Critical Groups in Saddle Point Theorems without a Finite Dimensional Closed Loop

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1 Math. Nachr ), Critical Groups in Saddle Point Theorems without a Finite Dimensional Closed Loop By Kanishka Perera ) of Florida and Martin Schechter of Irvine Received November 0, 000; accepted April 3, 00) Abstract. We obtain critical points with nontrivial critical groups in certain saddle point type theorems where there is no finite dimensional closed loop. An application to a semilinear elliptic boundary value problem with concave and jumping nonlinearities is given.. Introduction It is well known that critical groups play an important role in obtaining multiple solutions in a wide variety of variational problems see, e. g., Chang [3] or Mawhin and Willem [6]). Thus it is useful to have information about the critical groups of critical points produced by saddle point type theorems. We recall Theorem.. Let F be a C functional defined on a Banach space X. Assume that the critical values of F are isolated, and that each critical value corresponds to a finite number of critical points. Suppose that there are a direct sum decomposition X = X X,withX finite dimensional, ρ>0, ande X \{0} such that.).) sup B ρ X F < a := inf X F, b := inf R + e X F >, b := sup B ρ X F < +, and assume that F satisfies PS) c for all c [b ε, b + ε], forsomeε>0. ThenF has two critical points u,u such that.3).4) where j =dimx. b F u ) a > Fu ) b, C j F, u ) 0, C j F, u ) Mathematics Subject Classification. Keywords and phrases. ) Corresponding author/kperera@winnie.fit.edu c WILEY-VCH Verlag Berlin GmbH, 3086 Berlin, X/0/ $ /0

2 Perera and Schechter, Critical Groups in Saddle Point Theorems 57 The existence of a critical point u satisfying a F u ) b and C j F, u ) 0was proved in Liu [5], and a second critical point u with b F u ) <awas obtained in Schechter [0]. The fact that u can be chosen so that C j F, u ) 0wasshown in Perera [7]. Theorem. relies heavily upon the fact that the homology group H j B ρ X ) is nontrivial when j =dimx <. In the present paper we drop the assumption that X is finite dimensional so B ρ X may be contractible) and prove Theorem.. Suppose that there are a direct sum decomposition X = X X, with X finite dimensional, ρ>0, ande X \{0} such that.) and.) hold, and assume that F satisfies PS) c for all c [b ε, b + ε], forsomeε>0. ThenF has two critical points u,u such that.3) holds and.5) C j F, u ) 0, C j+ F, u ) 0 where j =dimx. We also prove Theorem.3. Suppose that there is a direct sum decomposition X = X X,with X finite dimensional, such that.6) a := inf F >, b := sup F < +, X X and assume that F satisfies PS) c for all c [a ε, b + ε], forsomeε>0. Then F has a critical point u such that.7) a F u) b, C j F, u) 0 where j =dimx. Theorems. and.3 improve some results in Perera and Schechter [9] where it was assumed that F is C and only Morse index estimates were obtained. We will prove Theorems. and.3 in the next section and give an application to a semilinear elliptic boundary value problem with concave and jumping nonlinearities in the last section.. Proofs of Theorems. and.3 We give the proof of Theorem.3 first since it is simpler. Since the critical values of F are isolated, there are a a ε, a), b b, b + ε) such that F has no critical values in [a,a) b, b ]. By PS), there are R 0,δ>0 such that.) F u) δ if F u) [a ε, b + ε] and u R 0. Set r =4b a )/δ, taker>r 0 + r, andletb = { u X : u R },S= B. Now let g : X [0, ] be a locally Lipschitz continuous function such that.) gu) = { if distu, S) r and F u) [a,b ], 0 if distu, S) R R 0 or F u) / a ε, b + ε),

3 58 Math. Nachr ) let V be a pseudo gradient for F satisfying.3) V u),f u) ) F u), V u) F u), and consider the flow ηt)u generated by V η) η = gη).4) V η), t > 0, η0) = u. The right hand side of the first equation in.4) is bounded by /δ, sothisflowexists for all time, and it is easily seen that F is nonincreasing along the flow lines. Let T = rδ/. We claim that.5) F ηt )u) a for all u S. If not, then there is a u S such that.6) F ηt)u) > a for all t [0,T]. Let.7) so that t u = inf { t>0:dist ηt)u, S ) = r },.8) dist ηt)u, S ) < r, t [0,t u ), dist ηt u )u, S ) = r. If t u T,then.9) so d dt F ηt)u) = η, F η) ) V η),f η) ) = V η),.0) F ηt )u) F u) T = b a ), which together with.6) implies that.) F u) > b, a contradiction. On the other hand, if t u <T,then tu tu dt.) ηt u )u u η dt 0 0 V ηt)u) t u < T δ δ contradicting the second equation in.8). So.5) holds. Now, since ηt ) is a homeomorphism of X that leaves X fixed and.3) sup F a < inf F, ηt )S X sup ηt )B F sup F < b, B we have the following commutative diagram induced by inclusions. = r, H j ηt )S) i i H j ηt )B) H j X \X ) H j F a ) i 3 Hj F b )

4 Perera and Schechter, Critical Groups in Saddle Point Theorems 59 Since the embedding i is nontrivial, so is i. Thisimpliesthati 3 is not injective because ηt )B is contractible, so it follows from the exact sequence of the pair F b,f a )thath j F b,f a ) 0. Hence F has a critical point u such that.4) a < Fu) < b, C j F, u) 0 see, e. g., Chang [3]), and the conclusion follows since F has no critical values in [a,a) b, b ]. Proof of Theorem.. Take a sup Bρ X F, a ),b b ε, b ),b b,b +ε) such that F has no critical values in [a,a) [b,b ) b,b ]. By PS), there are R 0 >ρ,δ>0 such that.5) F u) δ if F u) [b ε, a] and u R 0. Set r =4a b )/δ, taker>r 0 + r, andletd = R + e X ) BR,S= D, S = S\X,S = S X. Finally take a locally Lipschitz function g : X [0, ] such that.6) gu) = { if distu, S ) r and F u) [b,a ], 0 if distu, S ) R R 0 or F u) / b ε, a), and let ηt)u be the flow generated by.7) V η) η = gη) V η), t > 0, η0) = u where V is a pseudo gradient satisfying.3). Then an argument similar to the one in the proof of Theorem.3 shows that.8) F ηt )u) a for all u S where T = rδ/. Since F u) a for u S and F ηt )u) is nondecreasing in t, it follows that.9) F ηt )u) a for all u S. Let F = F.SinceηT ) is a homeomorphism of X that fixes B ρ X and.0).) sup F ηt )D sup F a < inf F, ηt )S B ρ X sup D F < b, inf B ρ X F > b,

5 60 Math. Nachr ) we have the following commutative diagram. H j ηt )S) i i H j ηt )D) i 4 ) i 3 ) H j X \ B ρ X )) H j F a H j F b i 5 H j X \B ρ X )) H j F b ) Since i is nontrivial, so are i and i 4. Therefore i 5 is not surjective because X \ B ρ X ) is contractible, so it follows from the exact sequence of the pair ) ) F a, F b that Hj F a, F b 0. Similarly, i3 is not injective because ηt )D is contractible, so it follows from the exact sequence of the pair F b, F a ) that H j+ F b, F a ) 0. So F has two critical points u,u such that.) b < F u ) < a < F u ) < b, C j F,u ) 0, Cj+ F,u ) 0..3) 3. An application Consider the semilinear elliptic boundary value problem 3.) { u = λ u q u + gx, u) in Ω, u = 0 on Ω where Ω is a bounded domain in R n with smooth boundary Ω, λ is a real parameter, <q<, and g CΩ R, R) satisfies 3.) 3.3) gx, t) = b 0 t + a 0 t + ot) uniformly as t 0, gx, t) = bt + at + ot) uniformly as t where t ± =max{± t, 0}. Recall that the set Σ of those points a, b) R for which the asymptotic problem 3.4) { u = bu + au in Ω, u = 0 on Ω has a nontrivial solution is called the Fučík spectrum. As is known, Σ consists, at least locally, of curves emanating from the points λ l,λ l )whereλ <λ <... denote the distinct Dirichlet eigenvalues of onω. ItwasshowninSchechter [] that in the square Q l =λ l,λ l+ ), Σ has two strictly decreasing curves C l,c l that

6 Perera and Schechter, Critical Groups in Saddle Point Theorems 6 pass through λ l,λ l ) such that the region I l in Q l below the lower curve C l and the region I l in Q l above the upper curve C l are free of Σ. Let II l = { a, b) R : a>a,b>b for some a,b } 3.5) ) C l, IIl = { a, b) R : a<a,b<b for some a,b } 3.6) ) Cl. We shall prove the following multiplicity results. Theorem 3.. Assume that a 0,b 0 ) I l, a, b) II l \Σ, and 3.7) Gx, t) λ l t, x Ω, t R, where Gx, t) = t 0 gx, s) ds. Then there is a λ > 0 such that 3.) has at least two nontrivial solutions for λ 0,λ ). Theorem 3.. Assume that a 0,b 0 ) I l, a, b) II l \Σ, and 3.8) Gx, t) λ l+ t, x Ω, t R. Then there is a λ λ λ, 0). < 0 such that 3.) has at least two nontrivial solutions for Corollary 3.3. Assume that a 0 = b 0 in 3.) and a = b/ σ ) in 3.3). i) If a 0 <λ l <aand 3.7) holds, then there is a λ > 0 such that 3.) has at least two nontrivial solutions for λ 0,λ ). ii) If a 0 >λ l >aand 3.8) holds, then there is a λ < 0 such that 3.) has at least two nontrivial solutions for λ λ, 0). Remark 3.4. Problem 3.) with λ > 0 and g superlinear at infinity was considered in Ambrosetti, Azorero, and Peral [], Ambrosetti, Brézis, and Cerami [], Li and Wang [4], and Perera [7]. The asymptotically linear case for λ<0was studied in Perera [8]. Before giving the proofs, we recall the following variational characterization of the curves C l and C l from Schechter []. From the variational point of view, solutions of 3.4) are the critical points of the functional 3.9) Iu) = Iu, a, b) = u au ) bu + ), u X = H0 Ω). Ω Let N l denote the subspace of X spanned by the eigenfunctions corresponding to λ,...,λ l and let M l = Nl.ItwasshowninSchechter [] that there are continuous and positive homogeneous maps θ : M l N l,τ: N l M l such that 3.0) θv) C v, τw) C w and v 0 = θw), w 0 = τv) are the unique solutions of 3.) 3.) Iv 0 + w) = Iv + w 0 ) = sup Iv + w), w M l, v N l inf Iv + w), v N l, w M l

7 6 Math. Nachr ) respectively. Let 3.3) 3.4) M l a, b) = m l a, b) = inf Iθw) +w), w M l w = sup Iv + τv)), v N l v = 3.5) ν l a) = sup { b : M l a, b) 0 }, 3.6) µ l a) = inf { b : m l a, b) 0 }. Then ν l and µ l are strictly decreasing continuous functions such that ν l λ l )= µ l λ l )=λ l,andc l : b = ν l a) andc l : b = µ l a) are the minimal and maximal curves of Σ in Q l, respectively. Now we prove Theorems 3. and 3.. We seek solutions of 3.) as critical points of 3.7) F λ u) = Ω u λ q u q Gx, u), u X where Gx, t) = t gx, s) ds, which satisfies PS) when a, b) / Σ. 0 Proof of Theorem 3.. We apply Theorem. to F λ = F λ with X = M l and X = N l. It is easily seen that in the statement of that theorem we can replace B ρ X with 3.8) Bρ = { w = w + θw, a 0,b 0 ):w M l, w ρ }. Take a point a 0,b 0 ) I l with a 0 >a 0,b 0 >b 0.Then 3.9) Gx, t) a 0 t ) + b 0 t + ) + C t, x Ω, t R where =n/n ) by 3.), so 3.0) F λ w) I w,a 0,b 0 )+ λ q w q L + C q w L M l a 0,b 0) w + C λ w q + w ). Since a 0,b 0) I l, M l a 0,b 0) > 0 by Lemma 3. of Schechter [], so it follows that there are λ,ρ>0 such that 3.) sup F λ < 0 B ρ for λ<λ. On the other hand, 3.7) implies that 3.) inf Fλ = 0 N l for all λ 0. To verify the first condition in.), take a point a,b ) Q l above C l with a <a,b <b.thenm l a,b ) < 0, so there is a w 0 M l \{0} such that 3.3) By 3.3), Iw 0 + θw 0 ),a,b ) 0.

8 Perera and Schechter, Critical Groups in Saddle Point Theorems ) Gx, t) a t ) + b t + ) C, x Ω, t R, so for s 0andv N l, F λ sw 0 + v) Isw 0 + v, a,b ) C 3.5) s Iw 0 + θw 0 ),a,b ) C C for all λ 0. Thus for λ [0,λ ), F λ has two critical points u λ,u λ such that 3.6) 3.7) F λ u λ ) 0 < Fλ u λ ), C dl Fλ,u λ ) 0, Cdl + Fλ,u λ ) 0 where d l =dimn l. On the other hand, an argument similar to the one in the proof of Lemma 4..3 of Perera [7] shows that 3.8) C q F λ, 0) = 0 for all q when λ>0. Proof of Theorem 3.. Here we apply Theorem. to F λ with X = N l, X = M l, and 3.9) B ρ = { ṽ = v + τv, a 0,b 0 ):v N l, v ρ } in place of B ρ X. Taking a point a 0,b 0) Il with a 0 <a 0,b 0 <b 0, 3.30) so 3.3) Gx, t) a 0 t ) + b 0 t + ) C t, x Ω, t R, F λ ṽ ) I ṽ, a ) λ 0,b 0 q ṽ q L + C q ṽ L m l a 0,b 0 ) v + C λ v q + v ). Since a 0,b 0) Il, m l a 0,b 0) < 0 by Lemma 3.7 of Schechter [], so there are λ < 0, ρ>0 such that 3.3) sup F λ B ρ < 0 for λ>λ. On the other hand, 3.33) inf F λ = 0 M l for all λ 0 by 3.8). To see that.) is satisfied, take a point a,b ) Q l below C l with a >a,b >b.thenm l a,b ) > 0, so there is a v 0 N l \{0} such that 3.34) and Iv 0 + τv 0 ),a,b ) 0,

9 64 Math. Nachr ) 3.35) Gx, t) a t ) + b t + ) + C, x Ω, t R, so for s 0andw M l, F λ sv 0 + w) Isv 0 + w, a,b ) C 3.36) s Iv 0 + τv 0 ),a,b ) C C for all λ 0. Thus F λ has two critical points u λ,u λ such that 3.37) 3.38) F λ u λ ) 0 > Fλ u λ ), C dl Fλ,u λ ) 0, Cdl Fλ,u λ ) 0 for λ λ, 0]. On the other hand, an argument similar to the one in the proof of Lemma. of Perera [8] shows that 0 is a strict local minimizer of F λ and hence 3.39) C q F λ, 0) = δ q0 Z when λ<0. References [] Ambrosetti, A., Azorero, J., and Peral, I.: Multiplicity Results for Some Nonlinear Elliptic Equations, J. Funct. Anal. 37 ) 996), 9 4 [] Ambrosetti, A., Brézis, H., and Cerami, G.: Combined Effects of Concave and Convex Nonlinearities in Some Elliptic Problems, J. Funct. Anal. ) 994), [3] Chang, K. C.: Infinite Dimensional Morse Theory and Multiple Solution Problems, Progress in Nonlinear Differential Equations and their Applications Volume 6, Birkhäuser Boston Inc., Boston, MA, 993 [4] Li, S. J., and Wang, Z. Q.: Mountain Pass Theorem in Order Intervals and Multiple Solutions for Semilinear Elliptic Dirichlet Problems, J. Anal. Math ), [5] Liu, J. Q.: The Morse Index of a Saddle Point, Systems Science and Mathematical Sciences ) January 989), 3 39 [6] Mawhin, J., and Willem, M.: Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences Volume 74, Springer Verlag, New York, 989 [7] Perera, K.: Critical Groups of Pairs of Critical Points Produced by Linking Subsets, J. Differential Equations 40 ) 997), 4 60 [8] Perera, K.: Multiplicity Results for Some Elliptic Problems with Concave Nonlinearities, J. Differential Equations 40 ) 997), 33 4 [9] Perera, K., and Schechter, M.: Morse Index Estimates in Saddle Point Theorems without a Finite Dimensional Closed Loop, Indiana Univ. Math. J. 47 3) 998), [0] Schechter, M.: Splitting Subspaces and Saddle Points, Applicable Analysis ), [] Schechter, M.: The Fučík Spectrum, Indiana Univ. Math. J. 43 4) 994), Department of Mathematical Sciences Florida Institute of Technology Melbourne, FL USA E mail: kperera@winnie.fit.edu Department of Mathematics University of California, Irvine Irvine, CA USA