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Perturbation by vanishing nonlinearities
Vonlmcnr Anaiyvsrr. Thcoq'..Ue~hodr c?Applicnnonr. Vol.7.No. 7.pp 73%746.1983 F'rimrd inGreatBritain.
PERTURBATION
BY VANISHING
0362-546X433 13 CO+ 03 0 1983PcrgamonRcss Ltd.
NONLINEARITIES
WERNER STORK FB Mathematik, J. W. Goethe-Universitlt,
Robert Mayer-Str. 6-10, D-6000 Frankfurt/M, Bundesrepublik Deutschland
(Receiued in revised
Key wor& and phmses: Nonlinear perturbations,
form 30 March 1981) vanishing nonlinearities,
Fredholm-operators
1. INTRODUCTION IN
[l] NONLINEAR perturbations of - A - & are considered, where -A is the negative Laplacian with Dirichlet boundary conditions over a bounded domain R C R” and & is an eigenvalue of - A. The nonlinear perturbations N are generated by continuous functions g : (w+ W which satisfy g( 30) = g( - 30) = 0. Provided g is odd and satisfies a decay condition which impedes ,g(r) to tend to zero too fast as r tends to infinity, it is shown that for sufficiently smooth 11E Lz(Q) the condition: /nh(x)~(x) dx = 0 for any weak solution g,of -Au = &u, is sufficient for the solvability of -Au(x) - &u(x) + g(u(x)) = h(x) with u EN’,‘.*(Q). In [2] the function g : [w+ W was assumed to be differentiable, with its derivative uniformly bounded from below by &-I - j\-k+ E and from above by kk+l - & - E, where Akel, i+, Ak+l are eigenvalues of a uniformly elliptic differential operator L and &_ I, Ak+, are the eigenvalues nearest to the left of the simple eigenvalue & and nearest to the right. Under various additional decay conditions for sufficiently smooth f+~ L?(R) II R(L - Ak) the set W(f) = {t E R: Lu - Aku + G(u) =f+ tq has a weak solution} where (G(rc))(x) = g(ic(x)) and ~1is a normalized eigenvector of L and I.k, is characterised to be a semiclosed interval (0, t*(f)] or [tl(f), 0) ([2] proposition 6.5) or a closed interval [tl(f), [z(f)] with rl(f) < 0 < t*(f) [2, theorem 5.2, theorem 6.71. In [3] g is assumed to be continuous, bounded and to obey those decay conditions which in [2] imply that W(f) is a closed interval. Essentially by Leray-Schauder degree it is then shown that in case of ker( - A - &) = span{q, . . . , qn} with orthonormal family {qj}, the corresponding set W(f) = {t = (cl,. . . , t,J: -Au - Aku + G(u) =f+ tlcpl + . . . + r,,q,, has a weak solution} is closed, contains 0 and contains a subset W,(f) such that for t E W,(f) there are at least two solutions. We drop all decay conditions, the differentiability assumption and the condition that g should be odd, but demand the Lipschitz-continuity of g. For general Fredholm-operators T we then prove the following proposition which at least in case of dim(ker(T)) = 1 is a contribution to the solution of the open problem 2.3.6 in [l] where the solvability of - Au(x) - &U(X) + g@(x)) = h( x ) in case of g with e.g. compact support is to consider. PROPOSITION 1.1. Let Q C R” be a bounded domain. Let T be a Fredholm-operator in L*(Q) over W with ker(T) @ R(T) = L*(Q) and ker(T) = span{q}, where Q,E L,(R) and 119;11 = 1. .4ssume that sets A C {u E D(T) II R(T) : IITullS M} are bounded sets in L,(n). Let bounded function with g(w) = g(- 00) = 0 and g : R + R be a Lipschitz-continuous, Lipschitz-constant q < y(T) where y(T) is the reduced minimum modulus of T. Define 739
740
W.
STORK
N:D(T)+ L?(Q) by (Nrr)(x) = g(+)). Th en for any fE R(T) there exists a connected set W(f) C W such that (*) TLC+ Nu =f+ ry has a solution in D(T) if and onlv if t E W(f). The set W(f) is (0) or [r,(f), 0) or (0, t?(f)] or [ri(f). r?(f)] where [i(f)’ + t?(f)’ f 0 and 4(f) s 0 s rz(f), andIt,(f)l~sup{lg(r)I:rE1W}In19?(x)Idrforj=1,2. In Section 2, for any type of W(f) stated in proposition 1.1, we will give conditions on g and T which suffice in order to prove that W(f) is of that type. Moreover, we show that for the closedness of R( T + N) the closedness of W(f) f or anyf E R(T) is necessary and sufficient. For the proof of proposition 1.1 we make use of the following proof and some of its consequences will be published separately. since we need some of the functions and estimates therein.
theorem. The theorem, its The proof is indicated here
1.2. Let T be a Fredholm-operator in a Hilbert space H over W with ker(T) @R(T) = H. Let {qi, . . qn} be an orthonormal base of ker(T) ‘and let P be the orthogonal projection P: H ---, ker( T). Let N: D(T) + H be a mapping which satisfies ]lNn - Null’ 5 L’(]]Tn - Tu]j’ + a’llPu Pull’) for K, u E D(T) with a’ > 0 and 0 5 L < 1. Then for any fE R(T) there exists a connected set W(f) C W” such that TLL+ NK =f+ tl@ +. . . + r,,q,, has a solution in D(T) if and only if t = (tl, . . . , f,J E W(f).
THEOREM
Proof. (i) We endow the domain D(T) with the norm ]/u]]-= (I] Tull* + a’llPull’)‘/’ and obtain a Hilbert space D(T). (ii) For fixed f E R(T) we consider the family of mappings {L(s,f):s E R”), L(s,f) defined by .L(s,f>u = ii”(lP>( - NK + f) + slql + . . . + s,P?, for K E D(T), where ?-’ is the right inverse of T and s = (si, . . . , s,) Each mapping L(s, f) is a contraction in d(T) with contraction-constant 0 5 L < 1 and hence has a unique fixed point u(s, f) E D(T). (iii) Th e estimates ]]u(s, f) - rc(s’, f)ll-S lal(l - I,*)-“‘Is - s’) and IIPNu(s, f) - PNu(s’, f)llS L]a](l - L2)-“2/s - s’] hold true. We define mappings j= 1,. . , n, by r,(s,f) = (N+,f), qj) and put r(f) = (r,(f), . . . , r,(f)), r(f):W”+ W”. By the estimates above r(f) is continuous. (iv) We put W(f) = R(T(f)). Since I(f) is continuous the set W(f) is connected. Finally we show that TM + Nu = f + tlg?l+. . . + r,,q~,,has a solution in D(T) iff (ti, . . . , r,) E W(f). rj(f):Wn+R,
Proof of proposirion 1.1. We choose the constant a in the norm ]I. II- in theorem a = y(T). Then for the mapping N in proposition 1.1 we get the estimate
1.2 by
IINK - Null2 5 ~*[IK - ullz % q2(jJ?‘-‘/12/(Tu - Tu(12 + IIPu - Pull_3
5 q’j/?---‘II’(/ITu - Toll’ + y(T)‘IIPu - Pullz) because I(f-‘li = (T)-‘. Since q < y(T) theorem 1.2 is applicable. Hence there exist a connected set W(f) C R such that (*) has a solution in D(T) iff t E W(f). The only nonvoid connected sets in W are intervals and sets consisting of a single point. 1.1. It remains to prove that W(f) must be of the type stated in proposition We consider the mapping I : = r(f) : R + W, defined in (iii) of the preceding proof. We know I-(s) = (N(u(s)), @,>, K(S):= u(s,f) the fixed point of L(s, f) in (i) of the preceding proof. Since g is bounded we obtain ]](I - P)u(s)lj- = IIT(Z - P)K(.s)/~ 5
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l~~(s>ll + llfil 5 M for any any s E R, x E R. Hence (I - P)U(S) + s(p. S’mce R these inequalities and
s E R. Then, by hypothesis, we get I(1 - P)u(s)(x)i 5 Mr for we know - Mi + s&x) d U(S)(X) 5 M,+ SF(X), because n(s) = is a bounded domain and g is bounded we obtain by use of theorem that lim T(s) = g(=)Jo-dx),& + by Lebesgue’s 5-x g(-=)Jo-~$x)dr=O, where Q+={xEQ: p7(x)>O} and Q-={xER:~(x)
2. COMPUTATION
OF R( 7- + A’)
In case of g = 0 in proposition 1.1 we clearly have W(f) = (0) which is the solvability condition for Fredholm operators. We prove some propositions which show that the other possibilities for W(f) listed in proposition 1.1 occur. A result similar to the following is proposition 6.5 in [2]. PROPOSITION 2.1. Let the hypotheses of proposition 1.1 be satisfied. If in addition Q:> a.e. in R and g(r) > 0 (g(r)
1.1. We know Proof. Consider W(f) = R(T), r as in the proof of proposition >Ofora.e.xE Q,wegetI(s) >O r(s) = hg(u(~9f)(4M4 h. s inceg,>Oandg(u(s,f)(x)) for any s E R. Hence the function r assumes its supremum f7(f) in so E W. Since r is continuous and !ly I’(s) =0 = ,lJm_ I’(s) we infer from the mean value theorem that for t E (0, t*(f)) there exist s1 < so < s2 with T(sJ = t = I&). Then u(si, f) + u(s2, f), because s1 f s1 and Pn(Sj, f) = Sjq, j = 1,2 and P the orthogonal projection on ker(T). Since PN(U(Jj, f)) = lI(sj)cp, and ru(+, f) + (I - P)Nu(+, f) = f the assertion follows. The following proposition shows in particular what may happen for g with compact support. Put C,(Q) = {u E C(Q): u is bounded in Q}. PROPOSITION 2.2. Let the hypotheses of proposition 1.1 be satisfied. Assume that the embedding J: 6(T) + C,(Q) is continuous with bound IIJ I=, where d(T) is the domain D(T) endowed ! and C,(Q) is endowed with the norm with the norm IIu/I-= (ll~ull~ + y(T)2~~~u~~2)12 lb& = ~UP~IWI: x E Q}. Assume further Q,> 0, g + 0. (a) If g(r) Z 0 (g(r) Z 0) for r E R then for any f E R(T) (0, h(f)] C W(f) C [O, h(f)] ([tl(f), 0) C W(f) C [tl(f), 01) and for t E (0, h(f)) (t E (t*(f), 0)) there exist at least two solutions in D( ?“) of Tu + Nu = f + tp (b) If g(r) E 0 (g(r) I 0) for r E Iwand g(r) = 0 for r E [ - r’, r’], then for any f E R(T) with
IJll4llfII+ supMr)I: r E WJQl&I < r’. W(f)
= [0, r2(f)] (W(f)
D(T) for Tu + Nu = f + tp
= [tl(f),
01) and for t = 0 there exist infinitely many solutions in
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Proof. (a) It is clear that T(s) IO for s E W. We show that r is not constant. Since g + 0 is continuous there is a ro E 52 and an E > 0 such that g(r) > 0 for r E [ro - E, r0 + E]. We prove that ro - ES u(s)(x) 5 ro + E for suitable, fixed s E W and x E R’, R’ C R and Lebesguemeasure meas > 0. This then implies T(s) > 0. The mapping F: W-t d(T) defined by F(s) = u(s), u(s) the fixed point of L(s,f), is continuous by the estimate in step (iii) of the proof of theorem 1.2. Since the embedding J is continuous and since point evaluation 6(x) is continuous on C,(Q) we conclude that 6(x) .J. F:R - G?, 6(x) .I. F(s) = u(s) (x), is continuous. From the proof of proposition 1.1 we know that --Ml + sq(x) I U(S)(X) 5 Ml + sq(x). Hence for fixed x0 E R we know lim (II( == and lim (u(s)(x~)) = - x. s-.-x S-X By the mean value theorem we infer that there is an SOsuch that U(SO)(XO) = ro. By the continuity of U(Q) there is a neighbourhood U(Q) = R’ of x0 such that U(SO)(X)E [ro - E, ro + E] for any x E R’. Clearly meas(U(xo)) > 0 and thus I+(Q) > 0 which implies that I assumes its supremum tl(f) in R. Analogously to proposition 2.1 one proves that for t E (0, t?(f)) there exist at least two solutions in D(T) for Tu + Nu = f + tq. (b) From part (a) we know that (0, tz(f)] C W(f) C [0, t?(f)]. Hence it remains to show that for I = 0 there are infinitely many solutions in D(T) for TM + Nu = f+ tq. From step (ii) in the proof of theorem 1.2 we know that (I - P)u(s) = T-‘(1 - P)( - Nu(s) ff). Since and since the embedding J is continuous we obtain the estimate (I - P)u(s) E D(T)
IIU- ~)~(~)Il~‘liJll~li~-‘(~-~)(-~~(~)
+f)ll-
This implies -Ml
+ s&x) 5 u(s)(x) I Ml + SF(X) for x E R. Since Q,E L,(Q), for s with I4 < (I’ - ~W)/lldl~ we get T(s) = 0. By the same arguments as in the proof of proposition Fl one shows that {u(s) : 1.~1< (r’ - Ml)/llqljx} are pairwise distinct solutions of Tu + Nu = The following example of the negative Laplacian with Neumann boundary condition shows that in particular for g with compact support the connected set W(f) may be closed for any f E R(T). Example 2.3. Let R c W3 be a bounded domain with smooth boundary. Let Tin L?(Q) over A be defined by D(T) = {u E W’.‘(Q) : du/h = 0 on dR} and TLI = - Au. Let g: A ---, W be Lipschitz-continuous with Lipschitz constant q < y(T) and g(r) = 0 for r < rl. g(r) 2 0 and g+O. LetN:D(T) += &(Q) be defined by (Nu)(x) = g(u(~r)). Then for any f E R(T) the set W(f) is closed, W(f) = [0, h(f)], for t E (0, h(f)) t h ere exist at least two solutions in D(T) of Tu + Nrc = f + tq, where q = c = (Jo ldx)-‘, and for Tu i Nrc = f there are infinitely many solutions in D(7). Proof. The embedding J:d(T)+ C,(Q), defined in proposition 2.2, is continuous by Sobolev’s lemma. From proposition 2.2(a) we then know that (0, t?(f)] c W(f) C [0, r?(f)] and that there are at least two solutions in D(T) of Trl + h’L1 = f i fq for f E (0, t?(f)). It remains to show that there are infinitely many solutions of Trl i IVU= f in D(T). Since Q:= c = (Jo1 du)-’ and -Ml + ST(X) Z u(s)(x) 5 Ml f SF(X) for any s E R we get u(s)(x) = 0 for s < (rl - MI)/c and r:(s) = 0. Similarly to the proof in proposition 2.1 one shows that {U(S): s < ( rl - Ml)/c} are pairwise distinct solutions of Tn + ~Vrt= f.
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The following proposition gives sufficient conditions for the closedness of W(f) for any f~ R(T). It is an extension with respect to the nonlinearity of the results in [2, theorem 5.21 and [3, theorem l] at the cost of an additional condition on the eigenfunctions of the linear operator T. In [2] and [3] only the choices O(s) = s and g,(p) = y+x,.(P) + y-x_(p) and g_(p) = y-xl(p) + y,x_(p) are allowed, where x= is the characteristic function of R =, ;J_ =Jiy O(s)g(s) and y- = lim 0 (s)g(s). s-+-x PROPOSITION 2.4. Let the hypotheses of proposition 1.1 be satisfied. Assume that there is a continuous function 0 : R -+ R with O(r) > 0 for r > ro and O(r) < 0 for r < - t-0 such that lim 0 (r)g(rp +r’)p = g+(p) and ,liy% O(r)g(rp +r’)p = g-(p) uniformly for r’ E [-K, K]
r-35
and any K > 0, where g_ and g_ are measurable functions with J&+(q(x)) dr > 0 and Jn&(F@)) dJZ’ 0. Th en for any f E R(T) the connected set W(f) is closed, W(f) = [h(f), h(f)1 with h(f) < 0 < tdf), and for any t E (tl(f), tz(f)), t f 0, there exist at least two solutions in D(T) of Tu + Nu = f + tq. Proof.
We know W(f)
= R(T)
f rom the proof of proposition
1.1. Since lim T(s) = 0 = s--t= lim r(s), it suffices to show that I’ attains values greater than zero and values smaller than s-+-1 zero. Consider the continuous function Or: IF?--* R. We know -M, + s&x) $ u(s)(x) 5 M1 + q(x) from the proof of proposition 1.1. Then by our assumptions on 0 and g and by Lebesgue’s theorem we get ,‘iir O(s)I$) =J&+(q(x)) dr and lim O(s)IY(s) = *-+--r m&(&x)) h. S’mce O(s) > 0 for s > r. and O(s) < 0 for s < - ro we have inf{I’(s) : s E W} = T(sl) = tl(f) < 0 and sup {T(s): s E R} = T(sZ) = h(f) > 0 with sl, s2 E iF&By the same arguments as in the proof of proposition 2.1 one shows that for f E (tl( f), rz(f )), t # 0, there are at least two solutions of Tu + Nu = f + tq in D(T). We give some examples of functions g and selfadjoint operators T such that the hypotheses of proposition 2.4 are satisfied. If g(r) = A$‘-‘/(1 + p) with 0 < h < y(T) and 0 < k < n, then 0 may be chosen by O(r) = ?“- 2k+1. The hypotheses on 0 in proposition 2.4 are satisfied with g+(p) = Apzk-*” = g_(p). Let T be the selfadjoint operator in LZ( - 1, 1) generated by the Legendre differential expression (-d/dt) (1 - 3) (d/dt) + 2$/(1 + t) + 22/(1 - t) - (2cr+ 1)2a for some (Y with 0 < a< (2n - 2k)-’ 5 2-l and boundary conditions A+(u) = lim {P”u’(l - E) E-+X &%(l - E)} = 0 and B,(u) = ~_mo{P”u’(~-l) - a~‘%(&-- 1)) = 0. Then T has discrete spectrum a(T) = ad(T) = {A,, = n = 0, 1,2, . . .} with corresponding simple eigenvectors P!,‘@.‘@ are ultraspherical polynomials [4, p. 15191 and 0 Clearly 9(f) = c0(1 - f2), spans ker(T) and J’&_(pa(t)) By the bounds 11 Pi2a*2aJll= 5 Cd” (5, th eorem 7.32.1 p. its eigenfunction expansion to
(n + 2cu + l)(n + 2~) - (2a + 1)2~:
qJt) = c,(l - tZ)“p(n?“.2n)(t)where < c,, < Cn@, n 2 1 [5, 4.3.4, p. 671. dr =Jilg-(&t)) dr > 0. 1631 for u E D(T) we may estimate
llu/z d C[n$ (n2n+112/n2)2(,,Tu~~2 + ~(T)‘I]Pu~/~)]@ < 13.
7-M
w. STORK
Since all eigenfunctions are continuous this shows that the embedding d( 7) ----,C[-1, l] is continuous. Hence T satisfies the hypotheses of propositions 1.1 and 2.4 respectively. The conditions in proposition 2.4 are sufficient for the closedness of R(T + N) as is shown in [3, theorem 21 in the special situation O(s) = s. We prove a necessary and sufficient condition for the closedness of R(T + N) from which the closedness of R(T + IV) for the operators in proposition 2.4 follows. PROPOSITION2.5. Let the hypotheses of theorem 1.2 be satisfied. Then R( T + N) is closed if and only if the connected set W(f) is closed for any f E R(T). Proof. h=f+tlql+...
closed for any f E R(T). Let h E R(T + N), W(f) is + f,,q,, where f E R(T) and tlql + . . . + f,,cp,,E ker( T). Then there is a sequence (hk) C R(T + N) with hk-, h in H and hk = fk + tl,kq)l + . . . + tn,kqn where fk E R(T) and hk - fk E ker(T). Since R(T) @ ker(T) = H we obtain fk+ f in H and hk - fk + h - f in H. We prove that hk-, h in H implies t = (tI, . . . , fn) E W(f) which by virtue of theorem 1.2 implies h E R(T + N). Suppose t 65 W(f). S ince W(f) is closed, there is an E> 0 such that II’(s, f) - t I,, > E for any s E R”. From the properties of U(S, f) and U(S, fk) respectively in step (ii) of the proof of theorem Assume
1.2 we infer the estimate b&f)
- +,fk)i/-
< (l - L)-‘iif
-fkll
and hence we get Ir(%f)
- r(s,fk)
In = lipNLL(%f) - PNu(%fk)
f) - rr(s,fk) I(- 5 L(1 - ,!,)
11g Ll/+,
-‘l,f -
fklj.
Combining these estimates we obtain the inequality s< ir(sk,f)
- tin 5 Ir(sk,f)
- mktfk)
In +
ir(sk,fk)
-
tin
of r(sk, fk) = (tl,k, . . . , f&k). Since 1r(&, fk) - tl, + 0 for k + 02, We k % kg(&) from this inequality and hence J2 5 infer Ir(Sk,f) r(Sk,fk)In 2 E/2 for oes not converge to f in Hand consequently (hk) L(1 -L)-‘IIf -fk\l for k2 kg(&). Thus (fk) d does not converge to h in H. The if-part of proposition 2.5 is proven. be fixed and consider Contrarily assume that R(T + N) is closed. Let f E R(T) t = (t*, . . . , fn) E W(f). Then there is a sequence (tk), tk = (fl,k,. . . , f&k), in W(f) with tk-+ t in R”. This implies
where sk is a
SdUtiOn
(f +
tl.kQ)l
+
. . . +
b,k%)
E R(T + N)
and Since R(T + N) is closed we then know that (f-t llql + . . . + t,,q,) E R( T + N) and by theorem 1.2 we infer that (tl, . . . , rn) E W(f). Thus W(f) is closed. This holds true for any f E R(T).
We present a class of linear operators T to which the nonlinear perturbations in propositions 1.1, 2.1, 2.2 and 2.4 may be applied and which covers uniformly elliptic differential and pseudodifferential operators of sufficiently high order and with sufficiently smooth coefficients.
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Example 2.6. Let Q C R” be a bounded domain having the cone property. Let T be a selfadjoint operator in LZ(S2) over i&Jwith domain D(T) C \vps’(S2) where 2p > m. Assume that 0 is a simple eigenvalue of T. Then the spectrum a(T) is discrete, a(T) = ad(T) = {Ak:k E N} the reduced minimum modulus y(T) = inf{l & 1:Ak # 0, k E N} > 0 and the embedding J: d(T) ---, C,(n) is continud(T) is the domain D(T) endowed with the norm 11~/1-= ous where * fl, P the orthogonal projection P: L?(R)-* ker(T), and C,(Q) is (11T@ + Y(T)*IIP~I~ ) endowed with the norm l/uII~ = sup{1u(x)11 x E Q}. Proof. We first prove that the embedding .I is continuous. The operator T is closed. Then by the closed graph theorem the embedding E: D(T) ---, WP.‘(Q) is continuous, where d(T) is the domain D(T) endowed with the graph norm. Since 2p > m and since R has the cone theorem. property the embedding J-: WP.*(Q) + C,(Q) is compact by the Rellich-Kondrachov We now prove that the graph norm and the nqrm Ij* II- are equivalent norms on D(T). This then implies that the embedding J = J-. E: D(T) + CB(1;2) is compact and in particular continuous. We know IIPull 5 Ilull and l/u/l 5 (II~-1~12~IT~~~2 + llP~11*)~‘* = ~(T)-‘ll+ Hence (IITuII’+ y(T)‘IIPulj*)”
s (max{L Y(T)*~)“(I~ Tull + ll4>
and /ITuII + /Iu[l s (1 + y(T)-‘)(IITujl*
+ ~(T)‘IIPK~I*)~~’
and the equivalence of the two norms is shown. We now prove that T has discrete spectrum, a(T) = ad(T) = {&: k E N}. This then implies that y(T)=inf{/A~~:kEN,&fO}>O
and
ker(T) @R(T)
= L2(R).
Consider the complexification Tc of T in the complexification of L*(Q) over 2 which is L2(Q) over C, II = {ul + iuz: ul, u2 E D(T)} and Tc(ul + iuz) = Tul + iTu?. The complexification Tc is selfadjoint ([6] e 5.32 p. 111) and since D(T) c Wp~‘(Q) the domain D(Tc) is contained in Wps’(Q) over C. By the same arguments as above one shows that the embedding JZ:* E,: d( Tc) + Wp,*(Q) over Q=is compact. Then the operator Tc has compact resolvent (Tc - i)-’ and hence has discrete spectrum a(Tc) = c~(Tc) = {Ak: k E N}. We show that ad(T) = od( Tc) = a(T). Let AkE ad(T) with corresponding eigenvector ek. Then Tc(ek + iek) = TQ + iTQ = &(ek + iek) and hence & E CQ(Tc). Conversely, consider & E ad(Tc) with corresponding D(T) and
eigenvector
fk + igk. Then 6, gkE
TC(fk+ igk) = Tfk+ iT& = &fk + i&gk. Since A, is real Tfk= & fk and Tgk = I$&. Thus & E ad(T) and ker(Tc - &) has an orthonormal base of real valued functions for any & E ad(Tc). Moreover, dim(ker(Tc)) = dim(ker(T)) = 1. We have shown ad(T) = od(Tc).
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Since the normalized eigenfunctions of TC form an orthonormal base in ,5,,(Q) over @ and since they may be chosen real valued, they form an orthonormal base of LZ(S2) over W. Then for r $?iad(T) the operator (T - r) -’ is a bounded operator, defined on L:(R), and thus u(T) = ad(T) = ud(Tc). The proof of example 2.6 is now complete. Results similar to those of proposition 1.1 hold true if we perturb, e.g. selfadjoint realisations T of -u” in L2(0, 1) over &?by N: D(T) + Lz(0. l), (Nn)(x)
= --&U(X) +&C(x)),
where Ak is a simple eigenvalue of T and g satisfies the hypotheses in proposition 1.1 with a suitable Lipschitz constant q. If we choose the perturbation N by (Nu(x)) = -&U(X) + g(u”(x)), then we can prove the results of proposition 1.1 merely for suitable ‘right hand’ sides f~ R(T - &) namely fE R(T - &) n L,(O, 1). REFERENCES 1. FU~IK S., Nonlinear noncoercive boundary value problems, in Equadiff IV, Proceedings Prague 1977, Lecture :Votes in Mathematics 703, 99-109, Springer. Berlin (1979). 2. XL~BROSEITIA. & MASCINI G., Existence and multiplicity results for nonlinear elliptic problems with linear part at resonance, /. diff. Eqns 28, 22C-245 (1978). 3. HESS P., sonlinear perturbations of linear elliptic and parabolic problems at resonance: existence of multiple solutions, Annali marh. Pba 5, 528-537 (1978). 4. DUNFORD N. & SCH~;\RTZ J. T., Linear operarors parr Il. Interscience, New York (1963). 5. SZEGO, G., Orrhogonal polynomials, American mathematical Society, New York (1939). 6. WEIDM.A%;NJ., Lineare Operaforen in Hilberrriiumen, Teubner, Stuttgart (1976).
PERTURBATION
BY VANISHING
0362-546X433 13 CO+ 03 0 1983PcrgamonRcss Ltd.
NONLINEARITIES
WERNER STORK FB Mathematik, J. W. Goethe-Universitlt,
Robert Mayer-Str. 6-10, D-6000 Frankfurt/M, Bundesrepublik Deutschland
(Receiued in revised
Key wor& and phmses: Nonlinear perturbations,
form 30 March 1981) vanishing nonlinearities,
Fredholm-operators
1. INTRODUCTION IN
[l] NONLINEAR perturbations of - A - & are considered, where -A is the negative Laplacian with Dirichlet boundary conditions over a bounded domain R C R” and & is an eigenvalue of - A. The nonlinear perturbations N are generated by continuous functions g : (w+ W which satisfy g( 30) = g( - 30) = 0. Provided g is odd and satisfies a decay condition which impedes ,g(r) to tend to zero too fast as r tends to infinity, it is shown that for sufficiently smooth 11E Lz(Q) the condition: /nh(x)~(x) dx = 0 for any weak solution g,of -Au = &u, is sufficient for the solvability of -Au(x) - &u(x) + g(u(x)) = h(x) with u EN’,‘.*(Q). In [2] the function g : [w+ W was assumed to be differentiable, with its derivative uniformly bounded from below by &-I - j\-k+ E and from above by kk+l - & - E, where Akel, i+, Ak+l are eigenvalues of a uniformly elliptic differential operator L and &_ I, Ak+, are the eigenvalues nearest to the left of the simple eigenvalue & and nearest to the right. Under various additional decay conditions for sufficiently smooth f+~ L?(R) II R(L - Ak) the set W(f) = {t E R: Lu - Aku + G(u) =f+ tq has a weak solution} where (G(rc))(x) = g(ic(x)) and ~1is a normalized eigenvector of L and I.k, is characterised to be a semiclosed interval (0, t*(f)] or [tl(f), 0) ([2] proposition 6.5) or a closed interval [tl(f), [z(f)] with rl(f) < 0 < t*(f) [2, theorem 5.2, theorem 6.71. In [3] g is assumed to be continuous, bounded and to obey those decay conditions which in [2] imply that W(f) is a closed interval. Essentially by Leray-Schauder degree it is then shown that in case of ker( - A - &) = span{q, . . . , qn} with orthonormal family {qj}, the corresponding set W(f) = {t = (cl,. . . , t,J: -Au - Aku + G(u) =f+ tlcpl + . . . + r,,q,, has a weak solution} is closed, contains 0 and contains a subset W,(f) such that for t E W,(f) there are at least two solutions. We drop all decay conditions, the differentiability assumption and the condition that g should be odd, but demand the Lipschitz-continuity of g. For general Fredholm-operators T we then prove the following proposition which at least in case of dim(ker(T)) = 1 is a contribution to the solution of the open problem 2.3.6 in [l] where the solvability of - Au(x) - &U(X) + g@(x)) = h( x ) in case of g with e.g. compact support is to consider. PROPOSITION 1.1. Let Q C R” be a bounded domain. Let T be a Fredholm-operator in L*(Q) over W with ker(T) @ R(T) = L*(Q) and ker(T) = span{q}, where Q,E L,(R) and 119;11 = 1. .4ssume that sets A C {u E D(T) II R(T) : IITullS M} are bounded sets in L,(n). Let bounded function with g(w) = g(- 00) = 0 and g : R + R be a Lipschitz-continuous, Lipschitz-constant q < y(T) where y(T) is the reduced minimum modulus of T. Define 739
740
W.
STORK
N:D(T)+ L?(Q) by (Nrr)(x) = g(+)). Th en for any fE R(T) there exists a connected set W(f) C W such that (*) TLC+ Nu =f+ ry has a solution in D(T) if and onlv if t E W(f). The set W(f) is (0) or [r,(f), 0) or (0, t?(f)] or [ri(f). r?(f)] where [i(f)’ + t?(f)’ f 0 and 4(f) s 0 s rz(f), andIt,(f)l~sup{lg(r)I:rE1W}In19?(x)Idrforj=1,2. In Section 2, for any type of W(f) stated in proposition 1.1, we will give conditions on g and T which suffice in order to prove that W(f) is of that type. Moreover, we show that for the closedness of R( T + N) the closedness of W(f) f or anyf E R(T) is necessary and sufficient. For the proof of proposition 1.1 we make use of the following proof and some of its consequences will be published separately. since we need some of the functions and estimates therein.
theorem. The theorem, its The proof is indicated here
1.2. Let T be a Fredholm-operator in a Hilbert space H over W with ker(T) @R(T) = H. Let {qi, . . qn} be an orthonormal base of ker(T) ‘and let P be the orthogonal projection P: H ---, ker( T). Let N: D(T) + H be a mapping which satisfies ]lNn - Null’ 5 L’(]]Tn - Tu]j’ + a’llPu Pull’) for K, u E D(T) with a’ > 0 and 0 5 L < 1. Then for any fE R(T) there exists a connected set W(f) C W” such that TLL+ NK =f+ tl@ +. . . + r,,q,, has a solution in D(T) if and only if t = (tl, . . . , f,J E W(f).
THEOREM
Proof. (i) We endow the domain D(T) with the norm ]/u]]-= (I] Tull* + a’llPull’)‘/’ and obtain a Hilbert space D(T). (ii) For fixed f E R(T) we consider the family of mappings {L(s,f):s E R”), L(s,f) defined by .L(s,f>u = ii”(lP>( - NK + f) + slql + . . . + s,P?, for K E D(T), where ?-’ is the right inverse of T and s = (si, . . . , s,) Each mapping L(s, f) is a contraction in d(T) with contraction-constant 0 5 L < 1 and hence has a unique fixed point u(s, f) E D(T). (iii) Th e estimates ]]u(s, f) - rc(s’, f)ll-S lal(l - I,*)-“‘Is - s’) and IIPNu(s, f) - PNu(s’, f)llS L]a](l - L2)-“2/s - s’] hold true. We define mappings j= 1,. . , n, by r,(s,f) = (N+,f), qj) and put r(f) = (r,(f), . . . , r,(f)), r(f):W”+ W”. By the estimates above r(f) is continuous. (iv) We put W(f) = R(T(f)). Since I(f) is continuous the set W(f) is connected. Finally we show that TM + Nu = f + tlg?l+. . . + r,,q~,,has a solution in D(T) iff (ti, . . . , r,) E W(f). rj(f):Wn+R,
Proof of proposirion 1.1. We choose the constant a in the norm ]I. II- in theorem a = y(T). Then for the mapping N in proposition 1.1 we get the estimate
1.2 by
IINK - Null2 5 ~*[IK - ullz % q2(jJ?‘-‘/12/(Tu - Tu(12 + IIPu - Pull_3
5 q’j/?---‘II’(/ITu - Toll’ + y(T)‘IIPu - Pullz) because I(f-‘li = (T)-‘. Since q < y(T) theorem 1.2 is applicable. Hence there exist a connected set W(f) C R such that (*) has a solution in D(T) iff t E W(f). The only nonvoid connected sets in W are intervals and sets consisting of a single point. 1.1. It remains to prove that W(f) must be of the type stated in proposition We consider the mapping I : = r(f) : R + W, defined in (iii) of the preceding proof. We know I-(s) = (N(u(s)), @,>, K(S):= u(s,f) the fixed point of L(s, f) in (i) of the preceding proof. Since g is bounded we obtain ]](I - P)u(s)lj- = IIT(Z - P)K(.s)/~ 5
Perturbation
by vanishing nonlinearities
741
l~~(s>ll + llfil 5 M for any any s E R, x E R. Hence (I - P)U(S) + s(p. S’mce R these inequalities and
s E R. Then, by hypothesis, we get I(1 - P)u(s)(x)i 5 Mr for we know - Mi + s&x) d U(S)(X) 5 M,+ SF(X), because n(s) = is a bounded domain and g is bounded we obtain by use of theorem that lim T(s) = g(=)Jo-dx),& + by Lebesgue’s 5-x g(-=)Jo-~$x)dr=O, where Q+={xEQ: p7(x)>O} and Q-={xER:~(x)
2. COMPUTATION
OF R( 7- + A’)
In case of g = 0 in proposition 1.1 we clearly have W(f) = (0) which is the solvability condition for Fredholm operators. We prove some propositions which show that the other possibilities for W(f) listed in proposition 1.1 occur. A result similar to the following is proposition 6.5 in [2]. PROPOSITION 2.1. Let the hypotheses of proposition 1.1 be satisfied. If in addition Q:> a.e. in R and g(r) > 0 (g(r)
1.1. We know Proof. Consider W(f) = R(T), r as in the proof of proposition >Ofora.e.xE Q,wegetI(s) >O r(s) = hg(u(~9f)(4M4 h. s inceg,>Oandg(u(s,f)(x)) for any s E R. Hence the function r assumes its supremum f7(f) in so E W. Since r is continuous and !ly I’(s) =0 = ,lJm_ I’(s) we infer from the mean value theorem that for t E (0, t*(f)) there exist s1 < so < s2 with T(sJ = t = I&). Then u(si, f) + u(s2, f), because s1 f s1 and Pn(Sj, f) = Sjq, j = 1,2 and P the orthogonal projection on ker(T). Since PN(U(Jj, f)) = lI(sj)cp, and ru(+, f) + (I - P)Nu(+, f) = f the assertion follows. The following proposition shows in particular what may happen for g with compact support. Put C,(Q) = {u E C(Q): u is bounded in Q}. PROPOSITION 2.2. Let the hypotheses of proposition 1.1 be satisfied. Assume that the embedding J: 6(T) + C,(Q) is continuous with bound IIJ I=, where d(T) is the domain D(T) endowed ! and C,(Q) is endowed with the norm with the norm IIu/I-= (ll~ull~ + y(T)2~~~u~~2)12 lb& = ~UP~IWI: x E Q}. Assume further Q,> 0, g + 0. (a) If g(r) Z 0 (g(r) Z 0) for r E R then for any f E R(T) (0, h(f)] C W(f) C [O, h(f)] ([tl(f), 0) C W(f) C [tl(f), 01) and for t E (0, h(f)) (t E (t*(f), 0)) there exist at least two solutions in D( ?“) of Tu + Nu = f + tp (b) If g(r) E 0 (g(r) I 0) for r E Iwand g(r) = 0 for r E [ - r’, r’], then for any f E R(T) with
IJll4llfII+ supMr)I: r E WJQl&I < r’. W(f)
= [0, r2(f)] (W(f)
D(T) for Tu + Nu = f + tp
= [tl(f),
01) and for t = 0 there exist infinitely many solutions in
742
w.
STORK
Proof. (a) It is clear that T(s) IO for s E W. We show that r is not constant. Since g + 0 is continuous there is a ro E 52 and an E > 0 such that g(r) > 0 for r E [ro - E, r0 + E]. We prove that ro - ES u(s)(x) 5 ro + E for suitable, fixed s E W and x E R’, R’ C R and Lebesguemeasure meas > 0. This then implies T(s) > 0. The mapping F: W-t d(T) defined by F(s) = u(s), u(s) the fixed point of L(s,f), is continuous by the estimate in step (iii) of the proof of theorem 1.2. Since the embedding J is continuous and since point evaluation 6(x) is continuous on C,(Q) we conclude that 6(x) .J. F:R - G?, 6(x) .I. F(s) = u(s) (x), is continuous. From the proof of proposition 1.1 we know that --Ml + sq(x) I U(S)(X) 5 Ml + sq(x). Hence for fixed x0 E R we know lim (II( == and lim (u(s)(x~)) = - x. s-.-x S-X By the mean value theorem we infer that there is an SOsuch that U(SO)(XO) = ro. By the continuity of U(Q) there is a neighbourhood U(Q) = R’ of x0 such that U(SO)(X)E [ro - E, ro + E] for any x E R’. Clearly meas(U(xo)) > 0 and thus I+(Q) > 0 which implies that I assumes its supremum tl(f) in R. Analogously to proposition 2.1 one proves that for t E (0, t?(f)) there exist at least two solutions in D(T) for Tu + Nu = f + tq. (b) From part (a) we know that (0, tz(f)] C W(f) C [0, t?(f)]. Hence it remains to show that for I = 0 there are infinitely many solutions in D(T) for TM + Nu = f+ tq. From step (ii) in the proof of theorem 1.2 we know that (I - P)u(s) = T-‘(1 - P)( - Nu(s) ff). Since and since the embedding J is continuous we obtain the estimate (I - P)u(s) E D(T)
IIU- ~)~(~)Il~‘liJll~li~-‘(~-~)(-~~(~)
+f)ll-
This implies -Ml
+ s&x) 5 u(s)(x) I Ml + SF(X) for x E R. Since Q,E L,(Q), for s with I4 < (I’ - ~W)/lldl~ we get T(s) = 0. By the same arguments as in the proof of proposition Fl one shows that {u(s) : 1.~1< (r’ - Ml)/llqljx} are pairwise distinct solutions of Tu + Nu = The following example of the negative Laplacian with Neumann boundary condition shows that in particular for g with compact support the connected set W(f) may be closed for any f E R(T). Example 2.3. Let R c W3 be a bounded domain with smooth boundary. Let Tin L?(Q) over A be defined by D(T) = {u E W’.‘(Q) : du/h = 0 on dR} and TLI = - Au. Let g: A ---, W be Lipschitz-continuous with Lipschitz constant q < y(T) and g(r) = 0 for r < rl. g(r) 2 0 and g+O. LetN:D(T) += &(Q) be defined by (Nu)(x) = g(u(~r)). Then for any f E R(T) the set W(f) is closed, W(f) = [0, h(f)], for t E (0, h(f)) t h ere exist at least two solutions in D(T) of Tu + Nrc = f + tq, where q = c = (Jo ldx)-‘, and for Tu i Nrc = f there are infinitely many solutions in D(7). Proof. The embedding J:d(T)+ C,(Q), defined in proposition 2.2, is continuous by Sobolev’s lemma. From proposition 2.2(a) we then know that (0, t?(f)] c W(f) C [0, r?(f)] and that there are at least two solutions in D(T) of Trl + h’L1 = f i fq for f E (0, t?(f)). It remains to show that there are infinitely many solutions of Trl i IVU= f in D(T). Since Q:= c = (Jo1 du)-’ and -Ml + ST(X) Z u(s)(x) 5 Ml f SF(X) for any s E R we get u(s)(x) = 0 for s < (rl - MI)/c and r:(s) = 0. Similarly to the proof in proposition 2.1 one shows that {U(S): s < ( rl - Ml)/c} are pairwise distinct solutions of Tn + ~Vrt= f.
Perturbation
by vanishing nonlinearities
743
The following proposition gives sufficient conditions for the closedness of W(f) for any f~ R(T). It is an extension with respect to the nonlinearity of the results in [2, theorem 5.21 and [3, theorem l] at the cost of an additional condition on the eigenfunctions of the linear operator T. In [2] and [3] only the choices O(s) = s and g,(p) = y+x,.(P) + y-x_(p) and g_(p) = y-xl(p) + y,x_(p) are allowed, where x= is the characteristic function of R =, ;J_ =Jiy O(s)g(s) and y- = lim 0 (s)g(s). s-+-x PROPOSITION 2.4. Let the hypotheses of proposition 1.1 be satisfied. Assume that there is a continuous function 0 : R -+ R with O(r) > 0 for r > ro and O(r) < 0 for r < - t-0 such that lim 0 (r)g(rp +r’)p = g+(p) and ,liy% O(r)g(rp +r’)p = g-(p) uniformly for r’ E [-K, K]
r-35
and any K > 0, where g_ and g_ are measurable functions with J&+(q(x)) dr > 0 and Jn&(F@)) dJZ’ 0. Th en for any f E R(T) the connected set W(f) is closed, W(f) = [h(f), h(f)1 with h(f) < 0 < tdf), and for any t E (tl(f), tz(f)), t f 0, there exist at least two solutions in D(T) of Tu + Nu = f + tq. Proof.
We know W(f)
= R(T)
f rom the proof of proposition
1.1. Since lim T(s) = 0 = s--t= lim r(s), it suffices to show that I’ attains values greater than zero and values smaller than s-+-1 zero. Consider the continuous function Or: IF?--* R. We know -M, + s&x) $ u(s)(x) 5 M1 + q(x) from the proof of proposition 1.1. Then by our assumptions on 0 and g and by Lebesgue’s theorem we get ,‘iir O(s)I$) =J&+(q(x)) dr and lim O(s)IY(s) = *-+--r m&(&x)) h. S’mce O(s) > 0 for s > r. and O(s) < 0 for s < - ro we have inf{I’(s) : s E W} = T(sl) = tl(f) < 0 and sup {T(s): s E R} = T(sZ) = h(f) > 0 with sl, s2 E iF&By the same arguments as in the proof of proposition 2.1 one shows that for f E (tl( f), rz(f )), t # 0, there are at least two solutions of Tu + Nu = f + tq in D(T). We give some examples of functions g and selfadjoint operators T such that the hypotheses of proposition 2.4 are satisfied. If g(r) = A$‘-‘/(1 + p) with 0 < h < y(T) and 0 < k < n, then 0 may be chosen by O(r) = ?“- 2k+1. The hypotheses on 0 in proposition 2.4 are satisfied with g+(p) = Apzk-*” = g_(p). Let T be the selfadjoint operator in LZ( - 1, 1) generated by the Legendre differential expression (-d/dt) (1 - 3) (d/dt) + 2$/(1 + t) + 22/(1 - t) - (2cr+ 1)2a for some (Y with 0 < a< (2n - 2k)-’ 5 2-l and boundary conditions A+(u) = lim {P”u’(l - E) E-+X &%(l - E)} = 0 and B,(u) = ~_mo{P”u’(~-l) - a~‘%(&-- 1)) = 0. Then T has discrete spectrum a(T) = ad(T) = {A,, = n = 0, 1,2, . . .} with corresponding simple eigenvectors P!,‘@.‘@ are ultraspherical polynomials [4, p. 15191 and 0 Clearly 9(f) = c0(1 - f2), spans ker(T) and J’&_(pa(t)) By the bounds 11 Pi2a*2aJll= 5 Cd” (5, th eorem 7.32.1 p. its eigenfunction expansion to
(n + 2cu + l)(n + 2~) - (2a + 1)2~:
qJt) = c,(l - tZ)“p(n?“.2n)(t)where < c,, < Cn@, n 2 1 [5, 4.3.4, p. 671. dr =Jilg-(&t)) dr > 0. 1631 for u E D(T) we may estimate
llu/z d C[n$ (n2n+112/n2)2(,,Tu~~2 + ~(T)‘I]Pu~/~)]@ < 13.
7-M
w. STORK
Since all eigenfunctions are continuous this shows that the embedding d( 7) ----,C[-1, l] is continuous. Hence T satisfies the hypotheses of propositions 1.1 and 2.4 respectively. The conditions in proposition 2.4 are sufficient for the closedness of R(T + N) as is shown in [3, theorem 21 in the special situation O(s) = s. We prove a necessary and sufficient condition for the closedness of R(T + N) from which the closedness of R(T + IV) for the operators in proposition 2.4 follows. PROPOSITION2.5. Let the hypotheses of theorem 1.2 be satisfied. Then R( T + N) is closed if and only if the connected set W(f) is closed for any f E R(T). Proof. h=f+tlql+...
closed for any f E R(T). Let h E R(T + N), W(f) is + f,,q,, where f E R(T) and tlql + . . . + f,,cp,,E ker( T). Then there is a sequence (hk) C R(T + N) with hk-, h in H and hk = fk + tl,kq)l + . . . + tn,kqn where fk E R(T) and hk - fk E ker(T). Since R(T) @ ker(T) = H we obtain fk+ f in H and hk - fk + h - f in H. We prove that hk-, h in H implies t = (tI, . . . , fn) E W(f) which by virtue of theorem 1.2 implies h E R(T + N). Suppose t 65 W(f). S ince W(f) is closed, there is an E> 0 such that II’(s, f) - t I,, > E for any s E R”. From the properties of U(S, f) and U(S, fk) respectively in step (ii) of the proof of theorem Assume
1.2 we infer the estimate b&f)
- +,fk)i/-
< (l - L)-‘iif
-fkll
and hence we get Ir(%f)
- r(s,fk)
In = lipNLL(%f) - PNu(%fk)
f) - rr(s,fk) I(- 5 L(1 - ,!,)
11g Ll/+,
-‘l,f -
fklj.
Combining these estimates we obtain the inequality s< ir(sk,f)
- tin 5 Ir(sk,f)
- mktfk)
In +
ir(sk,fk)
-
tin
of r(sk, fk) = (tl,k, . . . , f&k). Since 1r(&, fk) - tl, + 0 for k + 02, We k % kg(&) from this inequality and hence J2 5 infer Ir(Sk,f) r(Sk,fk)In 2 E/2 for oes not converge to f in Hand consequently (hk) L(1 -L)-‘IIf -fk\l for k2 kg(&). Thus (fk) d does not converge to h in H. The if-part of proposition 2.5 is proven. be fixed and consider Contrarily assume that R(T + N) is closed. Let f E R(T) t = (t*, . . . , fn) E W(f). Then there is a sequence (tk), tk = (fl,k,. . . , f&k), in W(f) with tk-+ t in R”. This implies
where sk is a
SdUtiOn
(f +
tl.kQ)l
+
. . . +
b,k%)
E R(T + N)
and Since R(T + N) is closed we then know that (f-t llql + . . . + t,,q,) E R( T + N) and by theorem 1.2 we infer that (tl, . . . , rn) E W(f). Thus W(f) is closed. This holds true for any f E R(T).
We present a class of linear operators T to which the nonlinear perturbations in propositions 1.1, 2.1, 2.2 and 2.4 may be applied and which covers uniformly elliptic differential and pseudodifferential operators of sufficiently high order and with sufficiently smooth coefficients.
Perturbation
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745
Example 2.6. Let Q C R” be a bounded domain having the cone property. Let T be a selfadjoint operator in LZ(S2) over i&Jwith domain D(T) C \vps’(S2) where 2p > m. Assume that 0 is a simple eigenvalue of T. Then the spectrum a(T) is discrete, a(T) = ad(T) = {Ak:k E N} the reduced minimum modulus y(T) = inf{l & 1:Ak # 0, k E N} > 0 and the embedding J: d(T) ---, C,(n) is continud(T) is the domain D(T) endowed with the norm 11~/1-= ous where * fl, P the orthogonal projection P: L?(R)-* ker(T), and C,(Q) is (11T@ + Y(T)*IIP~I~ ) endowed with the norm l/uII~ = sup{1u(x)11 x E Q}. Proof. We first prove that the embedding .I is continuous. The operator T is closed. Then by the closed graph theorem the embedding E: D(T) ---, WP.‘(Q) is continuous, where d(T) is the domain D(T) endowed with the graph norm. Since 2p > m and since R has the cone theorem. property the embedding J-: WP.*(Q) + C,(Q) is compact by the Rellich-Kondrachov We now prove that the graph norm and the nqrm Ij* II- are equivalent norms on D(T). This then implies that the embedding J = J-. E: D(T) + CB(1;2) is compact and in particular continuous. We know IIPull 5 Ilull and l/u/l 5 (II~-1~12~IT~~~2 + llP~11*)~‘* = ~(T)-‘ll+ Hence (IITuII’+ y(T)‘IIPulj*)”
s (max{L Y(T)*~)“(I~ Tull + ll4>
and /ITuII + /Iu[l s (1 + y(T)-‘)(IITujl*
+ ~(T)‘IIPK~I*)~~’
and the equivalence of the two norms is shown. We now prove that T has discrete spectrum, a(T) = ad(T) = {&: k E N}. This then implies that y(T)=inf{/A~~:kEN,&fO}>O
and
ker(T) @R(T)
= L2(R).
Consider the complexification Tc of T in the complexification of L*(Q) over 2 which is L2(Q) over C, II = {ul + iuz: ul, u2 E D(T)} and Tc(ul + iuz) = Tul + iTu?. The complexification Tc is selfadjoint ([6] e 5.32 p. 111) and since D(T) c Wp~‘(Q) the domain D(Tc) is contained in Wps’(Q) over C. By the same arguments as above one shows that the embedding JZ:* E,: d( Tc) + Wp,*(Q) over Q=is compact. Then the operator Tc has compact resolvent (Tc - i)-’ and hence has discrete spectrum a(Tc) = c~(Tc) = {Ak: k E N}. We show that ad(T) = od( Tc) = a(T). Let AkE ad(T) with corresponding eigenvector ek. Then Tc(ek + iek) = TQ + iTQ = &(ek + iek) and hence & E CQ(Tc). Conversely, consider & E ad(Tc) with corresponding D(T) and
eigenvector
fk + igk. Then 6, gkE
TC(fk+ igk) = Tfk+ iT& = &fk + i&gk. Since A, is real Tfk= & fk and Tgk = I$&. Thus & E ad(T) and ker(Tc - &) has an orthonormal base of real valued functions for any & E ad(Tc). Moreover, dim(ker(Tc)) = dim(ker(T)) = 1. We have shown ad(T) = od(Tc).
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Since the normalized eigenfunctions of TC form an orthonormal base in ,5,,(Q) over @ and since they may be chosen real valued, they form an orthonormal base of LZ(S2) over W. Then for r $?iad(T) the operator (T - r) -’ is a bounded operator, defined on L:(R), and thus u(T) = ad(T) = ud(Tc). The proof of example 2.6 is now complete. Results similar to those of proposition 1.1 hold true if we perturb, e.g. selfadjoint realisations T of -u” in L2(0, 1) over &?by N: D(T) + Lz(0. l), (Nn)(x)
= --&U(X) +&C(x)),
where Ak is a simple eigenvalue of T and g satisfies the hypotheses in proposition 1.1 with a suitable Lipschitz constant q. If we choose the perturbation N by (Nu(x)) = -&U(X) + g(u”(x)), then we can prove the results of proposition 1.1 merely for suitable ‘right hand’ sides f~ R(T - &) namely fE R(T - &) n L,(O, 1). REFERENCES 1. FU~IK S., Nonlinear noncoercive boundary value problems, in Equadiff IV, Proceedings Prague 1977, Lecture :Votes in Mathematics 703, 99-109, Springer. Berlin (1979). 2. XL~BROSEITIA. & MASCINI G., Existence and multiplicity results for nonlinear elliptic problems with linear part at resonance, /. diff. Eqns 28, 22C-245 (1978). 3. HESS P., sonlinear perturbations of linear elliptic and parabolic problems at resonance: existence of multiple solutions, Annali marh. Pba 5, 528-537 (1978). 4. DUNFORD N. & SCH~;\RTZ J. T., Linear operarors parr Il. Interscience, New York (1963). 5. SZEGO, G., Orrhogonal polynomials, American mathematical Society, New York (1939). 6. WEIDM.A%;NJ., Lineare Operaforen in Hilberrriiumen, Teubner, Stuttgart (1976).