On some elliptic interface problems with nonhomogeneous ... - My FIT

Adv. Nonlinear Anal. 2 (2013), 195 – 211 DOI 10.1515/anona-2013-0005

© de Gruyter 2013

On some elliptic interface problems with nonhomogeneous jump conditions T. Gnana Bhaskar and Kanishka Perera

Abstract. We obtain nontrivial solutions of some elliptic interface problems with nonhomogeneous jump conditions that arise in localized chemical reactions and nonlinear neutral inclusions. Our proofs in bounded domains use Morse theoretical arguments, in particular, critical group computations. An extension to the whole space is proved using concentration compactness arguments. Keywords. Elliptic interface problems, nonhomogeneous jump conditions, nontrivial solutions, Morse theory, critical groups, nonlinear eigenvalue problems, Yang index, anisotropic problems, concentration compactness. 2010 Mathematics Subject Classification. Primary 35J66; secondary 35J20, 35P30.

1

Introduction

In this paper we obtain nontrivial solutions of some elliptic interface problems with nonhomogeneous jump conditions. Such problems arise, for example, in localized chemical reactions and in nonlinear neutral inclusions. Let be a bounded domain in RN ; N 2, let 1 be a C 1 -subdomain of such that 1 and 2 D n 1 is connected, and let D @1 . First we consider the interface problem 8 u D 0 in n ; ˆ ˆ ˆ ˆ < @u Œu D 0; D g.x; u/ on ; ˆ @ ˆ ˆ ˆ : u D 0 on @;

(1.1)

where Œu D u2 u1 , ui D uji , i D 1; 2, Œ@u=@ D @u2 =@ @u1 =@, @=@ is the exterior normal derivative on , and g 2 C. R/ satisfies the subcritical growth condition jg.x; t/j C.jt jq

1

C 1/

8.x; t/ 2 R

(1.2)

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T. G. Bhaskar and K. Perera

N Here for some q 2 .1; 2/. ´

2.N 2N D 1;

1/=.N

N 3; N D 2;

2/;

is the critical trace exponent. This problem arises, for example, in chemical reactions that are localized due to the presence of a catalyst (see Chadam and Yin [1] and Pan [8]). A weak solution of problem (1.1) is a function u 2 H01 ./ satisfying Z Z g.x; u/ v D 0 8v 2 H01 ./; ru rv

where H01 ./ is the usual Sobolev space with kuk2

R D jruj2 . As noted by Tian and Ge [11] and Nieto and O’Regan [7] in the ODE case, weak solutions of this problem coincide with critical points of the C 1 -functional Z Z 1 jruj2 G.x; u/; u 2 H01 ./; ˆ.u/ D 2 Rt where G.x; t / D 0 g.x; s/ ds. We assume that the jump g is asymptotically linear near zero and satisfies the Ambrosetti–Rabinowitz superlinearity condition near infinity: g.x; t/ D t C o.t/

as t ! 0; uniformly on ;

(1.3)

for some 2 R, and 0 < G.x; t/ t g.x; t/;

x 2 ; jt j large;

(1.4)

for some > 2. Integrating (1.4) gives G.x; t/ c jt j

C

8.x; t/ 2 R

(1.5)

for some c > 0. A model example is g.x; t/ D t C jt jq

2

t;

N where q 2 .2; 2/. Assumption (1.3) leads us to consider the interface eigenvalue problem 8 u D 0 in n ; ˆ ˆ ˆ ˆ < @u Œu D 0; D u on ; ˆ @ ˆ ˆ ˆ : u D 0 on @: The spectrum ./ of this problem consists of an increasing and unbounded sequence of positive eigenvalues of finite multiplicities.

Elliptic interface problems with nonhomogeneous jump conditions

197

N (1.3), and (1.4) with > 2. If one Theorem 1.1. Assume (1.2) with q 2 .1; 2/, has … ./, then problem (1.1) has a nontrivial solution. Our proof of this theorem will be based on explicitly computing the critical groups of ˆ at zero and at infinity, and showing that they are nonisomorphic in some dimension. These groups are defined by Cq .ˆ; 0/ D Hq .ˆ0 \ U; ˆ0 \ U n ¹0º/; Cq .ˆ; 1/ D Hq .H01 ./; â /;

q 0;

where ˆ0 D ¹u 2 H01 ./ W ˆ.u/ 0º, U is any neighborhood of zero, a < 0 is such that ˆ has no critical points in â D ¹u 2 H01 ./ W ˆ.u/ aº, and H . ; / are the relative singular homology groups. We refer the reader to Chang [2] and Mawhin and Willem [6] for the necessary background on Morse theory. We also indicate how Theorem 1.1 can be extended to the p-Laplacian case 8 p u D 0 in n ; ˆ ˆ ˆ ˆ < @u Œu D 0; D g.x; u/ on ; jrujp 2 (1.6) ˆ @ ˆ ˆ ˆ : u D 0 on @; where p u D div.jrujp 2 ru/ is the p-Laplacian of u, p > 2, g satisfies (1.2) with q 2 .1; p/, N and pN D .N 1/p=.N p/ if N > p and pN D 1 if N p. 1; p A weak solution u 2 W0 ./ of this problem satisfies Z Z 1; p p 2 jruj ru rv g.x; u/ v D 0 8v 2 W0 ./;

and the associated variational functional is Z Z 1 ˆ.u/ D jrujp G.x; u/; p

1; p

u 2 W0

./:

We replace (1.3) with g.x; t / D jt jp

2

t C o.jt jp

1

/ as t ! 0; uniformly on ;

(1.7)

and take > p in (1.4). Our model example is g.x; t/ D jtjp 2 t C jt jq 2 t, where q 2 .p; p/. N The corresponding eigenvalue problem is 8 p u D 0 in n ; ˆ ˆ ˆ ˆ < p 2 @u Œu D 0; jruj (1.8) D jujp 2 u on ; ˆ @ ˆ ˆ ˆ : u D 0 on @;

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and the spectrum p ./ of this problem consists of the set of all 2 R for which it has a nontrivial solution. Theorem 1.2. Assume (1.2) with q 2 .1; p/, N (1.4) with > p, and (1.7). If one has … p ./, then problem (1.6) has a nontrivial solution. Our proof here will use an increasing and unbounded sequence of eigenvalues constructed via the Yang index as in Perera [9] (see also Perera, Agarwal and O’Regan [10]). The standard sequence of eigenvalues based on the Krasnosel0 ski˘ı genus does not provide sufficient information about the critical groups to prove this theorem. Next we consider the anisotropic problem 8 ˆ p u1 D 0 in 1 ; ˆ ˆ ˆ ˆ ˆ in 2 ; ˆ < u2 D 0 (1.9) @u @u 1 2 p 2 ˆ D g.x; u1 / on ; jru1 j u1 D u2 ; ˆ ˆ ˆ @ @ ˆ ˆ ˆ : u2 D 0 on @; where p > 2 and g satisfies (1.2) with q 2 .1; p/. N Problems of this type arise in nonlinear neutral inclusions, in particular, in assemblages of neutral coated spheres (see Jiménez, Vernescu and Sanguinet [3]). We work in the space ® ¯ X D u D .u1 ; u2 / 2 W 1; p .1 / H 1 .2 / W u1 D u2 on ; u2 D 0 on @ ; which is a closed subspace of W 1; p .1 / H 1 .2 /. A norm on X , equivalent to the standard norm of W 1; p .1 / H 1 .2 /, can be defined by Z kuk D

1

jru1 j

p

1=p

Z C

2

jru2 j

2

1=2 :

A weak solution u 2 X of problem (1.9) satisfies Z Z Z p 2 jru1 j ru1 rv1 C ru2 rv2 g.x; u1 / v1 D 0 1

2

and they coincide with critical points of the functional Z Z Z 1 1 jru1 jp C jru2 j2 G.x; u1 /; ˆ.u/ D p 1 2 2

8v 2 X;

u 2 X:

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Theorem 1.3. Assume (1.2) with q 2 .1; p/, N g.x; t/ D o.jt jp

1

/ as t ! 0; uniformly on ;

(1.10)

and (1.4) with > p. Then problem (1.9) has a nontrivial solution. Finally we consider an interface eigenvalue problem in the whole space RN , N 2, with RN 1 ¹0º as the interface, which we identify with RN 1 . We consider 8 N N 1 ˆ ; ˆ u C V .x/ u D 0 in R n R ˆ ˆ < @u (1.11) Œu D 0; D jujp 2 u on RN 1 ; ˆ @xN ˆ ˆ ˆ : u.x/ ! 0 as jxj ! 1; where V 2 L1 .RN / satisfies inf V .x/ > 0;

x2RN

lim V .x/ D V 1 > 0;

(1.12)

jxj!1

and where Œu D uC

u ;

uC D ujRN 1 .0;C1/ ; u D ujRN @u @uC @u D ; @xN @xN @xN

1 .

1;0/ ;

N A weak solution of this problem is a function u 2 H 1 .RN / 2 R, and p 2 .2; 2/. satisfying Z Z ru rv C V .x/ uv D jujp 2 uv 8v 2 H 1 .RN /; RN

RN

1

where H 1 .RN / is the usual Sobolev space with the norm k k induced by the inner product Z .u; v/ D

RN

ru rv C V 1 uv;

which is equivalent to the standard norm. Let Z Z I.u/ D jujp ; J.u/ D jruj2 C V .x/ u2 ; RN

1

RN

u 2 H 1 .RN /:

Then the eigenfunctions of (1.11) on the manifold ® ¯ M D u 2 H 1 .RN / W I.u/ D 1 and the corresponding eigenvalues coincide with the critical points and the critical values of the constrained functional J jM , respectively.

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We will first show that the autonomous problem at infinity, 8 ˆ u C V 1 u D 0 in RN n RN 1 ; ˆ ˆ ˆ < @u Œu D 0; D jujp 2 u on RN ˆ @x N ˆ ˆ ˆ : u.x/ ! 0 as jxj ! 1; has a least energy solution on M. Let Z 1 J .u/ D jruj2 C V 1 u2 ; RN

1

;

u 2 H 1 .RN /;

be the corresponding functional. Then 1 1 1 WD inf J .u/ > 0 u2M

by the Sobolev trace imbedding. N Then the infimum 1 is attained Theorem 1.4. Assume V 1 > 0 and p 2 .2; 2/. 1 at a function w11 > 0. p For the nonautonomous problem, J. / is an equivalent norm on H 1 .RN / since V 2 L1 .RN / and inf V > 0, so 1 WD inf J.u/ > 0: u2M

J1

By the invariance of with respect to the group D of .N 1/-dimensional shifts u 7! u. y/, y 2 RN 1 , we have 1 1 1 , and we will show that 1 is attained if the inequality is strict. N If 1 < 1 , then 1 is attained at Theorem 1.5. Assume (1.12) and p 2 .2; 2/. 1 a function w1 > 0. 1 The inequality 1 < 1 1 holds if V V , with the strict inequality on a set of positive measure. Indeed, in that case

1 J.w11 / < J 1 .w11 / D 1 1 since w11 > 0 a.e. So we have the following corollary. N If V .x/ V 1 for all x 2 RN and Corollary 1.6. Assume (1.12) and p 2 .2; 2/. the strict inequality holds on a set of positive measure, then 1 is attained at a function w1 that is positive in RN n RN 1 .


201

The main difficulty here is the lack of compactness inherent in this problem. This lack of compactness originates from the invariance of RN and RN 1 under the action of the noncompact group D, and manifests itself in the noncompactness of the Sobolev trace imbedding H 1 .RN / ,! Lp .RN 1 /, which in turn implies that the manifold M is not weakly closed in H 1 .RN /. Our proofs will use the concentration compactness principle of Lions [4,5], expressed as a suitable profile decomposition for minimizing sequences, to overcome this difficulty.

2 2.1

Proofs of Theorems 1.1–1.3 Proof of Theorem 1.1

Consider the linear interface problem 8 u D 0 in n ; ˆ ˆ ˆ ˆ < @u Œu D 0; D f .x/ ˆ @ ˆ ˆ ˆ : u D 0 on @;

on ;

where f 2 L2 ./. This problem has a solution u, obtained by minimizing the associated functional Z Z 1 jruj2 f .x/ u; u 2 H01 ./; ‰.u/ D 2 and it is unique by the maximum principle. Testing ‰ 0 .u/ D 0 with u and using the Hölder inequality gives kuk2 kf kL2 ./ kukL2 ./ : So the linear map L2 ./ ! H01 ./, f 7! u, is bounded by the boundedness of the trace imbedding H01 ./ ,! L2 ./, and hence the solution operator S W L2 ./ ! L2 ./;

Sf D u;

is compact by the compactness of the same imbedding. Thus, the spectrum ./ of the self-adjoint operator S 1 consists of isolated eigenvalues l % 1, of multiplicities dl < 1. Since … ./, zero is a nondegenerate critical point of the asymptotic functional Z Z 1 2 ˆ0 .u/ D jruj u2 ; u 2 H01 ./; 2 2 of Morse index X m0 D dl : l 0, Z Z 2 2 ˆ. u/ D jwj CC ! 1 as ! 1 (2.1) G.x; w/ c 2 2 by (1.5), and Z d ˆ. u/ D w g.x; w/; d Z 2 w ˆ. u/ g.x; w/ G.x; w/ D 2 2 ˆ. u/ a0 (2.2) for some a0 0 by (1.2) and (1.4). Fix a < a0 . Then ˆ has no critical points in â .


203

We have ˆ. u/ a for all sufficiently large by (2.1), and ˆ. u/ a H)

d ˆ. u/ < 0 d

by (2.2), so there is a unique u > 0 such that < (resp. D; >) u H) ˆ. u/ > (resp. D; p hold, then (i) ˆ satisfies the (PS) condition, (ii) Cq .ˆ; 1/ D 0 for all q. Proof. (i) If .uj / is a (PS) sequence for ˆ, then Z 0 p ˆ .uj /; uj C G.x; uj / 1 kuj k D ˆ.uj / p

uj g.x; uj /

o.kuj k/ C O.1/ by (1.2) and (1.4), so kuj k is bounded and a standard argument gives a convergent subsequence. (ii) The argument in the proof of Lemma 2.1 (ii) can be repeated with the direct sum decomposition 1; p W0 ./ D V ˚ W ; where 1; p

V D W0

1; p

.1 / ˚ W0 1; p

i.e., u 2 W0

.2 / and

® ¯ 1; p W D u 2 W0 ./ W u D 0 in n ;

./ belongs to W if and only if Z ru rv D 0 8v 2 V:

2.3 Proof of Theorem 1.3 By the elementary inequality .a C b/p 2p .ap C b 2 /, a 0, 0 b 1, (1.2), and (1.10), 1 C o.1/ kukp as kuk ! 0; ˆ.u/ 2p p


205

so zero is a strict local minimizer of ˆ and hence Cq .ˆ; 0/ D ıq0 G . Theorem 1.3 now follows from the following lemma as before. Lemma 2.3. If (1.2) with q 2 .1; p/ N and (1.4) with > p hold, then (i) ˆ satisfies the (PS) condition, (ii) Cq .ˆ; 1/ D 0 for all q. Proof. (i) If .uj / is a (PS) sequence for ˆ, then Z Z p 1 jruj1 j C 1 jruj 2 j2 p 2 1 2 Z G.x; uj1 / D ˆ.uj / ˆ0 .uj /; uj C

uj1 g.x; uj1 /

o.kuj k/ C O.1/ by (1.2) and (1.4), so kuj k is bounded and a standard argument gives a convergent subsequence. (ii) The argument in the proof of Lemma 2.1 (ii) can be repeated with the direct sum decomposition X DV ˚W; where 1; p

V D W0

.1 / ˚ H01 .2 / and

® ¯ W D u 2 X W u D 0 in n ;

i.e., u 2 X belongs to W if and only if Z Z ru1 rv1 C ru2 rv2 D 0 1

3

2

8v 2 V:

Proofs of Theorems 1.4 and 1.5

In the absence of a compact Sobolev trace imbedding, the main technical tool we use here for handling the convergence matters is the concentration compactness principle of Lions [4, 5]. This is expressed as the following profile decomposition for bounded sequences in H 1 .RN / (see Tintarev and Fieseler [12]). Proposition 3.1. Let .uk / H 1 .RN / be a bounded sequence, and assume that there is a constant ı > 0 such that if uk . C yk / * w ¤ 0 on a renumbered subsequence for some yk 2 RN 1 with jyk j ! 1, then kwk ı. Then there are

206

T. G. Bhaskar and K. Perera .n/

.1/

m 2 N, w .n/ 2 H 1 .RN /, yk 2 RN 1 , yk D 0 with k 2 N, n 2 ¹1; : : : ; mº, w .n/ ¤ 0 for n 2, such that, on a renumbered subsequence, .n/

uk . C yk / * w .n/ ; ˇ .n/ .l/ ˇˇ ˇy y ! 1 for n ¤ l; k k m X

.n/ 2

w lim inf kuk k2 ;

(3.1) (3.2) (3.3)

nD1

uk

m X

w .n/ .

.n/

yk / ! 0 in Lp .RN

1

N / 8p 2 .2; 2/:

(3.4)

nD1

Recall that .uk / M is a critical sequence for J jM at the level c 2 R if J 0 .uk /

k I 0 .uk / ! 0;

J.uk / ! c

for some sequence .k / R. The first limit is equivalent to Z ruk rv C V .x/ uk v RN Z D ck juk jp 2 uk v C o.kvk/ 8v 2 H 1 .RN / RN

(3.5)

(3.6)

1

p with ck D .p=2/ k . Since J. / is an equivalent norm on H 1 .RN /, uk is bounded by the second limit in (3.5), so taking v D uk shows that ck ! c. If one has uk . C yk / * w on a renumbered subsequence for some yk 2 RN 1 with jyk j ! 1, replacing v with v. yk / in (3.6), making the change of variable x 7! x C yk , and passing to the limit using (1.12) now gives Z Z rw rv C V 1 wv D c jwjp 2 wv 8v 2 H 1 .RN /: (3.7) RN

Taking v D w gives

RN

1

kwk2 D c kwkpp ;

where k kp denotes the Lp -norm in RN 1 , so if w ¤ 0, then it follows that c > 0 and 1 p=2 1=.p 2/ .1 / kwk c since kwk2 = kwkp2 1 1 .


207

Lemma 3.2. Let .uk / M be a critical sequence for J jM at the level c 2 R. Then it admits a renumbered subsequence that satisfies, in addition to the conclusions of Proposition 3.1, for n D 2; : : : ; m, Z rw .1/ rv C V .x/ w .1/ v RN Z (3.8) Dc jw .1/ jp 2 w .1/ v 8v 2 H 1 .RN /; RN

Z

1

rw .n/ rv C V 1 w .n/ v Z Dc jw .n/ jp 2 w .n/ v

RN

RN

1

J.w .1/ / D c I.w .1/ /; m X

I.w .n/ / D 1;

J 1 .w .n/ / D c I.w .n/ /;

J.w .1/ / C

m X

(3.10)

J 1 .w .n/ / D c;

(3.11)

in H 1 .RN /:

(3.12)

nD2

nD1

uk

(3.9) 8v 2 H 1 .RN /;

m X

w .n/ .

.n/

yk / ! 0

nD1 .1/

Proof. Since yk D 0, we have uk * w .1/ by (3.1), so (3.8) follows from (3.6). For n D 2; : : : ; m, .n/ uk . C yk / * w .n/ ; .n/

and taking l D 1 in (3.2) shows that jyk j ! 1, so (3.9) follows from (3.7). Taking v D w .1/ in (3.8) gives the first equation in (3.10), and taking v D w .n/ in (3.9) gives the second. Property (3.12) follows from (3.4), (3.6), and the continuity of the Sobolev imbedding. The second equation in (3.11) follows from (3.10) and the first, so it only remains to prove that m X

.n/ p

w D 1: p nD1

Since kuk kp D 1, it follows from (3.4) that

m

X

.n/ w .n/ . yk / ! 1

nD1

Let " there

p

> 0. Since C01 .RN / is dense in H 1 .RN / is a w e.n/ 2 C01 .RN / such that

.n/

w e

(3.13)

,! Lp .RN

" w .n/ p < : m

1 /,

for n D 1; : : : ; m,

208


Then, for all k, ˇ m ˇ X ˇ w e.n/ . ˇ ˇ nD1

m X

.n/

w e .

m

X

w .n/ .

.n/ yk /

.n/ yk /

nD1

p

.n/ yk /

w

.n/

.

p

.n/ yk / p

ˇ ˇ ˇ ˇ ˇ

m X

.n/

w D e

w .n/ p < " (3.14)

nD1

nD1

.n/

For sufficiently large k, the supports of the maps w e.n/ . yk / are pairwise disjoint by (3.2) and hence

m

p m m

X

X X

.n/

.n/ p

.n/ .n/ p .n/

w

w w e . yk / D e . yk / p D e p : (3.15)

nD1

nD1

p

nD1

Combining (3.13)–(3.15) gives ˇ m !1=p ˇ X

ˇ .n/ p

w e ˇ p ˇ nD1

ˇ ˇ ˇ 1ˇ ": ˇ

Let " ! 0. 3.1 Proof of Theorem 1.4 Specializing to the case V .x/ V 1 in Lemma 3.2 gives the following lemma. Lemma 3.3. Let .uk / M be a critical sequence for J 1 jM at the level c 2 R. Then it admits a renumbered subsequence that satisfies, in addition to the conclusions of Proposition 3.1, for n D 1; : : : ; m, Z rw .n/ rv C V 1 w .n/ v N R Z (3.16) Dc jw .n/ jp 2 w .n/ v 8v 2 H 1 .RN /; RN

1

J 1 .w .n/ / D c I.w .n/ /; m X

I.w .n/ / D 1;

nD1

uk

m X nD1

m X

J 1 .w .n/ / D c;

(3.17) (3.18)

nD1

w .n/ .

.n/

yk / ! 0

in H 1 .RN /:

(3.19)


209

Let .uk / M be a critical sequence for J 1 jM at the level 1 1 that satisfies the conclusions of Lemma 3.3. Then m X

I.w .n/ / D 1;

nD1

m X

J 1 .w .n/ / D 1 1

(3.20)

nD1

by (3.18), and writing I.w .n/ / D tn and using 2=p J 1 .w .n/ / 1 1 tn

gives m X

tn D 1;

nD1

m X

tn2=p 1:

nD1

Since p > 2, this implies that there is exactly one nonzero tn , say, tn0 . Then it follows from (3.20) that w .n0 / is a minimizer. Let w11 D jw .n0 / j and note that w11 0 is also a minimizer. If w11 .x0 / D 0 for some x0 2 RN n RN 1 , then by the strong maximum principle, w11 D 0 in the half-space determined by RN 1 that contains x0 , and hence also on RN 1 by the continuity of the trace imbedding H 1 ./ ,! Lp .RN 1 /. This is impossible since kw11 kp D 1, so w11 > 0 in RN n RN 1 . If w11 .x0 / D 0 at some x0 2 RN 1 , then by the Hopf Lemma, @.w11 /C .x0 / > 0 and @xN so

@.w11 / .x0 / < 0; @xN

@w11 .x0 / > 0: @xN

This violates the jump condition at x0 , so w11 > 0 on RN 3.2

1

as well.

Proof of Theorem 1.5

Let .uk / M be a critical sequence for J jM at the level 1 that satisfies the conclusions of Lemma 3.2. Then m X

I.w .n/ / D 1;

J.w .1/ / C

nD1

m X

J 1 .w .n/ / D 1

nD2

by (3.11), and writing I.w .n/ / D tn and using 2=p

J.w .1/ / 1 t1

;

2=p J 1 .w .n/ / 1 1 tn

for n D 2; : : : ; m

(3.21)

210


gives m X nD1

tn D 1;

2=p

1 t1

C 1 1

m X

tn2=p 1 :

nD2

Since p > 2 and 1 < 1 1 , this implies that m D 1. Then it follows from (3.21) .1/ that w1 D jw j is a minimizer, and w1 > 0 as in the proof of Theorem 1.4.

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Elliptic interface problems with nonhomogeneous jump conditions Received March 19, 2013; accepted March 30, 2013. Author information T. Gnana Bhaskar, Department of Mathematical Sciences Florida Institute of Technology Melbourne, FL 32901, USA. E-mail: [email protected] Kanishka Perera, Department of Mathematical Sciences Florida Institute of Technology Melbourne, FL 32901, USA. E-mail: [email protected]

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