In this paper we prove that a singularly perturbed Neumann problem with
potentials admits the existence of interior spikes concentrating in maxima and
minima of an auxiliary function depending only on the potentials.Comment: 7 page

Pomponio, Alessio

Secchi, Simone

English

arXiv

AbstractIn this paper, we prove that a singularly perturbed Neumann problem with potentials admits the existence of interior spikes concentrating in maxima and minima of an auxiliary functional depending only on the potentials

Pomponio, A.

Secchi, S.

Elsevier - Publisher Connector 

Available online at ww .sciencedirect.com Applied 
Mathematics 
.o,=.== ,,,.=c'r. Letters 
Applied Mathematics Letters 17 (2004) 1025-1031 EI_SEV~.R 
www.eIsevier.com/locate/aml 
Interior Spikes of a Singularly Perturbed 
Neumann Problem with Potent ia ls  
A.  POMPONIO 
SISSA, via Beirut 2/4, 1-34014 Trieste, Italy 
pomponio©sissa, it 
S. SECCHI 
Dipartimento di Matematica "F. Enriques" 
Universit~ degli Studi di Milano 
via C. Saldini 50, 1-20133 Milano, Italy 
secchi@mat, unimi, it 
(Received December 2003; accepted #anuary 200~) 
Communicated by A. Ambrosetti 
Abst rac t - - In  this paper, we prove that a singularly perturbed Neumann problem with potentials 
admits the existence of interior spikes concentrating in maxima nd minima of an auxiliary functional 
depending only on the potentials. © 2004 Elsevier Ltd. All rights reserved. 
Keywords-- Inter ior  spikes, Singularly perturbed Neumann problem. 
1. INTRODUCTION 
In this paper, we study the existence of interior spikes of the following problem: 
-s2div (J(x)Vu) + V(x)u = u p, in £1, 
Ou 
cg--u - O, on 0~, 
(1) 
where f~ is a smooth bounded domain of ]~N with external normal v, N > 3, 1 < p < (N + 
2) / (N - 2), j :~N ~ ~, and V:~ y --+ ~ are C 2 functions. 
In [1], the first author, emending the classical results by Ni and Takagi in [2,3], proved that there 
exist solutions of (1) that concentrate at maximum and minimum points of a suitable auxiliary 
function defined on the boundary Oft and depending only on J and V. Here we study the existence 
of solutions which concentrate in the interior of gt and we will show that the concentration occurs 
at maximum and minimum points of the same auxiliary function introduced in [1], but now 
defined in ~. We assume that the reader has familiarity with [1]. 
Both authors were supported by MIUR, national project Variational Methods and Nonlinear Differential Equa- 
tions. 
0893-9659/04/$ - see front matter (~) 2004 Elsevier Ltd. All rights reserved. Typeset by fitA/tS-TEX 
doi:10.1016/j.aml.2004.07.004 
1026 A. POMPONIO AND S. SECCHI 
When Y - 1 and V - 1, interior spikes have been found by Wei (see [4]) showing that 
concentration occurs at local maxima of the distance function dist (., OFt). 
On J and V, we will do the following assumptions. 
(J) J G Cl(gt, ]~), d and DI J  are bounded; moreover, d(x) _> C > 0, for all x E f~; 
(V) Y e Cl(f~, JR), V and DIV  are bounded; moreover, V(x) >_ C > 0, for all x C Ft. 
Let us introduce an auxiliary function which will play a crucial role in the study of (1). Let 
F: Ft --* R be the function defined by 
F (Q) = V (Q)(p+I)I(p-1)-N/2 ,l (Q)NI2. (2) 
Let us observe that by (J) and (V), F is well defined. Our main result is the following theorem. 
THEOREM 1.1. Let Qo c Ft. Suppose (d) and (V) hold. There exists eo > 0 such that ff 
0 < e < So, then (1) possesses a solution u6 concentrating at Q~ with Q~ --+ Qo, as e ~ O, 
provided that one of the two following conditions holds. 
(a) Qo is a nondegenerate critical point ofF. 
(b) Qo is an isolated local strict minimum or maximum of F. 
Notat ion  
• I f  u: R N --* ]~ and P E ]~N, we set up = u(. -- P). 
• If U Q is the function defined in (5), when there is no misunderstanding, we will often 
write U instead of U Q. Moreover, if P = Q/s, then Up - U Q (. - P). 
• If e > 0, we set ~t~ _~ f~/s ---- {x E ]~N : ~X e ~'~}. 
• With os(1), we denote a quantity which tends to zero as e --, 0. 
2. PREL IMINARY LEMMAS AND SOME EST IMATES 
First of all, we perform the change of variables x ~ ex and so problem (1) becomes 
-d iv  (d(sx)Vu) + V(ex)u = u p, in fl~, 
Ou 
O-T = O, on 0~,  
(3) 
where Ft¢ = e-lFt. Of course, if u is a solution of (3), then u(./s) is a solution of (1). 
Solutions of (3) are critical points u E H 1 (~)  of 
in l i v (ex)u ldx_  1 /~ luip+l 1 d(ex) IVu] 2 dx + ~ ~ f~ (u) = -~ ~ ~ p + 1 " 
We look for solutions of (3) near a U Q, the unique solution of the limiting problem 
-a  (Q) Au + V (Q) u = u p, in R N, 
u > 0, in ]~N, (4) 
~(0) = max., 
RN 
for an appropriate choice of Q ~ Ft. It is easy to see that 
(5) 
Neumann Problem with Potentials 1027 
where [J is the unique solution of 
-AU + U = 0 "p, in R N, 
0 > 0, in RN 
0(0) = ma~ O, 
AN 
which is radially symmetric and decays exponentially at infinity together with its first derivatives. 
For the sake of brevity, we will often write U instead of U Q. If P = E-1Q E ~,  we set 
Up = UQ (. - P) and 
Z ~ - {Up "P  e a~}. 
LEMMA 2.1. For alI Q E t2 and for a/ /e sufficiently small, if P = Q/e c t2~, then 
IlVf~(gp)il = o(~). (6) 
PROOF. Repeating the calculations of [1], we get 
= [ - J  (q) z~u + v (q) u - u~] ~_p + ] (q) -SV~ (V f~(Up) l v) -q)/e a~ 
L L (v (~x+ q) -v  (O))u~_,,. + -o,)/~ (J (Ez + q) - J (q)) vu .  w,_p  + -c~)/, 
Hence, since U = U q is a solution of (4), we get 
(v l~(up) I ~) = J (q) -0-b-~ + (J (~x + Q) - J(Q)) vu .  w_p  a~ -q)/~ (7) 
f(a (v (ex + q) - v (q)) u~_~. + -q)/e 
Let us estimate the first of these three terms 
[Joa, ~ _ CIIvllL~(oao) O. ] " J (Q,) 
First of all, we observe that there exist e0 > 0 and C > 0 such that, for all s E (0, Co) and for all 
v E H 1 (t2~), we have 
IlVllL~(Oao) < CIIvllm(a~). 
Using the exponential decay of U and its derivatives, it is easy to see that 
foa, OUp 2 ou 2 = fo((a-Q)/~)10~ = o(~). (8) 
Arguing as in [1], one can prove that 
fc~ -- [,_, (V @x-F Q) -V  (Q)) Uv_p = O(s)llvl,. -~)/~ (J (E~ + q) J (q)) vu .  w_p  + -,,o-~)/~ 
O) 
Now the conclusion follows immediately from (7)-(9). | 
We here present some useful estimates that will be used in the sequel. 
1028 A. POMPONIO AND S. SECOItI 
PROPOSITION 2.2. Let P = Q/~ E ~6. Then, we have 
~O cgUp rT 
~ --g; -~ ~ = o(~), 
J (Q)LoIVUpI2 +v  (Q)~ u~= fR~ (uq),+i +o(~), 
£ ~),~,~ = ~ ~)£ ,~,~+f~ ' ~)~ I~°1~÷°~) , (lO) 
(11) 
PROOF. Let us prove the first formula. We have 
L ~ =/~o_o~ o)~'1-- ~ ~o)~'1-/~,~o_o~/~o)  . 
Using the exponential decay of U Q, it is easy to see that 
/ / / /  (uo),+i < (uq) '+1 = c r N-i (u%)y  1 e~ = o(,). 
/v~ RN\(~t-Q)/e ~N\Sl/x/- £ 
Using the exponential decay of U Q, also the second formula can be proved in a similar way. The 
other equations can be proved as in [1]. | 
3. THE F IN ITE-DIMENSIONAL REDUCTION 
In this section, we perform a finite-dimensional reduction, following some ideas introduced 
in [5]. The symbol TueZ ~ denotes the tangent space to Z ~ at Up. Let L~,Q : (TupZe) ± 
(TupZ~) ± denote the operator defined by setting (L~,Qv IT) = D2fz(Up)[v, w]. 
LEMMA 3.1. There exists C > 0 such that for ~ small enough, one has that 
llL~,qvll > Ctlvii , V v e (T~pZ~)  ± . (12) 
PROOF. The proof of (12) is completely analogous to that of equation (21) in [1], so we omit the 
details. | 
LEMMA 3.2. For ~ > 0 small enough, there ex/sts a unique w = w (e, Q) E (TupZ~) ± such that 
Vfe(Up + w) E TueZ. Such a w(s,Q) is of c/ass C 2, respectively, C i,p-i, with respect o Q, 
provided that p >_ 2, respectively, 1 < p < 2. Moreover, the functional A~(Q) = fe(UQ/z + 
w(~, Q) ) has the same regularity o f t  and satisfies 
v.~, (0,o) = o <==> v£  (Uqo/, + ~ (~, 0,o)) = o. 
PROOF. Let P = P~,Q denote the projection onto (TupZ6) ±. We want to find a solution w E 
(Tu~Z~) ± of the equation PVf~(Up + w) = O. One has that Vf~(Up+w)  = Vf~(Up)+ 
D2fs(Up)[w] + R(Up, w) with IIR(Up, w)H -- °([[w[I), uniformly with respect o Up. Therefore, 
our equation is 
L~,Qw + PVf~ (Up) + PR (Up, w) = O. 
According to Lemma 3.1, this is equivalent to 
w = N~,Q (w), where N~,Q(w) = - (L~,Q) -1 (PVh  (Up) + PR (Up, w)). 
Neumann Problem with Potentials 1029 
By (6), it follows that 
I] N6,Q (w)I[ = O(~) + o([[w I[)" (13) 
Then, one readily checks that N~,Q is a contraction on some ball in (Tu, Z~) ± provided that 
e > 0 is small enough. Then, there exists a unique w such that w = Nc,Q(w). Given z > 0 
small, we can apply the implicit function theorem to the map (Q, w) H 79Vfz(Up + w). Then, 
in particular, the function w(¢, Q) turns out to be of class C 1 with respect to Q. Finally, it is a 
standard argument, see [5], to check that the critical points of A~(Q) = f~(Up + w) give rise 1;o 
critical points of f~. | 
REMARK 3.3. From (13), it follows immediately that 
1iwll = O(~). (14) 
Moreover, repeating the arguments of [1], if 7 = min{1,p - 1}, then, for i = 1,.. . ,  N, we infer 
that 
[1o.,~il = o (¢~). 
4. THE F IN ITE-D IMENSIONAL FUNCTIONAL 
THEOREM 4.1. Let Q E f~ and P = Q/¢ E f~s. Suppose (J) and (V). Then, for s sutNciently 
small, we get 
J' (Q)[x]ivul 2 + ~ v'  (Q) [x]u 2 + o(~), ~t~ (Q) = f~ (up + ~ (~, Q)) = c0r (Q) + ~ ~ 
where F is the auxiliary function introduced in (2) and 
1 1) ~ OP+I 
Moreover, for M1 i = 1,..., N, we get 
O~,A~ (Q) = ~c0Oq~r (Q) + o(~), (15) 
and hence, 
IIA~ - c0rllc~(a) : o(~). 06) 
PROOF. In the sequel, to be brief, we will often write w instead of w(E, Q). It is always understood 
that e is taken in such a way that nil the results discussed previously hold. 
First of all, reasoning as in the proofs of (10) and (11) and by (14), we can observe that 
s( x)vv , w = S(Q) [ vv . w + (17> 
¢ J~$e 
Recalling (14), we have 
fa 1L  V(zx)(Up+w) 2 1 L (Up+w)P+I 1 j(ex) iv(up +~)12 + ~ p+l  
= 1 J(sx)lVUvl 2+ ~ ~ ~ . + J(ex)VUp.Vw+ V(ex)Up w 
1 [ [  p+l Ja L(UP+ - - (p+l )Uf ,  o(s) 
(by Proposition 2.2, (17), and (18), and with our notations) 
s s 1 
= 1 ~v Up+I-l-2 ~N J/(Q) [x]lVV[2 -{- 2 ~N Vt (Q) [x]g2 - 2 LN Up+I 
1 1)~vup+l+o(~). +J(Q) fa VUp'Vw+V(Q) fa Upw- /a  Uf'w+(~ p+ 
1030 A. POMPONIO AND S. SEOCHI 
Since that U is solution of (4), we infer 
J (Q)~ VUp 'Vw+V(Q) /a  Upw- /a  U~w 
= [ - J  (Q) ~u~ + v (Q) u~ - u~] ~ + J (Q) - ~  = -5~-~ = 
By these considerations, we can say that 
A~(Q)= p+l  . 5 .  (Q)[x]lVU[2+2 . (Q)[x]U2+°(s)" 
Now the conclusion of the first part of the theorem follows observing that, since by (5), 
UQ(z) = V (Q)~/(p-1) ~ / , - - z~, ,  
V k J (~¢) ]
then ~ N UP+I = V (Q)W+I)/(~-I)-N/2 j (Q)N/2 ~N ~p+l. 
Estimate (15) on the derivatives of A~ follows easily by repeating the arguments of [1]. | 
5. PROOF OF THEOREM 1.1 
In this section, we will state and prove two multiplicity results for (1). Theorem 1.1 will follow 
as a particular case. 
THEOREM 5.1. Let (J) and (V) hold and suppose F has a nondegenerate smooth manifold of 
critical points M C ~. There exists so > 0 such that if 0 < s < so, then (1) has at least l(M) 
solutions that concentrate near points of M. Here l(M) denotes the cup long of M (for a precise 
definition, see [6]). 
PROOF. Fix a 5-neighborhood M~ of M such that the only critical points of F in M~ are those 
in M. We will take U = M~. For z sufficiently small, by (16) and [6, Chapter II, Theorem 6.4], 
~4~ possesses at least l(M) critical points, which are solutions of (3) by Lemma 3.2. Let Qs E M 
be one of these critical points, then uE Q~ = UQ~/6 + w(s, Q~) is a solution of (3). Therefore, 
(x) (x) = 
is a solution of (1). | 
Moreover, when we deal with local minima (respectively, maxima) of F, the preceding results 
can be improved because the number of positive solutions of (1) can be estimated by means of 
the category and M does not need to be a manifold. We will give only the statement of the 
theorem; for the proof, see [1]. 
THEOREM 5.2. Let (J) and (V) hold and suppose F has a compact set X C ~ where F achieves 
a strict local minimum (respectively, maximum), in the sense that there ex/st 5 > 0 and a 
5-neighborhood X~ C ~ of X such that 
b = inf{Y (Q): Q ~ axe} > a - FI~, (respectively~ sup(y (Q): Q ~ aXe} < FL~ ) . 
Then, there exists So > 0 such that (1) has at least cat (X, X~) solutions that concentrate near 
points of X~, provided s 6 (0, So). Here cat ( X, X~ ) denotes the Lusternik-Schnirehnan category 
of X with respect o X~. 
REMARK 5.3. Let us observe that Part (a) of Theorem I.i is a particular case of Theorem 5.1, 
while Part (b) of Theorem I.i is a particular case of Theorem 5.2. 
Neumann Problem with PotentiMs 1031 
REFERENCES 
1. A. Pomponio, Singularly perturbed Neumann problems with potentials, to appear on TopoA Meth. Noni',in. 
Analysis, http : / /~  .mat. uniroma3, it/AnalisiNonLineare/preprints/preprints, html, (2003). 
2. W.M. Ni and I. Takagi, On the shape of the least-energy solutions to a semilinear Neumann problem, Comm. 
Pure Appl. Math. 44, 819-851, (1991). 
3. W.M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke 
Math. J. 70, 247-281, (1993). 
4. J. Wei, On the interior spike layer solutions to a singularly perturbed Neumann problem, Tohoku Math. J. 
50, 159-178, (1998). 
5. A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear SchrSdinger equations 
with potentials, Arch. Rational Mech. Anal. 159, 253-271, (2001). 
6. K.C. Chang, Infinite Dimensional Morse Theory and Multiple Solutions Problems, Birkh£user, (1993). 


Interior spikes of a singularly perturbed Neumann problem with potentials 

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Interior spikes of a singularly perturbed Neumann problem with potentials

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