The authors prove that a semilinear elliptic boundary value problem has five solutions when the range of the derivative of the nonlinearity includes at least the first two eigenvalues. Extensive use is made of Lyapunov–Schmidt reduction arguments, the mountain pass lemma, and characterizations of the local degree of critical points

Castro, Alfonso

Cossio, Jorge

Scholarship@Claremont

Multiple Solutions for a Semilinear Dirichlet Problem

Keck Graduate Institute

English

The authors prove that a semilinear elliptic boundary value problem has five solutions when the range of the derivative of the nonlinearity includes at least the first two eigenvalues. Extensive use is made of Lyapunov-Schmidt reduction arguments, the mountain pass lemma, and characterizations of the local degree of critical points

Claremont Colleges Scholarship @ Claremont All HMC Faculty Publications and Research HMC Faculty Scholarship 11-1994 Multiple Solutions for a Nonlinear Dirichlet Problem Alfonso Castro Harvey Mudd College Jorge Cossio Universidad Nacional de Colombia Follow this and additional works at: https://scholarship.claremont.edu/hmc_fac_pub  Part of the Mathematics Commons Recommended Citation Castro, Alfonso and Cossio, Jorge, "Multiple Solutions for a Nonlinear Dirichlet Problem" (1994). All HMC Faculty Publications and Research. 1165. https://scholarship.claremont.edu/hmc_fac_pub/1165 This Article - postprint is brought to you for free and open access by the HMC Faculty Scholarship at Scholarship @ Claremont. It has been accepted for inclusion in All HMC Faculty Publications and Research by an authorized administrator of Scholarship @ Claremont. For more information, please contact scholarship@cuc.claremont.edu. SIAM J. MATH. ANAL.Vol. 25, No. 6, pp. 1554-1561, November 1994() 1994 Society for Industrial and Applied Mathematics0O7MULTIPLE SOLUTIONS FOR ANONLINEAR DIRICHLET PROBLEM*ALFONSO CASTRO# AND JORGE COSSIO$Abstract. The authors prove that a semilinear elliptic boundary value problem has five solu-tions when the range of the derivative of the nonlinearity includes at least the first two eigenvalues.Extensive use is made of Lyapunov-Schmidt reduction arguments, the mountain pass lemma, andcharacterizations of the local degree of critical points.Key words, nonlinear elliptic equations, multiplicity of solutions, local degree, mountain passlemmaAMS subject classifications. 35J65, 35J201. Introduction. Let f" R R be a differentiable function such that f(0) 0,and(1.1) ff(oc) lim f(u) e .Let f be a smooth bounded region in n, and A the Laplacian operator. Let/1 < ’2 -- _ k -- be the eigenvalues of-A with Dirichlet boundary conditionin f.The solvability of the boundary value problemAu + f(u) 0 in 12,(1.)u 0 on 0,has proven to be closely related to the position of the numbers if(0), ff(cx)) withrespect to the spectrum of-A. In fact, Castro and Lamer in [11] showed that ifthe interval (f’(0),ff(oc))t3 (ff(oc),f’(0))contains the eigenvalues Ak,...,zkj andif(t) < Aj+l for all t E then (1.2) has at least three solutions. The proofs in [11]are based on global Lyapunov-Schmidt arguments applied to variational problems.Subsequently Chang (see [12]) approached the same problems using Morse theory,and Hofer (see [14]) obtained the existence of five solutions when ff(oc) < 1. Forother results in the study of this problem we refer the reader to [3], [4], [6], [8], [10],[17], [18], and [19], among others.Here we prove the following.THEOREM A. /f if(0) < , f’(c) e (k,k+)with k >_ 2, andif(t) <_ " < Ak+, then (1.2) has at least five solutions. Moreover, one of the fol-lowing cases occur.(a) k is even and (1.2) has two solutions that change sign.(b) of oI *he(c) k is odd and (1.2) has two solutions that change sign.Received by the editors April 27, 1992; accepted for publication (in revised form) August 6,1993. Partially supported by National Science Foundation grant DMS-8905936, the Texas AdvancedResearch Program, and Colciencias Grant 168-93.Department of Mathematics, University of North Texas, Denton, Texas 76203-5116(acastrount. edu).Departamento de Matemticas, Universidad Nacional de Colombia, Apartado A(reo 3840,Medellin Colombia (jcossio(C)s+/-gma. eafit, edu. co).1554MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM 1555(d) k is odd and (1.2) has three solutions of the same sign.The assumption k _> 2 is sharp; Theorem B of [11] gives sufficient conditionsfor (1.2) to have exactly three solutions when k 1. We prove Theorem A by usingLyapunov-Schmidt arguments to reduce the solvability of (1.2) to a finite-dimensionalproblem, and then we use degree and index theories applied to critical points. Wemake intensive use of the fact that the Leray-Schauder degree is invariant under theLyapunov-Schmidt reduction process. In order to calculate various indices and degreeswe prove that in large regions the Leray-Schauder degree of maps arising in problemslike (1.2) where f crosses the first eigenvalue(__lim f(u) < 1 <limu,-, f(U))uis equal to zero. We also use "mountain pass arguments" of the Ambrosetti-Rabino-witz type (see [5]).In 2 we recall the framework that allows studying solutions to (1.2) in terms ofvariational functionals and the Lyapunov-Schmidt reduction method. In 3 we calcu-late the index of the trivial solution when the nonlinearity crosses the first eigenvalue,establish the existence of positive and negative solutions, and compute their indices.In 4 we prove Theorem A.2. Preliminaries and notation. First we state a global version of theLyapunov-Schmidt method. For the sake of completeness we recall that if (I) is aiunctional of class C and u0 is a critical point of (I) then u0 is called of mountainpass type if for every open neighborhood U of uo (I)-l(-cx),(I)(uo)) U 0 and(I)- (-cx, (I)(uo)) g V is not path connected.LEMMA 2.1. Let M be a real separable Hilbert space. Let X and Y be closedsubspaces of M such that M X Y. Let j" M I be a functional of class C. Ifthere are m > 0 and a > 1 such that(2.1) (Vj(x + y) Vj(x + yl),y- y) >_ mI[y- YII[ for all x E X, y, y E Ythen we have the following.(i) There exists a continuous function " X -, Y such thatj(x + (x)) minj(x + y).yEYMoreover, (x) is the unique member of Y such that(2.3)(Vj(x+(x)),y>=O for all y e Y.(ii) The function " X ] defined by (x) j(x + (x)) is of class C1, and(V(x),xl} (Vj(x / (x)),Xl} for all x, 1 e X.(iii) An element x X is a critical point of if and only if x + (x) is a criticalpoint of j.(iv) Let dimX < oc and P be the projection onto X across Y. Let S c X andE C M be open bounded regions such that{x + (); x e s} r {x + (x); x e x}.If V)(x) # 0 for x OS thend (V, S, O) d (Vj, E, 0),1556 ALFONSO CASTRO AND JORGE COSSIOwhere d denotes the Leray-Schauder degree.(v) If uo xo + (x0) is a critical point of mountain pass type of j then Xo is acritical point of mountain pass type of.Proof. The reader is referred to [9] for the proof of parts (i)-(iii). The proof ofpart (iv) follows by arguing as in Lemma 2.6 of [16]. Now we proceed with the proofof part (v).Suppose xo is not of mountain pass type of . Let V be an open neighbor-hood of xo in X such that either -(-x,(x0)) V is empty or path connected.If )-l(-oc,(x0))n V is empty, by part (i) we see that {x + y;x E V,y Y} Nj-l(-oc, j(uo)) is also empty. Thus u0 is not of mountain pass type for j. Onthe other hand if -l(-c,(x0))n V is path connected, letting W {x + y;x eV, IlY (x)ll < 1} and using again part (i) it is easily seen that W j-(-oc, j(uo))is also path connected. This concludes the proof of Lemma 2.1.For each positive integer m let m denote an eigenfunction corresponding to theeigenvalue Am. Let H be the Sobolev space H(D) which is the completion of theinner product space consisting of real C functions having support contained in Ftwith inner product<u, v> ] Vu(x) Vv(x)dx.As it is well known, the set {(m) can be assumed to be complete and orthonormal inH.We say that u H is a weak solution to (1.2) if for every H(Vu.Vf(u) ) dx 0By standard regularity for elliptic operators (see [11]) it follows that weak solutionsare classical solutions when f is continuous and sublinear, i.e., when f is continuousand there is a positive constant a such thatIf( )l < +Let J" H denote the functional defined by(2.5) J(u) J Il]Vu]]2 F(u)) dx,where F() fo f(s)ds. SinceJ e C(H,) (see [19]) ande (Ak,Ak+l), f satisfies (2.4). Thus(2.6) (VJ(u), ) jn(Vu.V f(u) ) dx for H.In particular u is a weak solution of (1.2) if and only if u is a critical point of J.Let X denote the subspace of H spanned by {1, 2,..., k}, Y its orthogonalcomplement, and J the functional defined by (2.5). We claim J satisfies hypothesis(2.1). Indeed, from (2.6) and the mean value theorem(2.7) (VJ(x + y) VJ(x + yl),y yl) IlY Ylll 2 /fl f’()(Y Y):"MULTIPLE SOLUTIONS FOIL A NONLINEAR DIRICHLET PROBLEM 1557Denoting by I10 the usual L2() norm and using that f’() < 7 < Ak+l, we have(VJ(x + y) VJ(x + yl), y y} >_ IlY YII2 9/IlY YII(> 1 Ak+where we have used that [[z[[ 2 Ak+l[[Z[[ for all z e Y. Thus (2.1) holds withm 1- 7/(Ak+) and a 2.3. Index of the trivial solution when the nonlinearity crosses the firsteigenvalue. For /> A let p(/) :-p be the homogeneous function defined byTx for x > 0,p(x)f’(O)x for x < 0.Let P be the primitive of p with P(0) 0, and r: H be the functional definedby(3.1) r(u)= f (llVul,2 P(u)) dx.As observed in 2 (see (2.4)) r is a functional of class C1, and its critical points arethe weak solutions toAu + p(u) 0 in(3.2)u 0 on 0.Because f’(0) < and the principal eigenvalue of the Laplacian in any subregion ofF is bigger than or equal to A1, we see that if u # 0 is a weak solution to (3.2) then uis a positive eigenfunction. Since this contradicts that 7 > A1, we conclude that u 0is the only critical point of r.LEMMA 3.1. If B is a ball in H containing zero then d (Vr, B, 0) 0.Proof. By the definition of the Leray-Schauder degree if Z denotes the subspacespanned by p, 2,..., Pz with big enough(3.3) d (Vr, B, 0) d (P VTr, B 3 Z, 0),where P denotes the orthogonal projection onto Z. Since 7 > we see that h(t) "=p(t) Xlt > 0 for t # 0. Because l is in Z we have(P VTr(x), l} (Vr(x), l)(3.4) f(Vx.V x h(x)) dz]a(-h(x)) dz < O if x E Z N OB,where we have used that is positive in t. From (3.4) we have now, for each s E [0, 1]and x Z f30B,(3.5) <sPVTr(x) + (1 s)(-l), 1> < 0.Hence by invariance under homotopy of the Leray-Schauder degree we have(3.6) d (P Vr, B 3 Z, 0) d (g, B t3 Z, 0) 0,1558 ALFONSO CASTRO AND JORGE COSSIOwhere K(x) -1 for all x E Z. From (3.3) and (3.6) the lemma is proven.Let f+ be the function defined byf+() { ff!) if>_ 0,(0) if<0.Let F+() fo f+ (s)ds, and J+" g If be the functional defined by(3.7) J+ (u) IIVu]l 2 F+(u) dx.Imitating the proof of Corollary 2.23 of [19] it readily follows that J+ satisfies thehypotheses of the mountain pass theorem. Hence J+ has a critical point u+, whichby the maximum principle is a positive solution to (1.2). Therefore, by Theorems 1and 2 of [15], if the set of critical points of J+ is discrete then at least one of themis of mountain pass type and has local degree -1. Similar arguments produce eitherinfinitely many negative solutions to (1.2) or a negative solution u- which is a criticalpoint of mountain pass type and has local degree -1.Let , f’(oc) and r as in Lemma 3.1. Since f’(x) is not an eigenvalue of-A with zero Dirichlet boundary conditions, for p > 0 big enough and s E [0, 1] thefunction sVJ+ + (1 s)Vr has no zero on the sphere centered at 0 with radius p.Hence by Lemma 3.1 we have(3.8) d(VJ+, Bp, 0) 0for p big enough. For future reference we summarize the above discussion into thefollowing lemma.LEMMA 3.2. Under the hypotheses of Theorem A, (1.2) possesses a positive(respectively, a negative) solution. If the set of positive (respectively, negative) so-lutions is discrete then at least one of them is a critical point of mountain pass typeand its local degree is -1.Since 0 is an isolated local minimum of J+ and J we have(3.9) d(Vg+, B, O) 1 d(VJ, B, 0),where B is a ball centered at zero containing no other critical point (see [2]). Henceif is a bounded region containing the positive solutions and no other critical pointof J we have(3.10) d(VJ, , 0) d(VJ+, , 0)d(VJ+, Bp-, 0)d(VJ, 0)- d(VJ, B, O)-1.Similarly we see that if 1 is a bounded region containing the negative solutions to(1.2) and no other critical point of J then(3.11) d(VJ, , O) -1.4. Proof of Theorem A. First, we show that there exists u0 H such thatVJ(u0) 0 and, if isolated, then(4.1) d(VJ, V,O)--(-1)kMULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PROBLEM 1559in any region V containing no other critical point of J. In fact, by Lemma 2.1 and(2.8) there exists " X --+ Y such thatJ(x + (x)) min J(x + y).yEYMoreover, (x) is the unique member of Y such that(4.2) (VJ(x + (x)), y> 0 for all y e Y,the function 07. X --* N defined by (x) J(x + (x)) is of class C1, and(4.3) (VJ(x),xl/= (VJ(x + (x)),Xl/ for all x, xl e X.We now claim that for x E X(4.4) g(x)Because f’(oo) E (Ak,Ak+) there exists b IR and > Ak such that F() >_ (2/2) +b. Hence1 / 1 fax2_b[fJ(x) - Ilxll F(x) < - Ilxll 2Since (x, x) _< Ak (x, X}o for x X, we obtainJ(x)<-llxll 1-Because J(x) <_ J(x), (4.4)implies that(4.5) .](x)Since dimX < oo there exists x0 X such that(xo) max J(x + (x)).xEXTaking uo x0 +(x0) we have (see Lemma 2.1) VJ(u0) 0. Suppose now that xo isan isolated critical point of J, hence u0 is an isolated critical point of J. Since -J hasa local minimum at x0, taking W {x e X; x+(x) e V} then d (V7, W, 0) (-1)k.Therefore by part (iv) of Lemma 2.1 we have (4.1).Suppose k is even. Let R be large enough so that if V](x) 0 then Ilxll < R.Because f(t) _< - < Ak+, there exist positive numbers c and c2 such that for allX e X II)(x)ll Cl 2t- C211XlI. Thus if u x + y is a critical point of J then Ilxll <_ Rand IlYll <- Cl + c211xll. Because _7 is coercive, d (V7, BR, 0) (-1)k 1. Thus bypart (iv) of Lemma 2.1 d(VJ, C, 0)= 1 where C {x + y; Ilxll < R, IlYll < c + c2R}.Suppose that K, the set of critical points of J, is finite. Let $1, $2 and $3 be disjointopen bounded regions in H such that $1 C K {0}, $2 N K is the set of positivesolutions to (1.2), and 3 C g is the set of negative solutions to (1.2). By (3.10) and(3.11) we have(4.6) d (VJ, $2,0) d (VJ, $3,0) -1.If u0 xo + (x0) t $2 U $3 we let $4 denote an open bounded region disjoint from$1 U $2 U $3 such that S4CK {no}. By the excision property of the Leray-Schauder1560 ALFONSO CASTRO AND JORGE COSSIOdegree we have1 d (VJ, C, 0) d (VJ, $1,0) + d (VJ, $2,0) + d (VJ, $3,0) -- d (VJ, $4,0)+ d (VJ, C ($1 t2 $2 U $3 t2 $4), 0)1 1 1 + 1 + d(VJ, C- (S u $2 U $3 t9 $4),0).Thus, by the existence property of the Leray-Schauder degree we see that there existsu E C (S U $2 U $3 U $4) such that VJ(u) 0, which proves that (1.2) has atleast five solutions. In this case both u0 and U change sign.Suppose now that uo E $2 t2 $3; without loss of generality we can assume thatu0 $2. Let $4 be a neighborhood of u0 such that $4 K {uo}. By Lemma3.2 there exists a critical point of mountain pass type ul $2 such that if $5 is aneighborhood of Ul containing no other critical point of J+ then d (VJ, $5,0) -1.Thus-1 d (vg, $2,0) d (VJ, $4,0) + d (vg, $5,0) + d (VJ, $2 $4 U $5,0)1 l+d(VJ, S2 $4 USh,0).Thus, by the existence property of the Leray-Schauder degree there existsu2 E $2 $4 U $5 with VJ(u2) 0. Finally,1 d (VJ, C, 0) d (VJ, S1, 0) + d (VJ, $2,0) + d (VJ, $3,0)+ d (VJ, C (S t2 $2 t2 $3), 0)1 1 1 + d (VJ, C (S1 [-J $2 [-J $3), 0).Thus there exists u3 C- (S t2S2 S) with VJ(u3) 0. Thus the set{O, uo, ul,u2,u} together with a critical point of J in $3 shows that (1.2) has sixsolutions. Since u3 $2 t2 $3 and uo, ul, u2 E $2, u3 is a sign changing solution anduo, ul, u2 have the same sign. This completes the proof of Theorem A when k is even.Suppose k is odd. Let Si, 1,2,3 be as above. Ifuo $2$3 the prooffollows very closely that of the case k even; the details are left to the reader. SupposeU0 $2 I..J $3, say, u0 e $2. Because u0 > 0 in gt and Ouo/O? < 0 in OFt (here0/07 denotes the outward unit normal derivative), using that X is finite-dimensionMand standard regularity theory of elliptic operators it follows that for some e > 0x + (x) > 0 in t if IIx- x011 < e. Thus and + coincide in {x; IIx- xoll < e}. Thus]+ has a local maximum at x0. Since we are assuming (1.2) to have only finitely manysolutions, x0 is a strict local maximum of +. Let 5 > 0 be such that +(x) < ]+ (x0)if IIx-x011 < 5. Since k > 2, {x;0 < IIx-x011 < 5} is connected. Thus xo is nota critical point of mountain pass type. By Lemma 3.2 J+ has a critical point ofmountain pass type ul x +(x) such that if V is a neighborhood of u containingno other critical point of J+ in its closure then d (VJ+, V, 0) -1. In particular, bypart (v) of Lemma 2.1 x0 - x. Let V0 (respectively, V1) be a neighborhood of u0(respectively, Ul x ++(x)) containing no other critical point in its closure. Thus-1 d(VJ+,S2,0) d(VJ+, Vo,0) + d(VJ+, V1,0) + d(VJ+,S2 (Vo V),O)-2 + d (VJ+, 32 (V0 t2 V1), 0).Thus by the existence property of the Leray-Schauder degree there exists a thirdpositive solution u2 $2- (Vo W V). Since by the existence property of the Leray-Schauder degree (1.2) has a solution u3 e $3, we see that (1.2) has five solutions,MULTIPLE SOLUTIONS FOR A NONLINEAR DIRICHLET PRO.BLEM 1561namely 0, Uo, ul, u2, u3. Since Uo, ul, u2 E $2 they have the same sign. This provesTheorem A. ]REFERENCES[1] S. AGMON, The Lp approach to the Dirichlet problem, Ann. Scuola Norm. Sup. Pisa, 13 (1959),pp. 405-408.[2] H. AMMAN, A note on degree theory for gradient mappings, Proc. Amer. Math. Soc., 85 (1982),pp. 591-595.[3] S. AMMAN AND E. ZEHNDER, Nontrivial solutions for a class of nonresonance problems andapplications to nonlinear differential equations, Anon. Scuola Norm. Sup. Pisa, 7 (1980),pp. 539-603.[4] A. AMBROSETTI AND G. 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Multiple Solutions for a Nonlinear Dirichlet Problem

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Multiple Solutions for a Semilinear Dirichlet Problem

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