zn⩾0 on ∂Br(0). Moreover, zn⩾0 on ∂Ω. If Ω is bounded, the maximum principle says that zn⩾0 in Ω∖Br(0) (see [20, 21]). If Ω is unbounded, by virtue of w(x) tending to 0 as |x|→∞ by construction, thus for any γ>0, we may pick R~>0 such that zn⩾-γ in Ω∖BR~(0). Moreover, applying regularity theory to un∈W01,p(Ω), we can get un(x)→0 as |x|→∞. Now the same maximum principle is applied on Ω∩(BR~∖Br(0)¯), which implies that zn⩾-γ in all of Ω∖Br(0). Letting γ→0, we obtain zn>0 again. By analogy we obtain un⩾-w (take zn:=w+un); hence
(36)|zn|⩾winΩ∖Bϵ(0),∀n⩾Nϵ/2.
Hence (a) follows from above arguments with the fact that w is continuous in Ω∖Bϵ(0).

Next, the hypotheses in (b) imply that there is δ>0 such that Qn⩽-δ on Ω∖B1/n(0) for each n large enough. Let wn be a positive solution to
(37)-div(|∇u|p-2∇u)=-δ|u|q-2u,x∈Rn∖B1/n(0)lim|x|→1/nwn(x)=+∞,lim|x|→∞wn(x)=0.
Then the sequence wn is monotone decreasing, by using the maximum principle to wn⩾wn+1 on ∂B1/n(0) for every n∈N. Therefore, wn converges locally and uniformly to a nonnegative solution w to (37) on Rn∖{0}. It follows from our hypotheses on N and p that w is an entire solution to (37) by applying the argument as in [24]. And then, due to [25], w≡0. For another, the function wn dominates the solution un on Ω¯∖Br(0) for some r∈(ϵ/2,ϵ), as seen in the proof of (a). Thus, un also converges to 0 locally and uniformly in Ω∖Br(0); that is, limn→∞|un|s,Ω∖Br(0)=0.

For (c), we first consider the case s(⩾1)∈(N(q-p)/p,q]. By interpolation inequality, we have the following estimate for solution un:
(38)∥un∥p=∫ΩQn|un|qdx⩽c1|un|qq⩽c1|un|sqθ|un|p*q(1-θ)⩽c2|un|sqθ∥un∥q(1-θ).
Here c1, c2 are independent of n, and θ satisfies that
(39)1q=θs+1-θp*.
According to Lemma 2, it suffices to impose that q(1-θ)<p or equivalent s>N(q-p)/p. This and (a) prove the case s∈(N(q-p)/p,q]. And then, (38) and (a) yield |un|q,Bϵ(0)→∞; hence |un|s,Bϵ(0)→∞ for every s∈(q,∞] as n→∞. Using (a) again we get (30).

Note that (38) implies (30) for s=N(q-p)/p, so case (d) is easily followed.

4. Concentration at Several Points

Now we assume that the function Qn is positive in a neighbourhood of two distinct points x1,x2∈Ω (indeed, the following argument is also valid for any finite number of points in Ω). More precisely, we assume.

(**)Qn>0 in a neighbourhood of {x1}∪{x2}, and there exists a constant C such that |Qn|L∞(Ω)⩽C for all n. Moreover, for each ϵ>0 there exist constants δϵ>0 and Nϵ such that Qn⩽-δϵ whenever x∉Bϵ(x1)∪Bϵ(x2) and n⩾Nϵ.

As in Section 2, we put Jn(u)=∫ΩQn|u|qdx:
(40)sn:=infJn(u)∥u∥pJn(u)p/q≡infJn(u)∫Ω(|∇u|p+V|u|p)dx(∫ΩQn|u|qdx)p/q.

Theorem 6.

Suppose Qn satisfies (**) and q∈(p,p*), and un is a ground state solution to (1). Then, for n large, un concentrates at x1 or x2. More precisely, for each ϵ>0 we have by passing to a subsequence
(41)limn→∞∫Ω∖Bϵ(xj)(|∇u|p+Vunp)dx∫Ω(|∇u|p+Vunp)dx=0,limn→∞∫Ω∖Bϵ(xj)Qnunpdx∫ΩQnunpdx=0
for j=1 or 2 (but not for j=1 and 2).

Remark 7.

Note that, in view of the obvious modification of Theorem 4, the limits in (41) are 0 if Ω∖Bϵ(xj) is replaced by Ω∖Bϵ(x1)∪Bϵ(x2). So if j=1 in (41), then concentration occurs at x1 and if j=2, it occurs at x2.

Proof.

As in [19], we may assume that Jn(un)=∫ΩQn|un|pdx=1 by renormalizing (un may not be a solution to (1), but we still have sn:=∥un∥p/Jn(un)p/q). Let ξj∈C0∞(Ω,[0,1]) be a function such that ξj=1 on Bϵ/2(xj) and ξj=0 on Ω∖Bϵ(xj), j=1,2, where ϵ is so small that Bϵ(xj)¯⊂Ω and Bϵ(x1)¯∩Bϵ(x2)¯=∅. Set vn:=ξ1un, wn:=ξ2un, and zn:=un-vn-wn. Since suppzn⊂Ω∖(Bϵ/2(x1)∪Bϵ/2(x2)) and the conclusion of Theorem 4 remains valid after a modification, we have
(42)∥un∥p=∫Ω(|∇un|p+Vunp)dx=(∫Ω(|∇vn|p+Vunp)dxhhlh+∫Ω(|∇wn|p+Vwnp)dx)(1+o(1))=(∥vn∥p+∥wn∥p)(1+o(1)),1=Jn(un)=∫ΩQn|un|qdx=∫ΩQn|vn|qdx+∫ΩQn|wn|qdx+o(1)=Jn(vn)+Jn(wn)+o(1).
First, we assume that limsupn→∞Jn(vn)⩾0 and limsupn→∞Jn(wn)⩾0. By passing to a subsequence, we may assume that Jn(vn)→c0∈[0,1] and Jn(wn)→1-c0∈[0,1]. If c0∈(0,1), recalling that q>p, we get a contradiction from the following inequality:
(43)sn=∥un∥pJn(un)p/q=(∥vn∥p+∥wn∥p)(1+o(1))(Jn(vn)+Jn(wn)+o(1))p/q>∥vn∥p+∥wn∥pJn(vn)p/q+Jn(wn)p/q⩾min{∥vn∥pJn(vn)p/q,∥wn∥pJn(wn)p/q}⩾sn.
So c0=0 or 1. If c0=1 (say), then the second limit in (41) is 0 for j=1 because suppvn⊂Bϵ(x1). The first limit is 0 as well, since ∥wn∥p/∥vn∥p is otherwise bounded away from 0 for large n, and we obtain a contradiction again from
(44)sn=(∥vn∥p+∥wn∥p)(1+o(1))(Jn(vn)+Jn(wn)+o(1))p/q>∥vn∥pJn(vn)p/q⩾sn.

Finally, suppose limsupn→∞Jn(wn)<0 (the case limsupn→∞Jn(vn)<0 is of course analogous); it passes to a subsequence Jn(wn)⩽-η for some η>0 when n is large enough. Then a contradiction (44) holds for such n because Jn(vn)>Jn(vn)+Jn(wn)+o(1).

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 p=2, 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).

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