A Concentration Phenomenon for p-Laplacian Equation

x ∉ B ε (0) and n ⩾ N ε . As it is known,u ≡ 0 is the only solution to (1) ifQ n (x) ⩽ 0 for all x ∈ Ω. In addition, if Q n (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 = Q n be such that Q n > 0 on the ball B 1/n (0) and Q n < 0 on Ω \ B 2/n (0) and u n ̸ ≡ 0 are the solutions to (1) associated with Q n (x). Accordingly, the question is what happens to u n 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:

And an assumption of is as follows. ( * ) The set { | ( ) > 0} contained in the neighborhood of zero has positive measure, and | | ∞ (Ω) ⩽ with the constant is independent of . Moreover, for each > 0 there exist constants (> 0) and such that ⩽ − whenever ∉ (0) and ⩾ .
As it is known, ≡ 0 is the only solution to (1) if ( ) ⩽ 0 for all ∈ Ω. In addition, if ( ) > 0 is based on a bounded set of positive measures, it is clear that there exists a solution ̸ ≡ 0 (see Theorem 1). Hence, without loss of generality, we assume that 0 ∈ Ω and let = be such that > 0 on the ball 1/ (0) and < 0 on Ω \ 2/ (0) and ̸ ≡ 0 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 ( > 0) or dark ( < 0) solitons.
Equations of these types have been studied extensively in many monographs and lectures (e.g., [2][3][4][5][6][7][8][9][10] for = 2, [11][12][13][14][15][16][17][18] for general ). In [2], Byeon and Wang considered the standing wave solutions ( , ) ≡ exp (− /ℏ)V( ) for the nonlinear Schrödinger equation: Thus, they needed only to discuss the function V which satisfies and rewrote it in the following form: Let the zero set Z ≜ { ∈ R | ( ) = 0} and be an isolated component of Z, and they distinguished three cases of to prove the concentration as → 0. And then, in [3] by replacing V with a fairly general class nonlinearity (V), 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, −Δ + ( ) = ( )| | −2 . 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 1, 0 (Ω) 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.
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 > 0 independent of , such that ‖ ‖ ⩾ .
Proof. As in [19], let (V) = ∫ Ω |V| and Suppose that (V ) is a minimizing sequence for , normalized by (V ) = 1; then ‖V ‖ is bounded. Hence, V ⇀ V in and V ( ) → V( ) a.e. in Ω (by choosing a subsequence). Note that < 0 on | | > 1 for large. The Rellich-Kondrachov Theorem and Fatous's Lemma say that Thus V is a minimizer. And then, the lagrange multiple rule implies that = V is a solution to (1) for some appropriate constant > 0. Moreover, since V may be replaced by |V |, V ⩾ 0 (and hence ⩾ 0). To show that > 0, we note that satisfies it follows from the strong maximum principle (see [20,21]) that > 0. If ̸ = 0 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 > 0 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 loc (Ω) after passing to a subsequence. Multiplying (1) (with = ) by , integrating by parts, and recalling that < 0 for each > 0 and ⩾ , it holds that Journal of Applied Mathematics 3 If → 0, → 0 in . It is a contradiction to ‖ ‖ ⩾ > 0 given in Theorem 1.

Lemma 3. Consider
Proof. We prove it by contradiction as well. We may assume that Due to Lemma 2 with > , ∫ Ω | | → 0.
On the other hand, we have for 0 < < 1 We may choose small 1 such that the second integral on the right-hand side above is positive as ̸ ≡ 0. Then we get the contradiction as → 0.
Multiplying (1) (with = ) by we obtain namely, Given > 0, we have ⩽ − on supp , provided that is large enough. Hence for all such , where is a constant independent of . Since = / ‖ ‖ → 0 in loc (Ω) 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 > 0 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 Journal of Applied Mathematics

Concentration in the -Norm
The next is to consider the concentration in other norms. (1) for each ∈ N. Suppose that the assumption ( * ) holds and there exists , > 0 such that ⩾ whenever ∈ Ω \ (0), and there exists > 0 such that (0) ⊂ Ω; then one can get that Proof. There is clearly a positive classical solution to the equation
For (c), we first consider the case (⩾ 1) ∈ ( ( − )/ , ]. By interpolation inequality, we have the following estimate for solution : Here 1 , 2 are independent of , and satisfies that According to Lemma 2, it suffices to impose that (

Concentration at Several Points
Now we assume that the function is positive in a neighbourhood of two distinct points 1 , 2 ∈ Ω (indeed, the following argument is also valid for any finite number of points in Ω). More precisely, we assume.