Calc. Var. (2017) 56:20 DOI 10.1007/s00526-017-1111-2

Calculus of Variations

Multiple solutions for a class of semilinear elliptic problems via Nehari-type linking theorem Chong Li1,2 · Yanyan Liu1

Received: 25 July 2016 / Accepted: 9 January 2017 © Springer-Verlag Berlin Heidelberg 2017

Abstract This paper is concerned with multiplicity results for semilinear elliptic equations of the type −u + a(x)u = |u| p−2 u + f (x, u) in , u = 0 on ∂. We obtain at least two nontrivial solutions for this type of equations with the case that 0 is not a local minimizer of the corresponding functional under suitable hypotheses. The method we used here is based on a linking structure relevant to the Nehari manifold. Mathematics Subject Classification 35P05 · 35A15

1 Introduction In this paper, we study semilinear Dirichlet problems of the type  −u + a(x)u = |u| p−2 u + f (x, u) , x ∈ , u = 0, x ∈ ∂,

(1.1)

Communicated by P. Rabinowitz. Chong Li is supported by NSFC (11471319) and BCMIIS. Yanyan Liu is supported by NSFC (11471319).

B

Chong Li [email protected] Yanyan Liu [email protected]

1

Institute of Mathematics, AMSS, Academia Sinica, Beijing 100190, China

2

Beijing Center for Mathematics and Information Interdisciplinary Sciences (BCMIIS), Beijing, China

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C. Li, Y. Liu N

 a bounded domain with smooth boundary ∂ in R N (N ≥ 3), a(x) ∈ L 2 () and p ∈ (2, 2∗ ), where 2∗ = N2N −2 for N ≥ 3. The corresponding functional of the Eq. (1.1) is    1 1 |∇u|2 + a(x)u 2 d x − |u| p d x − J (u) = F(x, u)d x, where 2  p   s F (x, s) = 0 f (x, t) dt. The nonlinearity f (x, s) satisfies some of the following hypotheses: ( f 1 ) f (x, s) ∈ C 1 ( × R, R) and f (x, s) = o (s) uniformly in x as s → 0.   p −2 p ∈ (2, p], such that  f s (x, s) ≤ C 1 + |s|  ( f 2 ) There are constants C > 0 and any  for x ∈  and s ∈ R. ( f 3 ) 0 ≤ s f (x, s) ≤ p F (x, s) , for x ∈  and s ∈ R. ( f 4 ) f s (x, s) > f (x,s) s , s = 0. For more general nonlinear term h(x, s) instead of |s| p−2 s + f (x, s), we assume (h 1 ) h ∈ C ( × R, R).   (h 2 ) There are constants C1 > 0 and q ∈ (2, 2∗ ) such that |h (x, s)| ≤ C1 1 + |s|q−1 for x ∈  , s ∈ R. (h 3 ) h (x, s) = o (s) uniformly in x as s → 0. (h 4 ) There are ζ > 2 and M > 0, such that for all x ∈ , |s| ≥ M, s

0 < ζ H (x, s) ≤ sh (x, s) ,

where H (x, s) = 0 h (x, t) dt. (h 5 ) h ∈ C 1 ( × R, R) and h s (x, s) >

h(x,s) s ,

s = 0.

Note that it is easy to verify that ( f 1 ) ⇒ (h 1 ) + (h 3 ) , ( f 2 ) ⇒ (h 2 ) , ( f 4 ) ⇒ (h 5 ) as h(x, s) = |s| p−2 s + f (x, s). The study of multiplicity solutions for superlinear elliptic equations  −u + a(x)u = h (x, u) , x ∈ , (1.2) u = 0, x ∈ ∂, with energy functional 1 J(u) = 2

 

|∇u|2 + a(x)u 2 − H (x, u)d x,

has attracted immense attention in recent decades. In the pioneer work of Ambrosetti and Rabinowitz [1] in 1973, the authors got infinitely many solutions with odd symmetry of the nonlinearity h(x, s). But for the more general type of nonlinearity, they obtained two nontrivial solutions under the situation that 0 is a local minimizer of J. More precisely, with the assumptions (h 1 )–(h 4 ), Ambrosetti and Rabinowitz showed that the Eq. (1.2) as a(x) = 0 admits at least two nontrivial solutions in [1] (see also [5]) by Mountain-pass theorem. Under the same hypotheses, further progress was achieved by Wang [2], who found the third one in 1991 by employing linking approach and Morse theory. Recently, Li and Li [3] found a fourth nontrivial solution for (1.1) as a(x) = 0 under the hypotheses ( f 1 ) − ( f 4 ) and (h 4 ). The Nehari-type linking method which they introduced in [3] would be useful to obtain more solutions for (1.1) as a(x) = 0, which inspired us to consider superlinear Dirichlet problems with the case that 0 is not a local minimizer of J . For the case a(x) = 0, if 0 is not a local minimizer of J, (1.2) provokes some mathematical difficulties that make the study of the question particularly interesting. Benci and Rabinowitz

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proved some abstract linking theorems for finding saddle point of J in [4], which yielded one nontrivial solution for (1.2) under additional assumption H (x, s) ≥ 0. Li and Liu got the similiar result in [6] by local linking method without the additional assumption that H (x, s) maintains the sign. Furthermore, if a(x) = λ is a constant function and λ lies in a small enough left neighborhood of λi ∈ σ (−) for i ≥ 2, people obtained more than one solution with suitable assumptions of h(x, s). For instance, in 2004, Mugnai [7] gave an interesting result for −u − λu = h (x, u). He obtained two nontrivial solutions by Theorem 3 of [7], and found the third one by an additional linking structure with additional assumption that h (x, u) behave exactly like |u| p−2 u for |u| large. In 2007, Rabinowitz et al. [8] got an attracting consequence by using a combination of bifurcation analysis and minimax methods under more natural and weaker conditions. Let us briefly sketch Nehari-type linking method in [3]. As we all know, (1.1) with a(x) = 0 possesses at least three nontrivial solutions including two Mountain-pass points u 1 , u 2 , and a sign-changing solution u 3 , and critical groups of u 1 , u 2 , u 3 are of the form Cq (J, u 1 ) ∼ = Cq (J, u 2 ) ∼ Li and Li [3] showed that the Nehari = δq1 G, Cq (J, u 3 ) ∼ = δq2 G respectively. manifold N = u ∈ E\ {0} : J  (u) , u = 0 is differentiable homeomorphic to the unit sphere in H01 (), which could offer a homological linking structure. Then they got a new nontrivial solution u 4 with Ck+1 (J, u 4 ) = 0 for some k ≥ 2. In view of a(x) = 0, N may become invalid in some directions, the linking structure of N does not exist. Moreover, due to the vanishing Mountain-pass structure, we just know one nontrivial solution of (1.1) with very little information about critical groups. Hence, we need to find new linking structure and distinguish between nontrivial solutions by known information. Before stating our results, we recall some notions. Let −∞ < λ1 ≤ · · · ≤ λl < 0 = λl+1 = · · · = λm < λm+1 ≤ λm+2 ≤ · · · be the sequence of



−u + a(x)u = λu, u = 0,

x ∈ , x ∈ ∂,

(1.3)

where each eigenvalue is repeated according to its multiplicity. Let e1 , e2 , . . . , em , em+1 , . . . be the corresponding orthonormal eigenfunctions in L 2 (). Define E = H01 () for simplicity and let E k = span {e1 , e2 , . . . , ek } , where E k is the subspace of E spanned by the eigenfunctions corresponding to λ1 , . . ., λk and E k⊥ is the orthogonal complement of E k in E. Denote E − = span {e1 , e2 , . . . , el } , E 0 = span {el+1 , . . . , em } , E + = span {em+1 , em+2 , . . .} , / σ (− + a(x)). then we have E = E − ⊕ E 0 ⊕ E + . Evidently, E 0 = {0} when 0 ∈ If we set  < u, v >a =



∇u + ∇v + +a(x)u + v + d x −

 

∇u − ∇v − +a(x)u − v − d x +

 

∇u 0 ∇v 0 d x,

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for u, v ∈ E, u = u + + u 0 + u − , v = v + + v 0 + v − , u + , v + ∈ E + , u 0 , v 0 ∈ E 0 , u − , v − ∈ E − , then we get a new norm ·a : E → R satisfying  ua2 = (|∇u|2 + a(x)u 2 )d x, u ∈ E + ,   2 ua = − (|∇u|2 + a(x)u 2 )d x, u ∈ E − , 

ua2 = ∇u2L 2 , u ∈ E 0 ,

2 2 2 ua2 = u + a + u − a + u 0 a , u ∈ E,

2 2 where u = u + + u 0 + u − , u + ∈ E + , u 0 ∈ E 0 , u − ∈ E − . Note that ua2 = u + a + u − a as E 0 = {0}. Furthermore, it is easy to check that < . . . >a is an inner product of E. Based on the definitions above, J is reduced to the following form:  

− 2  1

1  + 2 p

|u| d x − u a− u a − J (u) = F(x, u)d x. (1.4) 2 p   We set



N + = u ∈ E + \ {0} : J  (u) , u = 0 , SE =  u ∈ E :  u a = 1 , + + SE =  u ∈ E :  u a = 1 ,

where N + is the subset of N . Our main theorem concerning the existence of multiplicity solutions of (1.1) reads: Theorem 1.1 (a) If f satisfies ( f 1 )-( f 4 ), with p ∈ (2, 2∗ ) and h(x, s) satisfies (h 4 ), then (1.1) has a nontrivial solution u 1 with J (u 1 ) > 0 and Cm+1 (J, u 1 ) = 0. (b) Furthermore, suppose that for fixed k ∈ N, k ≥ m +1, λk+1 > λk . If there exist constants pk > 2, C > 0 s.t. f (x, s) satisfies ( f 1 )-( f 4 ) with p ∈ (2, pk ] and h(x, s) satisfies (h 4 ) and (1 − C) λk+1 > λk + C, then (1.1) has at least two nontrivial solutions u 1 , u 2 for p ∈ (2, p]. In particular, we have Cm+1 (J, u 1 ) = 0, Ck+1 (J, u 2 ) = 0 and p ∈ (2, pk ],  J (u 1 ) = J (u 2 ). If J (u 1 ) = J (u 2 ), (1.1) admits at least three nontrivial solutions. Remark 1.2 The structure we established in the proof of (a) is similiar to the usual linking subset in [9] to some extent. Remark 1.3 In (b) of Theorem 1.1, the critical point u 1 is based on (a) of Theorem 1.1 while the existence of the second solution u 2 depends on the gaps of consecutive eigenvalues of − + a(x). Fortunately, it is possible for us to find k ∈ N satisfying the condition of (b) if C is properly small and Li and Li in [3] gave an example for a = 0. Remark 1.4 From (b) of Theorem 1.1, we also know that the more k that satisfiy the conditions of (b) we have, the more solutions we might find.

2 Proof of the main results For R > r > 0, let Ba (0, R) = {u ∈ E : ua ≤ R}. We define the subset Dr,R = u ∈ E k+1 (e) : u = w + μek+1 , w ∈ E k , μ ∈ [r, R] , ua ≤ R

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of E k+1 (e) = span {E k , ek+1 } and the subset Q = N + ∩ E k⊥ of E k⊥ . It is easy to see that ∂ Dr,R can be written as ∂ Dr,R = D1 ∪ D2 , (2.1) where D1 = {u = w + r ek+1 : w ∈ E k , ua < R} and D2 = {u = w + μek+1 : w ∈ E k , μ ∈ [r, R] , ua = R}. Next, in view of [12], we have a property for operater − + a(x): 

Lemma 2.1 δ =

in f

u∈E + ,|∇u|2 =1

ua > 0.

Proof (See Willem [12] for more here. By definition,   details) We just sketch the arguments ∀ u ∈ E + , we have ua2 =  (|∇u|2 + a(x)u 2 )d x ≥ λm+1  u 2 d x. For a minimizing   sequence {u n } ⊂ E + , χ(u n ) =  a(x)u 2n d x, |∇u n |2 = 1, 1 + χ(u n ) → δ 2 , we have 1 χ(·) is weakly u 0 ∈ E + satisfying u n u 0 in  H0 for2a subsequence. As the functional 2 continuous: δ = 1+ χ(u 0 ) ≥  |∇u 0 | d x+ χ(u 0 ) ≥ λm+1  u 20 d x. If u 0 = 0, δ = 1    and if u 0 = 0, δ 2 ≥ λm+1  u 20 d x > 0. With the aid of Lemma 2.1, we have the following properties about N + under some hypotheses, which are more general than what we need. Lemma 2.2 For h(x, s) = |s| p−2 s + f (x, s), suppose that ( f 3 ), (h 2 ), (h 3 ), (h 5 ) hold. Then there exist unique C 1 function t (·) : S E+ → R+ , α > 0 and β > 0, such that (i) t (·) · : S E+ → N + is a diffeomorphism. (ii) sup J (t u ) = J (t ( u)  u ) for  u ∈ S E+ . t∈(0,+∞)

(iii) inf t ( u ) ≥ α > 0.  u ∈S E+

(iv) inf J (u) ≥ β > 0. u∈N +

Proof Let



u ) , t u g (t,  u ) = J  (t  |∇t = u |2 + a(x)(t u )2 − h(x, t u ) · t ud x   h(x, t u ) · t ud x = t u a2 −   h(x, t u ) · t u d x, = t2 − 

(2.2)

for all  u ∈ S E+ , t ∈ R + . By ( f 3 ), sup g (t,  u ) > 0 and g (t,  u ) < 0 for t large enough, then there exists t ( u) ∈ R+ t∈R +

satisfying

   h(x, t ( u)  u) ·  ud x = 0 g (t ( u) ,  u ) = 0 ⇔ t ( u ) t ( u) −   h(x, t ( u)  u) ·  u d x. ⇔ t ( u) = 

(2.3)

By (2.3), we have ∀ u ∈ S E+ , ∃t ( u ) ∈ R + satisfying t ( u) ·  u ∈ N + ⇔ g (t ( u) ,  u ) = 0.

(2.4)

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Moreover, by (2.2) and (h 5 ), gt (t ( u) ,  u ) = 2t ( u) −

 



< 2t ( u) − 2 

+

u)  u ) t ( u)  u 2 − h (x, t ( h s (x, t ( u)  u)  ud x



h (x, t ( u)  u)  u d x = 0.

(2.5)

u 0 ) ∈ R+ , S E , if t0 u 0 ∈ N + , then g (t0 ,  u 0 ) = 0, gt (t0 ,  u0) < i.e. for each (t0 ,  0. By using the implicit function theorem, there exist r , r > 0 and ∃|t u) ∈ ( 1 2   u 0 , r1 ) ∩ S E+ , B (t0 , r2 ) such that t ( u) ,  u ) < 0. u 0 ) = t0 , t ( C 1 Ba ( u)  u ∈ N + and gt (t ( This implies that t (·) · is a diffeomorphism from S E+ onto N + . Thus (i) is proved. For all  u ∈ S E+ , t ∈ R + , consider  1 2 H (x, t u )d x. (2.6) J (t u) = t − 2  We have

 d h(x, t u) ·  ud x J (t u) = t − dt   1 ≥ 0, t ≤ t ( u) , = g (t,  u) = ≤ 0, t ≥ t ( u) , t

by (2.2) and (2.5). Therefore, (2.7) shows that

(2.7)

J (t u ) = J (t ( u)  u ). We complete

sup t∈(0,+∞)

the proof of (ii). By way of negation, ∃ tn ·  u n ∈ N + , tn > 0,  u n ∈ S E+ , tn → 0. Combining (h 2 ) and (h 3 ) we obtain |h (x, s)| ≤ ε |s| + Cε |s| p−1 , ∀s ∈ R, (2.8) for ∀ε ∈ (0, λm+1 ), ∃ Cε > 0. By (2.8), we get |h (x, tn u n ) tn u n | ≤ ε |tn u n |2 + Cε |tn un | p and hence tn2 ≤

ε λm+1

p

tn2 + Cε tn

 

| un | p ,

by (2.3) and (2.9). Therefore, using Lemma 2.1 and Sobolev inequality, we have  ∼ ∼ | ∃C > 0, u n | p ≤ C, 

and thus 0<1−

ε λm+1

≤

p−2 Cε tn

(2.9) (2.10)

(2.11)

 

| u n | p → 0, n → +∞.

(2.12)

We get a contradiction. This proves (iii). As ∀ε ∈ (0, λm+1 ), ∃Cε > 0, for ∀s ∈ R, by (2.8), we have |H (x, s)| ≤ Thus for u ∈ E + ,

ε 2 Cε p |s| + |s| . 2 p

 1 ua2 − H (x, u)d x 2    1 ε Cε |u|2 d x − |u| p d x. ≥ ua2 − 2 2  p 

J (u) =

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(2.13)

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Observe that ua2 ≥ λm+1 |u|2 for u ∈ E + . Combining with Sobolev inequality and Lemma 2.1, we can choose C  > 0, satisfying   1 ε J (u) ≥ (2.14) ρ2 − C ρ p . 1− 2 λm+1 

Take ρ > 0 sufficiently small for (2.14), we have β =

inf

v∈E + , v=ρ

J (v) ≤ inf J (u) . The u∈N +

proof of Lemma 2.2 is completed.

 

Remark 2.3 Notice that N vanishes in some directions because of the existence of E − ⊕ E 0 , so we consider the property of N + instead of N . We get a homeomorphism between N + and S E+ , which could offer a linking structure we need. Lemma 2.4 Suppose ( f 3 ) holds, k ≥ m, then there exists Rk > 0, for u ∈ D2 , J (u) < 0, where D2 = {u = w + μek+1 : w ∈ E k , μ ∈ [r, R] , ua = R}. Proof By the definition of D2 , as u ∈ D2 , we have u = u+ + u0 + u− By (2.15) and ( f 3 )

and

ua = R.

(2.15)

 



1  p

u + 2 − u − 2 − 1 |u| d x − F(x, u)d x a a 2 p    1 1 |u| p d x. ≤ ua2 − 2 p 

J (u) =

(2.16)

As D2 is compact and p > 2, we can choose R large enough, for Rk ≥ R, such that J (u) < 0 for u ∈ D2 . The assertion follows.   For k ≥ m, Rk > 0 given by Lemma 2.4 and α, β appearing on Lemma 2.2, we get the property of value separation about ∂ Dr,Rk and Q as follows: √ Lemma 2.5 Suppose ( f 1 )-( f 4 ) hold, 0 < r < min{ α2 , β}, then p ∈ (2, p] , inf J (u) > (1) for k = m, p ∈ (2, 2∗ ),  u∈Q

sup

J (u);

u∈∂ Dr,Rk

(2) for k ∈ N, k ≥ m + 1 s.t. λk+1 > λk , (1 − C) λk+1 > λk + C, where C is given by ( f 2 ), there exists pk > 2 for p ∈ (2, pk ],  p ∈ (2, p], s.t. inf J (u) >

u∈Q

Proof Note that k = m and r < min{ α2 ,

sup

J (u) .

(2.17)

u∈∂ Dr,Rk

√

β 2,

β} ⇒ sup J (u) < u∈D1

and also derive

sup J (u) < 0 according to Lemma 2.4. By (iv) of Lemma 2.2, we have inf J (u) ≥ β > 0. u∈Q

u∈D2

This leads to inf J (u) ≥ β > u∈Q

β 2

> sup J (u). u∈∂ Dr,R

In order to prove (2), we need to make estimates for inf J (u) and u∈Q

tively. For u ∈ Q,



 

|∇u|2 + a(x)u 2 d x =



sup

J (u) respec-

u∈∂ Dr,Rk

 |u| p +



f (x, u) u.

(2.18)

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In virtue of the definition of J and by (2.18), we have     1 1 1 |u| p + J (u) = − f (x, u) u − F (x, u) . 2 p 2   

(2.19)

Let u = t ( u)  u , t ( u ) > 0,  u ∈ S E+ , (2.18) yields   f (x, t ( u)  u)  u | u) u| p +  1 = t p−2 ( t u ( )  and this gets

⎛ t ( u) = ⎝

1−

⎞

 





1

p−2 f (x,t( u ) u ) u t( u) ⎠ || p

u

.

(2.20)

g (x, t ( u)  u ) d x,

(2.21)

Combining (2.19) with (2.20),  J (u) =

  1− 1

1 − 2 p

 



f (x,t( u ) u ) u t( u)

u|  |

p



p  p−2

 +

2 p−2



where g (x, s) = 21 f (x, s) s − F (x, s), s ∈ R. Observe that ( f 4 ) ⇒ g (x, s) ≥ 0 for ∀x ∈  and ∀s ∈ R, so (2.21) yields   p  u ) u ) u p−2   1 −  f (x,t( t( u) 1 1 J (u) ≥ − . (2.22)   2 2 p u | p p−2  | By ( f 2 )

 | f (x, s) s| ≤ C s 2 +

 1 p |s|  .  p−1

(2.23)

Hence, we have    u)  u)  u C  p  f (x, t ( | u) t p−2 (  u2 + u | ≤C t ( u)  p−1     p  p p 1 C 1−  p p | | || ·  u ≤C  u2 +  p −2   p−1   u | p p−2  |   2( p−p)  p( p−2) p C 1−  2 p p | u| ≤C  u + . (2.24) · ||  p − 1   By Hölder inequality  u κL 2 ()  , κ= u 1−κ u  L p () ≤  ∗ L 2 ()

N N −2 − . p 2

Notice that  u  L 2∗ () ≤ C0 , thus, (2.24) can be written as   p )[2 p−( p−2)N ] p− p ( p−  p N u)  u)  u C  f (x, t ( ||1− p C0 p  u  L 2 ()( p−2) p  u2 + . ≤C t ( u)  p−1 

123

(2.25)

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Thereby, for all u ∈ Q, based on (2.22) and (2.25), we get ⎤ p ⎡     ( p−p)[2 p−( p−2)N ] p−2 p− p 2( p−2) p  p N 1 1 C 1 ⎣ C ⎦ ||1− p C0 p − − 1− J (u) ≥ 2 p λk+1  p−1 λk+1 p

p−2 ×C0−N λk+1



N p

− N 2−2



.

(2.26)

On the other hand, for u ∈ ∂ Dr,Rk , set   = R+u+ ⊕ E 0 ⊕ E −. E(u)

(2.27)

   Then according to the definition of E(u), we have u ∈ E(u). As dim(span( E(u))) ≤ m +1 0 −   for 0 < sup J < +∞ and J (v) < 0 in [ E(u)\B Rk (0)] ∪ E ⊕ E , there exists vu ∈ E(u)  E(u)      satisfying sup J = J (vu ) ≥ J (u) and J (vu ), vu = 0. Define θ 2 =  |∇vu |2 + a(x)vu2 d x,  E(u)    vu  vu = θ . Combining with J (vu ), vu = 0, we get   vu f (x, θ vu ) p−2 p | 1=θ vu | +  . θ  Therefore we conclude that ⎛ θ =⎝

1−



vu f (x,θ vu ) θ  vu | p  | 

⎞ ⎠

1 p−2

.

(2.28)

By the assumption ( f 3 ), (2.18) and (2.28), for all u ∈ ∂ Dr,Rk , we have J (u) ≤ J (vu )     1 1 1 |∇vu |2 + a(x)vu2 d x + = f (x, vu ) vu − F (x, vu ) − 2 p p    ⎞ 2 ⎛  p−2     vu vu )  f (x,θ 1 − 1 1 1 1 ⎝ θ ⎠  − − θ2 = ≤ 2 p 2 p vu | p  |   − p p−2 1 1 2 || · | − vu | ≤ 2 p    p 1 1 || λkp−2 . ≤ − 2 p 

(2.29)

Now, combining (2.26) with (2.29), we show that ∃ pk > 2, the following inequality holds ⎤ ⎡   ( p−p)[2 p−( p−2)N ] p− p 2( p−2) p  p N − p−2 N N − N −2 1 C C 1− p ⎦C p λ p 2 ⎣1 − || p C0 − 0 k+1 λk+1  p−1 λk+1 > ||

p−2 p

λk ,

(2.30)

p ∈ (2, p]. Obviously, as p =  p = 2, (2.30) ⇔ (1 − C) λk+1 > λk + C. for p ∈ (2, pk ],  p ∈ (2, p] and so (2.17) holds as Thereby, ∃ pk > 2 , such that (2.30) holds for p ∈ (2, pk ],  well. We complete the proof.  

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Remark 2.6 For the further quatitative analysis of pk , we refer the reader to [3]. Indeed, in contrast with Theorem 2.1 in [3], we have a more general version of Nehari-type linking theorem. Lemma 2.7 Under the hypotheses of Lemma 2.5 and (h 4 ), for k ≥ m, ∂ Dr,Rk and Q homotopically link. Moreover, 

Ck =

inf   sup h∈h Dr,Rk ,E\Ba (0, r2 ) u∈Dr,R

J (h (u)) > k

sup

J (u)

u∈∂ Dr,Rk

is a critical value of J , where  r   ⎧ ⎫  r  ⎨ h ∈ C Dr,Rk , E\Ba 0, 2 : h|∂ Dr,Rk = id|∂ Dr,Rk , ⎬ = h Dr,Rk , E\Ba 0, h is a continuous map   ⎩ ⎭ 2 from Dr,Rk into E\Ba 0, r2 . 

Proof By the definition of homotopical linking, we first show Q ∩ ∂ Dr,Rk = ∅. Suppose by contradiction, Q ∩ ∂ Dr,Rk = ∅. Since ∂ Dr,Rk = D1 ∪ D2 , J (u) < 0 for u ∈ D2 and inf J (u) ≥ β > 0, we have Q ∩ D1 = ∅. Then for u ∈ Q ∩ D1 , by (iii) of Lemma 2.2 and u∈Q

) which is impossible as r < α2 ≤ t (ek+1 the definition of Q, we get u = rek+1 , r = t (ek+1 2 .  ),r   Next, we claim that ∀h ∈ h Dr,Rk , E\Ba θ, 2 , h Dr,Rk ∩ Q = ∅. Let Pk be the orthogonal projection of E onto E k and define % &  (s, w + μek+1 ) = (1 − s) w + s Pk h (w + μek+1 ) F & % + (1 − s) μ + s (I − Pk ) h (w + μek+1 )a − t (ek+1 ) ek+1 ,

for all s ∈ [0, 1], w + μek+1 ∈ Dr,Rk . Then, for s = 0 and s = 1,we obtain  (0, w + μek+1 ) = w + (μ − t (ek+1 )) ek+1 , F

(2.31)

%  (1, w + μek+1 ) = Pk h (w + μek+1 ) + (I − Pk ) h (w + μek+1 )a F & −t (ek+1 ) ek+1 .

(2.32)

and

Clearly, for s ∈ [0, 1], w + μek+1 ∈ ∂ Dr,Rk ,  (s, w + μek+1 ) = w + (μ − t (ek+1 )) ek+1 = 0, F

(2.33)

due to r and Rk we choose. By homotopy invariance of degree, we have      (0, ·) , Dr,Rk , 0  (1, ·) , Dr,Rk , 0 = deg F deg F   = deg id| E k , E k ∩ B (θ, Rk ) , 0 · deg (1 − t (ek+1 ), (r, Rk ) , 0) = 1. Consider the map H : [0, 1] × S E → R 1 ,  '

+   u+ u a t  + st (ek+1 ), for ∀s ∈ [0, 1] ,  u a = 1,  u + = 0, (1 − s)  + u a H (s,  u) = for ∀s ∈ [0, 1] ,  u a = 1,  u + = 0, st (ek+1 ), and it is easy to verify H ∈ C([0, 1] × S E ).

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20

Let F (s, w + μek+1 ) = Pk h (w + μek+1 ) % + (I − Pk ) h (w + μek+1 )a   h (w + μek+1 ) −H s, (2.34) ek+1 . h (w + μek+1 )a    Notice that h ∈ h Dr,Rk , E\Ba 0, r2 , so (2.34) is well-defined and % & F (1, w + μek+1 ) = Pk h (w + μek+1 ) + (I − Pk ) h (w + μek+1 )a − t (ek+1 ) ek+1 . We get

 (1, w + μek+1 ) . F (1, w + μek+1 ) = F

Since F (s, w + μek+1 ) is continuous in [0, 1] × Dr,Rk , and a similar argument to (2.33) gets F (s, w + μek+1 ) = 0, ∀s ∈ [0, 1] , ∀w + μek+1 ∈ ∂ Dr,Rk , we obtain

    deg F (0, w + μek+1 ) , Dr,R , 0 = deg F (1, w + μek+1 ) , Dr,Rk , 0    (1, w + μek+1 ) , Dr,Rk , 0 = deg F = 1.

(2.35)

Notice that (2.35) is equivalent to finding u 0 ∈ Dr,Rk satisfying ' Pk h (u 0 ) = 0,   h(u 0 ) h (u 0 )a = t h(u , 0 ) a

so (2.35) implies that ∂ Dr,Rk and Q homotopically link. It is easy to know that J satisfies (P S) condition (See Willem [12] for detailed proofs). Following from (iv) of Lemma 2.2, we have Ck ≥ inf J (u) > u∈Q

sup

J (u) > β.

u∈∂ Dr,Rk

Now we show that Ck is a critical value of ϕ. Let K = u ∈ E : J  (u) = 0 and K Ck = {u ∈ K : J (u) = Ck}. Suppose by contradiction, K Ck = ∅. We can take ε > 0 sufficiently  small such that K ∩ J Ck +2ε \J Ck −2ε = ∅ and Ck − 2ε >

sup u∈∂ Dr,Rk

J (u) > β >

sup

u∈Ba (0,r )

J (u) .

   Let  h ∈ h Dr,Rk , E\Ba 0, r2 s.t.

  sup J  h (u) < Ck + ε.

u∈Dr,Rk

Then by standard deformation lemma, we know that  r     , ∀t ∈ [0, 1] , η t,  h (·) ∈ h Dr,Rk , E\Ba 0, 2 where η (t, ·) are 1-parameter family of homeomorphism of E with η (t, ·) | E\ J Ck +2ε \J Ck −2ε  = id| E\ J Ck +2ε \J Ck −2ε 

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       and η 1, J Ck +ε ⊂ J Ck −ε . Define ξ (u) = η 1,  h (u) , then ξ (u) ∈ h Dr,Rk , E\Ba 0, r2 . We get Ck ≤ sup J (ξ (u)) < Ck − ε, u∈Dr,Rk

 

a contradiction! The proof is completed.

Corollary 2.8 Furthermore, the sets ∂ Dr,Rk and Q in Lemma 2.7 homologically    link and Hk+1 J d , J a = 0 , where a = sup J (u), d > max J (u), Hk+1 J d , J a denotes u∈Dr,Rk

u∈∂ Dr,Rk

  the singular k + 1-relative homology group of topological pair J d , J a . Proof The proof is completed immediately by applying Definition 2.2, Proposition 2.3 and Lemma 2.4 of [3].  

3 Proof of Theorem 1.1 Proof (a) follows directly from (1) of Lemma 2.5, Lemma 2.7 and Corollary 2.8. p ∈ (2, p], Observe that if (1 − C) λk+1 > λk + C and k ≥ m + 1, for p ∈ (2, pk ],  according to (2) of Lemma 2.5, Lemma 2.7 and Corollary 2.8,     Hm+1 J dm , J am = 0, Hk+1 J dk , J ak = 0, am =

J (u), dm > max J (u), ak =

sup

u∈Dr,Rm

u∈∂ Dr,Rm

sup

J (u), dk > max J (u). So using

u∈∂ Dr,Rk

the Morse inequalities, there are nontrivial critical points u, v with

u∈Dr,Rk

Cm+1 (J, u) = 0, Ck+1 (J, v) = 0 and u might be equal to v. Suppose that there exists unique nontrivial solution u 1 for the equation (1.1) and J (u 1 ) > 0. Then we know that Cq (J, u 1 ) = 0 when q = m +1, k +1. We can choose  > 0 satisfying

2   J (u 1 ) > . Since J (u) = − 21 u − a − 1p  |u| p d x −  F(x, u)d x ≤ 0 = J (0) for all u ∈ E − ⊕ E 0 and we can choose  small enough satifying   1 1 |u| p d x − J (u) = ua2 − F(x, u)d x > 0 = J (0) 2 p   on E + ∩ Ba (0, ε) \{0} as |∇u| L 2 ≤ 1δ ua , thus J has a local linking at 0. As dim(E − ) = m for 0 ∈ / σ (−+a(x)) and dim(E − ⊕ E 0 ) = m for 0 ∈ σ (−+a(x)), by Proposition 2.3 of [13], we have Cq (J, 0) ∼ = δqm G. Consider the exact sequence of the pair (E, J  ): i∗

j∗

∂∗

i∗

j∗

· · · → Hq (E) → Hq (E, J  ) → Hq−1 (J  ) → Hq−1 (E) → · · ·. Since Hq (E) ∼ = 0 for q ≥ 1, we have Hq (E, J  ) ∼ = Hq−1 (J  ), q ≥ 2. Consider the exact sequence of the pair (J  , J − ) : j∗ ∂∗ i∗ ∂∗ i∗ q (J − ) → q (J  ) → q−1 (J − ) → ···→ H H Hq (J  , J − ) → H ·· .

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(3.1)

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For q ≥ 1, we get q (J  ) ∼ H = Hq (J  , J − ) ∼ = Cq (J, 0) = Thus, combining (3.1) with (3.2), for q ≥ 1, we have q (J  ) = Hq+1 (E, J  ) ∼ = Hq (J  ) ∼ =H This shows that for q ≥ 2, Cq (J, u 1 ) ∼ = Hq (E, J  ) =



G, 0,





G, 0,

q = m, q = m.

G, 0,

q = m, q = m.

20

(3.2)

q = m + 1, q = m + 1,

contradicting Cm+1 (J, u 1 ) = 0 and Ck+1 (J, u 1 ) = 0. This is the desired result. Furthermore, suppose Theorem 1.1 admits only two nontrivial solutions u 1 = u 2 satisfying J (u 1 ) = J (u 2 ) = c > 0, then for Cm+1 (J, u 1 ), Ck+1 (J, u 1 ), Cm+1 (J, u 2 ), Ck+1 (J, u 2 ), two of them are nontrivial at the least. Similar to the proof above, for q ≥ 2, we have  G, q = m + 1,  ∼ c+  Hq (E, J ) = Hq (J , J ) = 0, q = m + 1. By Theorem I.4.2 of [9], we get H∗ (J c+ , J  ) ∼ = C∗ (J, u 1 ) ⊕ C∗ (J, u 2 ). Thus Ck+1 (J, u 1 ) ∼ = Ck+1 (J, u 2 ) ∼ = 0 and either Cm+1 (J, u 1 ) ∼ = 0 and Cm+1 (J, u 2 ) ∼ = G ∼ ∼ or Cm+1 (J, u 1 ) = G and Cm+1 (J, u 2 ) = 0, so we get a contradiction. That’s the precise statement (b). The proof of Theorem 1.1 is thereby completed.   Acknowledgements We are grateful to the referees for perusing this manuscript and giving valuable suggestions. Moreover, the first author would like to appreciate for strong supports by NSFC (11471319) and BCMIIS (Beijing Center for Mathematics and Information Interdisciplinary Sciences).

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