Fučík, Svatopluk

Hess, Peter

English

Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaeSvatopluk Fučík; Peter HessNonlinear perturbations of linear operators having nullspace with strong uniquecontinuation property (Preliminary communication)Commentationes Mathematicae Universitatis Carolinae, Vol. 19 (1978), No. 2, 403--407Persistent URL: http://dml.cz/dmlcz/105863Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1978Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czCOMffiNTATlONBS mTHEMATICAB UNIVIRSITATIS CAROLINA! 19,2 (1978) NONLINEAR PERTURBATIONS OF LINEAR OPERATORS HAVING NULL-SPACE WITH STRONG UNIQUE CONTINUATION PROPEHTX (Preliminary Communication) Svatopluk KJČÍK, Praha, Peter HESS, ZOrich Abstract: We are concerned with the existence of one or multiple solutions of various problems for nonlinear differential equations which can be reduced to an abstract operator equation of the form Lu + G(u) s f ina real Hil-bert space H, with L:HaG(L)—> H being linear ani nonin-vertible, G:H—p H nonlinear and f € H given. Key words: Nonlinear operator equations, strong uni-que continuation property. AMS: 47H15 Let Q denote a bounded domain in R^ (N£l)t and let 2 H » L (Q), with norm 1-1 and inner product (.,.)• Let L: :H3 D(L)—*• H be a closed linear operator with dense dom-ain D(L) and closed range R(L). We assume that 0 is an ei-genvalue of L and of its adjoint operator L* , and that for the corresponding eigenspaces , E(L) « N(L*) and dim N(L)< + oo « Hence H admits the orthogonal decompo-sition H a N(L)® R(L). We set H-̂:-- N(L)f H2:= R(L), and denote by Pĵ  the orthogo-403 -nal projection of H onto H^ (i » 1,2). An element fe H may thus be decomposed into f « f-, + fgt whe*** % s ̂ i^# The restrictioa Ls* LjH t with .0(11) « DtDnHji is a» al-gebraic isomorphism in H** Its inverse (the so-called ri^it inverse of L) will be denoted by T• We assume that T;H2—* H 2 is compact. Our main assumption on the functions in N(L) is the following "strong unique continuation property": (SUCP); H(L)cL*(Q), and there exists f > 0 such that for the function y t E H-^jp(e) = sup meas 4xe Q; I w(x) t < e I * wcN(L) ftwfl^l y( e ) = o(e* ) as e—** o+. Remark. The usually imposed "unique continuation pro-perty" demands that the only function w«N(L) vanishing on a set of positive measure in Q is w » 0. As dim N(L)< < + $0 this implies that y(e>)— * 0 as e—^0+. Thus the (SUCP) prescribes the speed of convergence. Let g;R—*R be a continuous function with finite li-mits g.;« lim g(s). Without restriction we may assume g_4 0i4g+., Suppose the-re exists (f > 0 such that g(a)^g+ V ®2<T g(s)ig^ Vs|-<f. For a £ of we set (with the same ® as in (SUCP)) 404 ^(aK:** lim inf f l4f min (g(s) - g . ) , 8 • §-**i» > A«C%f 3 ^ (a ) s« lim inf f 1 + f min (g . - g ( s ) ) . 8 - I-***© f /i€C-|ra,3 Let G:H—«• H be the Neu^rtskii operator associated with g: G(u)(x):» g(u(x)) f x#Q f for any function u defined on Q. The napping G is conti-nuous and has bounded range in H. Let S:=4f1€H1t (flfw) £ J (g^w* - gjT)dx, V w i ^ l . Here w (w~) is the positive (negative) part of the func-tion wf i.e. w « w -w~. Note that Sc H^ is nonempty, boun-ded, closed and convex. Theorem 1. Suppose (A) D{L)cL*(Q)f and T:H2~-* L***(Q) is continuous. Suppose further that either (oO) the functions in N(L) have constant sign in Q and y(a) + « T^aL s * °® (for a suitable a i < f ) , ©3* (($) the functions in N(L) change sign in Q and at least one of y(a)+i 9b(a). is = * CD (for a suitable a & Z<T)* Then to each f2* H2 there exists an open set Bf c H-̂ , Sf 3 S , such that r2 (i) the equation (1) Lu * G(u) = f has at least one solution for f * f^ + f2 with f., c % ; - 405 -(ii) the equation (1) has at least two solutions for f « f 1 + fg with f-̂ t Sf \ S. Theorem 2, Under the assumptions of Theorem 1, the range of L + 0 is closed in H. Hence the assertion (i) of theorem 1 is in fact valid for f. t S ^ • A variant of Hie or em 1 is Theorem 3. Instead of (A) let the following regula-rity assumption be satisfied: (A#) ISiere exists m»0 such that for any solution u€H of Lu * f with f c Lm (Q) we have u« L°°(Q) and 1 P̂ u L A m If B_ # £ C Q 00 Suppose either (oC ) or ((I). Then the assertions of theo-rem 1 hold, provided f2f L*>(Q). It is possible to apply the above abstract theorems to a large variety of examples, such as the boundary value problem for ordinary and elliptic differential equations, and the problem of existence of periodic solutions of the nonlinear heat equation and the nonlinear telegraph equa-tion. The proofs and the investigation of these applicat-ions will appear elsewhere. The main part of the results was obtained while the second author was visiting the Charles University, the de-tailed paper has been submitted to "Nonlinear Analysis. Theory and applications1*. - 406 Matematieko-f^zikální fakulta Mathematisches Institut Universita Karlova der Universitlt ZCirich Sokolovská 83 ireiestrasse 36 18600 Praha 8 8032 Zurich Československo Schweiz (Oblátům 24.3. 1978) - 407 -

Nonlinear perturbations of linear operators having nullspace with strong unique continuation property (Preliminary communication)

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Nonlinear perturbations of linear operators having nullspace with strong unique continuation property (Preliminary communication)

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Institute of Mathematics AS CR, v. v. i.