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Existence Theorem for One Class of Strongly Resonance Boundary-Value Problems of Elliptic Type with Discontinuous Nonlinearities

We consider the Dirichlet problem for an equation of the elliptic type with a nonlinearity discontinuous with respect to the phase variable in the resonance case; it is not required that the nonlinearity satisfy the Landesman-Lazer condition. Using the regularization of the original equation, we est...

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Дата:2005
Автори: Pavlenko, V. N., Chizh, E. A., Павленко, В. Н., Чиж, Е. А.
Формат: Стаття
Мова:Російська
Англійська
Опубліковано: Institute of Mathematics, NAS of Ukraine 2005
Онлайн доступ:https://umj.imath.kiev.ua/index.php/umj/article/view/3577
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Назва журналу:Ukrains’kyi Matematychnyi Zhurnal
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Ukrains’kyi Matematychnyi Zhurnal
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author Pavlenko, V. N.
Chizh, E. A.
Павленко, В. Н.
Чиж, Е. А.
Павленко, В. Н.
Чиж, Е. А.
author_facet Pavlenko, V. N.
Chizh, E. A.
Павленко, В. Н.
Чиж, Е. А.
Павленко, В. Н.
Чиж, Е. А.
author_sort Pavlenko, V. N.
baseUrl_str https://umj.imath.kiev.ua/index.php/umj/oai
collection OJS
datestamp_date 2020-03-18T19:59:02Z
description We consider the Dirichlet problem for an equation of the elliptic type with a nonlinearity discontinuous with respect to the phase variable in the resonance case; it is not required that the nonlinearity satisfy the Landesman-Lazer condition. Using the regularization of the original equation, we establish the existence of a generalized solution of the problem indicated.
first_indexed 2026-03-24T02:45:07Z
format Article
fulltext UDK 517.95 V. N. Pavlenko, E. A. ÇyΩ (Çelqbyn. nac. un-t, Rossyq) TEOREMA SUWESTVOVANYQ DLQ ODNOHO KLASSA SYL|NO REZONANSNÁX KRAEVÁX ZADAÇ ∏LLYPTYÇESKOHO TYPA S RAZRÁVNÁMY NELYNEJNOSTQMY We consider the Dirichlet problem for an elliptic-type equation with nonlinearity discontinuous with respect to a phase variable in the resonance case. It is not necessary that this nonlinearity satisfy the Landesman – Lazer condition. Using the regularization of the initial equation, we establish the existence of generalized solution of the problem considered. Rozhlqnuto zadaçu Dirixle dlq rivnqnnq eliptyçnoho typu z rozryvnog za fazovog zminnog nelinijnistg v rezonansnomu vypadku, pryçomu nelinijnist\ moΩe ne zadovol\nqty umovu Landesmana – Lazera. Za dopomohog rehulqryzaci] poçatkovoho rivnqnnq vstanovleno isnuvannq uzahal\nenoho rivnqnnq vkazano] zadaçi. 1. Vvedenye. Pust\ Ω — ohranyçennaq oblast\ v R m s hranycej Γ klassa C 2, µ , 0 < µ < 1, A u ( x ) ≡ – i j m i ij jx a x u x, ( ) = ∑ ∂ ∂ ∂     1 ∂ + c ( x ) u ( x ) — ravnomerno πllyptyçeskyj dyfferencyal\n¥j operator na Ω s koπffy- cyentamy ai j ∈ C1,µ Ω( ), ai j ( x ) = aj i ( x ), c ∈ C0,µ Ω( ), c ( x ) ≥ 0 v Ω. Rassmotrym zadaçu A u ( x ) – λ1 u ( x ) + g ( x, u ( x ) ) = h ( x ), x ∈ Ω, (1) u Γ = 0, (2) hde λ1 — naymen\ßee sobstvennoe znaçenye operatora A s hranyçn¥m uslo- vyem (2), h ∈ L q ( x ), q > m. Nelynejnost\ g ( x, ξ ) udovletvorqet sledugwym uslovyqm: 1) g : Ω × R → R boreleva ( mod 0 ) [1], t. e. suwestvuet boreleva funkcyq g̃ : Ω × R → R, otlyçagwaqsq ot g lyß\ na podmnoΩestve l ⊂ Ω × R, proek- cyq kotoroho na Ω ymeet meru nul\; 2) dlq poçty vsex x ∈ Ω funkcyq g ( x, ⋅ ) ymeet razr¥v¥ tol\ko pervoho roda, g ( x, ξ ) ∈ g x g x− +[ ]( , ), ( , )ξ ξ dlq lgboho ξ ∈ R, hde g x− ( , )ξ = lim ( , )η ξ η→ g x , g x+ ( , )ξ = lim ( , )η ξ η→ g x . Krome toho, predpoloΩym, çto suwestvuet funkcyq a ∈ L q ( Ω ), dlq koto- roj spravedlyva ocenka | g ( x, ξ ) | ≤ a ( x ) ∀ ξ ∈ R y poçty vsex x ∈ Ω. (3) Opredelenye 1. Obobwenn¥m reßenyem zadaçy (1), (2) budem naz¥vat\ funkcyg u ∈ Wq 2( )Ω ∩ Wq 1� ( )Ω , udovletvorqgwug dlq poçty vsex x ∈ Ω vklgçenyg – A u ( x ) + λ1 u ( x ) + h ( x ) ∈ g x u x g x u x− +( ) ( )[ ], ( ) , , ( ) . Yzvestno [2], çto podprostranstvo reßenyj zadaçy © V. N. PAVLENKO, E. A. ÇYÛ, 2005 102 ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1 TEOREMA SUWESTVOVANYQ DLQ ODNOHO KLASSA SYL|NO … 103 A u ( x ) = λ1 u ( x ), x ∈ Ω, (4) u Γ = 0, (5) odnomerno, pryçem bazysnug funkcyg ϕ πtoho podprostranstva moΩno sçytat\ poloΩytel\noj v Ω y ∂ϕ ∂n Γ < 0, hde ∂ ∂n — proyzvodnaq po vneßnej normaly k hranyce Γ. PoloΩym L u = A u – λ1 u, a çerez Ker ( L ) oboznaçym prostranstvo reßenyj zadaçy (4), (5). Poskol\ku K e r ( L ) ≠ { 0 }, a dlq nelynejnosty g verna ocenka (3), zadaça (1), (2) qvlqetsq rezonansnoj. Systematyçeskoe yssledovanye rezonansn¥x kraev¥x zadaç πllyptyçeskoho typa naçalos\ s rabot¥ E. Landesmana y A. Lazera [3] v 1970 h. V ukazannoj stat\e predpolahalos\, çto nelynejnost\ g ( x, ξ ) ≡ g ( ξ ) neprer¥vna na R y su- westvugt lim ( )ξ ξ→±∞ g = g ( ± ∞ ). Pry takyx dopuwenyqx b¥lo pokazano, çto zadaça (1), (2) ymeet reßenye, esly h udovletvorqet neravenstvu g x dx( ) ( )−∞ ∫ Ω ϕ < Ω ∫ h x x dx( ) ( )ϕ < g x dx( ) ( )+∞ ∫ Ω ϕ . (6) Zdes\ y dalee ϕ — bazysnaq funkcyq Ker ( L ), ϕ > 0 v Ω. Krome toho, esly g ( – ∞ ) < g ( ξ ) < g ( + ∞ ) dlq lgb¥x ξ ∈ R, to uslovye Landesmana – Lazera (6) qvlqetsq y neobxodym¥m dlq suwestvovanyq reßenyq yssleduemoj zadaçy. V dal\nejßem poqvylos\ bol\ßoe çyslo statej po πtoj tematyke, v kotor¥x avtor¥ naklad¥valy na nelynejnost\, vxodqwug v uravnenye, uslovyq typa Landesmana – Lazera. Dlq zadaç s razr¥vn¥my po fazovoj peremennoj nely- nejnostqmy naybolee obwye rezul\tat¥ b¥ly poluçen¥ N. Basile, M. Mininni [4], I. Massabo [5], K. C. Chang [6] y V. N. Pavlenko, V. V. Vynokurom [7, 8]. VKposlednee vremq bol\ßoj ynteres v¥z¥vaet yzuçenye kraev¥x zadaç πllypty- çeskoho typa v sluçae syl\noho rezonansa, t. e. kohda uslovyq Landesmana – La- zera ne v¥polnqgtsq. Otmetym rabotu P. Bartolo, V. Benci, D. Fortunato [9] dlqKuravnenyj s hladkoj nelynejnost\g y R. Iannacci, M. N. Nkashama, J. R. Ward [10] dlq uravnenyj s karateodoryevoj nelynejnost\g. V [11] dokaza- no suwestvovanye obobwennoho reßenyq zadaçy (1), (2) s razr¥vnoj nelynej- nost\g v predpoloΩenyy, çto dlq poçty vsex x ∈ Ω y dlq lgboho ξ ∈ R g ( x, ξ ) ξ ≤ 0 (7) y funkcyq h ( x ) ortohonal\na Ker ( L ) v L 2 ( Ω ). V dannoj rabote poluçen¥ uslovyq razreßymosty zadaçy (1), (2), obobwag- wye rezul\tat¥ rabot [10, 11]. Opredelenye 2. Budem hovoryt\, çto dlq nelynejnosty g y funkcyy h v uravnenyy (1) v¥polneno (i)-uslovye, esly verno lybo neravenstvo Ω ∫ ∈ + sup ( , ) ( ) ξ ξ ϕ R g x x dx ≤ Ω ∫ h x x dx( ) ( )ϕ ≤ Ω ∫ ∈ − inf ( , ) ( ) ξ ξ ϕ R g x x dx , (8) lybo neravenstvo Ω ∫ ∈ − sup ( , ) ( ) ξ ξ ϕ R g x x dx ≤ Ω ∫ h x x dx( ) ( )ϕ ≤ Ω ∫ ∈ + inf ( , ) ( ) ξ ξ ϕ R g x x dx , (9) hde R ± = { ξ ∈ R : ± ξ > 0 }, a ϕ — bazysnaq funkcyq Ker ( L ), ϕ > 0 v Ω. Kak budet pokazano nyΩe, esly v¥polneno (i)-uslovye, to zadaça (1), (2) yme- et, po krajnej mere, odno obobwennoe reßenye. Zameçanyq. 1. Esly g ( x, ξ ) ≡ g ( ξ ) y sup ( ) ξ ξ ∈ −R g = g ( – ∞ ), inf ( ) ξ ξ ∈ +R g = g ( + ∞ ), ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1 104 V. N. PAVLENKO, E. A. ÇYÛ to neravenstvo (9) v (i)-uslovyy daet bolee ßyrokoe, çem uslovye Landesmana – Lazera (6), mnoΩestvo funkcyj h ∈ L q ( x ), dlq kotor¥x zadaça (1), (2) ymeet reßenye. 2. Esly funkcyq h ( x ) ortohonal\na Ker ( L ), to yz neravenstva (8) v (i)- uslovyy ymeem Ω ∫ ∈ + sup ( , ) ( ) ξ ξ ϕ R g x x dx ≤ 0 ≤ Ω ∫ ∈ − inf ( , ) ( ) ξ ξ ϕ R g x x dx , çto, v çastnosty, v¥polnqetsq, esly verno (7). 2. Osnovn¥e rezul\tat¥. Teorema. PredpoloΩym, çto: 1) funkcyq g ( x, ξ ) udovletvorqet uslovyqm 1, 2; 2) verna ocenka (3) s q > m; 3) dlq nelynejnosty g y funkcyy h ∈ L q ( Ω ) v uravnenyy (1) v¥polneno (i)-uslovye. Tohda zadaça (1), (2) ymeet, po krajnej mere, odno obobwennoe reßenye. Dokazatel\stvo. Rassmotrym approksymyrugwug zadaçu A u – ( λ1 + δn ) u + g ( x, u ) = h, x ∈ Ω, (10) u Γ = 0, (11) hde δn → 0 — parametr rehulqryzacyy. Opredelym operator G : L p ( Ω ) → ( L p ( Ω ) ) * = L q ( Ω ), 1 / q + 1 / p = 1, raven- stvom Gu = g ( x, u ( x ) ), a lynejn¥j dyfferencyal\n¥j operator Ln : D ( Ln ) ⊂ ⊂ L p ( Ω ) → L q ( Ω ) ravenstvom L n u = A u – ( λ1 + δn ) u , hde D ( Ln ) = = u W uq∈ ={ }2 0( )Ω Γ ⊂ L p ( Ω ). Tohda zadaçu (10), (11) moΩno zapysat\ v ope- ratornoj forme Ln u + G u = h. (12) Rassmotrym Ln kak operator yz E = Wq 2( )Ω ∩ Wq 1� ( )Ω s normoj prostran- stva Wq 2( )Ω v L q ( Ω ). Yzvestno [12], çto λ1 — yzolyrovannaq toçka spektra Ln kak operatora v L 2 ( Ω ) s oblast\g opredelenyq W2 2( )Ω ∩ W2 1� ( )Ω . Poπtomu najdetsq natural\noe çyslo n0 takoe, çto pry n > n0 operator Ln neprer¥vno obratym. Yz πtoho zaklgçaem o byektyvnosty Ln : E → L q ( Ω ) (s uçetom teorem o rehulqrnosty reßenyj πllyptyçeskyx kraev¥x zadaç [2] y vloΩenyq L q ( Ω ) ⊂ ⊂ L2 ( Ω )). Otsgda y yz zamknutosty Ln sleduet ohranyçennost\ Ln −1 : L q ( Ω ) → E. Poskol\ku Wq 2( )Ω kompaktno vloΩeno v L p ( Ω ), Ln −1 : L q ( Ω ) → → L p ( Ω ) kompakten. Operator G ohranyçen na vsem L p ( Ω ) v sylu uslovyq (3). Rassmotrym eho sekvencyal\noe zam¥kanye S G : L p ( Ω ) → 2Lq ( )Ω [13] (znaçenye S G u dlq u ∈ ∈ L p ( Ω ) opredelqetsq kak zamknutaq v¥puklaq oboloçka mnoΩestva vsex sla- bo predel\n¥x toçek v L q ( Ω ) posledovatel\nostej vyda { G un }, hde un → u vKKL p ( Ω )). ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1 TEOREMA SUWESTVOVANYQ DLQ ODNOHO KLASSA SYL|NO … 105 Sohlasno [13], S G sovpadaet s G � [1], hde G � — ov¥puklyvanye operatora G, t. e. otobraΩenye yz L p ( Ω ) v 2Lq ( )Ω , znaçenyem kotoroho v proyzvol\noj toçke u ∈ L p ( Ω ) qvlqetsq G � u = ε ε > = − <{ } 0 ∩ conv z Gy y u: , a conv M ( M ⊂ L q ( Ω ) ) oboznaçaet zamknutug v¥puklug oboloçku mnoΩe- stvaKKM. V [1] pokazano, çto vklgçenye z ∈ G � u ravnosyl\no tomu, çto z ( x ) ∈ ∈ g x u x g x u x− +( ) ( )[ ], ( ) , , ( ) poçty vsgdu na Ω . Otsgda poluçaem, çto esly u ∈ D ( Ln ) udovletvorqet vklgçenyg h – Ln u ∈ S G u, to u — obobwennoe re- ßenye (10), (11). Verno y obratnoe. Takym obrazom, suwestvovanye obobwennoho reßenyq zadaçy (10), (11) pry n > n0 ravnosyl\no suwestvovanyg u ∈ L p ( Ω ), udovletvorqgwemu vklgçenyg u ∈ L h SGun − −1( ) . Rassmotrym otobraΩenye Φn ( u ) = L h SGun − −1( ) , n > n0, y dokaΩem, çto suwestvuet zamknut¥j ßar Bn ⊂ L p ( Ω ), kotor¥j Φn otobra- Ωaet v sebq y qvlqetsq na πtom ßare mnohoznaçn¥m kompaktn¥m operatorom. Poslednee oznaçaet, çto Φn : Bn → K Bn poluneprer¥ven sverxu na Bn ( K Bn — semejstvo vsex nepust¥x v¥pukl¥x kompaktn¥x podmnoΩestv Bn ) y obraz ßara Bn predkompakten v L p ( Ω ) [14]. Dejstvytel\no, dlq lgboho u ∈ L p ( Ω ) v sylu svojstv sekvencyal\noho za- m¥kanyq [13] y ohranyçennosty operatora G mnoΩestvo B = h – S G u — zamk- nutoe, v¥pukloe y ohranyçennoe v L q ( Ω ). Poskol\ku lynejn¥j operator v¥pukl¥e mnoΩestva perevodyt v v¥pukl¥e, to Φn ( u ) — v¥pukloe mnoΩestvo v L p ( Ω ). PokaΩem, çto L Bn −1( ) — zamknuto. Pust\ ( yk ) — nekotoraq posledo- vatel\nost\ v L Bn −1( ) , t. e. yk = L xn k −1 , hde xk ∈ B, y pust\ yk → y. Posledo- vatel\nost\ ( xk ) soderΩytsq v ohranyçennom mnoΩestve B, y poπtomu ona ohranyçena. Otsgda v sylu refleksyvnosty prostranstva L q ( Ω ) sleduet, çto suwestvuet podposledovatel\nost\ xkl( ), slabo sxodqwaqsq v L q ( Ω ) k neko- toroj toçke x. V¥puklost\ y zamknutost\ mnoΩestva B vleçet eho slabug zamknutost\, otkuda poluçaem, çto x ∈ B. Dalee, tak kak operator Ln zamk- nut¥j, xkl = Ln ykl ⇀ x y ykl → y, to y ∈ D ( Ln ) y x = L n y. Sledovatel\no, y = L xn −1( ) ∈ L Bn −1( ) , çto y dokaz¥vaet zamknutost\ mnoΩestva L Bn −1( ) . Operator Ln −1 kompakten, y poπtomu on ohranyçennoe mnoΩestvo B perevodyt v predkompaktnoe mnoΩestvo L Bn −1( ) . Otsgda y yz zamknutosty L Bn −1( ) sleduet kompaktnost\ πtoho mnoΩestva v L p ( Ω ). Takym obrazom, dlq lgboho u ∈ L p ( Ω ) znaçenye Φ n ( u ) qvlqetsq nepust¥m kompaktn¥m v¥pukl¥m podmnoΩestvom v L p ( Ω ). Poskol\ku operator G v sylu ocenky (3) ohranyçen na vsem prostranstve L p ( Ω ), najdetsq zamknut¥j ßar Bn , soderΩawyj Φn ( L p ( Ω ) ) = u L n p u ∈ ( ) ( ) Ω Φ∪ . ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1 106 V. N. PAVLENKO, E. A. ÇYÛ Predkompaktnost\ Φn nB( ) sleduet yz kompaktnosty operatora Ln −1 . Takym obrazom, Φn nB( ) ⊂ Bn y mnoΩestvo Φn nB( ) predkompaktno v L p ( Ω ). Ustanovym poluneprer¥vnost\ sverxu na L p ( Ω ) otobraΩenyq Φn . Dosta- toçno pokazat\, çto operator L SGn −1( ) poluneprer¥ven sverxu. PredpoloΩym, çto πto ne tak, tohda najdutsq u ∈ L p ( Ω ) y otkr¥toe mnoΩestvo V ⊃ ⊃ L SG un −1( )( ) takye, çto dlq proyzvol\noho natural\noho k suwestvugt uk s u uk − < 1 / k y w k ∈ L SG un k −1( )( ), no w k ∉ V. ∏lement wk = Ln −1bk , hde bk ∈ SG uk( ). Yz ohranyçennosty otobraΩenyq S G sleduet ohranyçennost\ po- sledovatel\nosty ( bk ) v L q ( Ω ). Otsgda y yz refleksyvnosty L q ( Ω ) zaklg- çaem o suwestvovanyy podposledovatel\nosty bkl( ), slabo sxodqwejsq k neko- toromu b v L q ( Ω ). V sylu svojstv sekvencyal\noho zam¥kanyq [13] b ∈ ∈ SG u( ) . Poskol\ku Ln −1 lynejn¥j kompaktn¥j, posledovatel\nost\ wkl = = L bn kl −1 → Ln −1b ∈ L SG un − ( )1 ( ) ⊂ V v L p ( Ω ). Yz πtoho y otkr¥tosty V zaklg- çaem, çto dlq dostatoçno bol\ßyx l πlement wkl ∈ V, no πto protyvoreçyt v¥boru wk . Takym obrazom, otobraΩenye Φn pry n > n0 qvlqetsq mnohoznaçn¥m kom- paktn¥m operatorom, otobraΩagwym zamknut¥j ßar Bn v sebq. Otsgda sle- duet, çto dlq lgboho n > n0 otobraΩenye Φn ymeet nepodvyΩnug toçku un ∈ Bn [14], a πto ravnosyl\no suwestvovanyg obobwennoho reßenyq zadaçy (10), (11). Zametym, çto, sohlasno opredelenyg obobwennoho reßenyq, pry lgbom n > > n0 suwestvugt funkcyy un ∈ Wq 2( )Ω ∩ Wq 1� ( )Ω y zn ∈ L q ( Ω ) takye, çto dlq poçty vsex x ∈ Ω znaçenye zn ( x ) ∈ g x u x g x u xn n− +( ) ( )[ ], ( ) , , ( ) y A un ( x ) – ( λ1 + δn ) un ( x ) + zn ( x ) = h ( x ). (13) Poskol\ku q > m, Wq 2( )Ω kompaktno vklad¥vaetsq v C1 Ω( ), y, znaçyt, un ∈ C1 Ω( ). DokaΩem ohranyçennost\ v C1 Ω( ) posledovatel\nosty ( un ) obob- wenn¥x reßenyj zadaçy (10), (11). PredpoloΩym protyvnoe. Tohda suwestvuet podposledovatel\nost\, kotorug, po-preΩnemu, budem oboznaçat\ ( un ), takaq, çto un C1 Ω( ) → + ∞, un C1 Ω( ) ≠ 0. Oboznaçym vn = u un n C1 . Zametym, çto vn C1 Ω( ) = 1 y vn qvlqetsq reßenyem zadaçy A vn – ( λ1 + δn ) vn + z u n n C1 = h un C1 , x ∈ Ω, (14) vn Γ = 0. (15) Oboznaçym gn ( x ) = z x u n n C ( ) 1 , h̃n = h un C1 y f n = ˜ ( )h gn n n n+ + −( )λ δ1 v ∈ ∈ L q ( Ω ). Tohda zadaça (14), (15) prymet vyd A vn = fn , vn Γ = 0. Poskol\ku vn C1 = 1, suwestvuet poloΩytel\naq konstanta M0 takaq, çto ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1 TEOREMA SUWESTVOVANYQ DLQ ODNOHO KLASSA SYL|NO … 107 fn Lq ≤ h̃n Lq + ( λ1 + δn ) M0 + gn Lq ≤ ≤ h u L n C q 1 Ω( ) + ( λ1 + δn ) M0 + a u L n C q 1 Ω( ) . (16) Dalee, tak kak δn → 0 y un C1 Ω( ) → + ∞, najdetsq konstanta M > 0 takaq, çto fn Lq ( )Ω ≤ M dlq lgboho n ∈ N. Otsgda y yz teorem ob ocenke syl\n¥x reßenyj zadaçy Dyryxle dlq uravne- nyj πllyptyçeskoho typa [2] sleduet, çto suwestvuet konstanta C > 0 takaq, çto vn Wq 2 ( )Ω ≤ C pry lgbom natural\nom n. Poskol\ku Wq 2( )Ω refleksyv- no, yz ohranyçennosty posledovatel\nosty ( vn ) v πtom prostranstve v¥tekaet suwestvovanye podposledovatel\nosty, slabo sxodqwejsq k nekotoromu v v Wq 2( )Ω . ∏tu podposledovatel\nost\, po-preΩnemu, budem oboznaçat\ ( vn ). Yz kompaktnosty vloΩenyq Wq 2( )Ω v C1 Ω( ) pry q > m sleduet syl\naq sxody- most\ ( vn ) k v v C1 Ω( ). Krome toho, A vn ⇀ A v v L q ( Ω ). Uçyt¥vaq, çto gn Lq ( )Ω = z u n L n C q 1 ≤ a u L n C q 1 → 0, h u L n C q 1 → 0 pry n → ∞, y perexodq v (14), (15) k predelu pry n → ∞, poluçaem A v = λ1 v, (17) v Γ = 0. (18) Zametym, çto v � 0, tak kak v C1 = 1. Sledovatel\no, v qvlqetsq sob- stvennoj funkcyej operatora A, sootvetstvugwej sobstvennomu znaçenyg λ1 . Takym obrazom, v = K ϕ ( x ), hde K ≠ 0 — nekotoraq konstanta. VozmoΩn¥ dva sluçaq: a) K > 0 y, sledovatel\no, v > 0 v Ω, a ∂ ∂ v n < 0 na Γ; b) K < 0 y, sledovatel\no, v < 0 v Ω, a ∂ ∂ v n > 0 na Γ. Sxodymost\ posledovatel\nosty ( vn ) k v v C1 Ω( ) s uçetom a) y b) vleçet suwestvovanye nomera n1 ∈ N takoho, çto dlq vsex n > n1 v¥polnqetsq nera- venstvo vn ( x ) > 0 na Ω, esly K > 0, y vn ( x ) < 0 na Ω, esly K < 0. Poskol\ku un = vn un C1 , dlq vsex n, bol\ßyx n1 , poluçaem, çto un ( x ) > 0 na Ω (un ( x ) < 0 na Ω sootvetstvenno). Voz\mem n > n1 , tohda, domnoΩyv na v obe çasty uravnenyq (13) y proyn- tehryrovav po Ω, poluçym Ω ∫ Au dxnv – Ω ∫ +( )λ δ1 n nu dxv + Ω ∫ z dxnv = Ω ∫ h dxv . V pervom yntehrale dvaΩd¥ prymenym formulu yntehryrovanyq po çastqm y s uçetom toho, çto v Γ = 0 y un Γ = 0, budem ymet\ ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1 108 V. N. PAVLENKO, E. A. ÇYÛ Ω ∫ −( )A u dxnv vλ1 – δn nu dx Ω ∫ v + Ω ∫ z dxnv = Ω ∫ h dxv . Poskol\ku v — sobstvennaq funkcyq operatora A, sootvetstvugwaq sob- stvennomu znaçenyg λ1 , perv¥j yntehral raven nulg. Poπtomu Ω ∫ z dxnv = δn n C nu dx1 Ω ∫ v v + Ω ∫ h dxv . Pust\ v (i)-uslovyy v¥polnqetsq neravenstvo (8) (neravenstvo (9)), tohda voz\mem δn → + 0 ( δn → – 0 sootvetstvenno). Tak kak vn → v v C1 Ω( ), to Ω ∫ v vn dx → Ω ∫ v2 dx > 0 y, sledovatel\no, dlq dostatoçno bol\ßyx n ∈ N Ω ∫ z dxnv > Ω ∫ h dxv (19) Ω Ω ∫ ∫<     z dx h dxnv v . (20) Rassmotrym dva sluçaq: 1. PredpoloΩym, çto K > 0. Tohda v ( x ) > 0 y pry n > n1 funkcyy un ( x ) > > 0 na Ω. Yz neravenstva (8) (neravenstva (9)) poluçaem Ω ∫ z dxnv – Ω ∫ h dxv ≤ Ω ∫ z dxnv – Ω ∫ ∈ + sup ( , ) ξ ξ R g x dxv ≤ 0 Ω ∫   z dxnv – Ω ∫ h dxv ≥ Ω ∫ z dxnv – Ω ∫ ∈ + ≥   inf ( , ) ξ ξ R g x dxv 0 , tak kak un ( x ) > 0 y zn ( x ) ≤ g x u xn+( ), ( ) = lim ( , )( )η η→u xn g x ≤ sup ( , ) ξ ξ ∈ +R g x   zn ( x ) ≥ g x u xn−( ), ( ) = lim ( , )( )η η→u xn g x ≥ inf ( , ) ξ ξ ∈ +  R g x . Poluçyly protyvoreçye s (19) ((20)). 2. Esly predpoloΩyt\, çto K < 0, to v, un , n > n1 , otrycatel\n¥ v Ω . Yz neravenstva (8) (neravenstva (9)) poluçaem Ω ∫ z dxnv – Ω ∫ h dxv ≤ Ω ∫ z dxnv – Ω ∫ ∈ − inf ( , ) ξ ξ R g x dxv ≤ 0 Ω ∫   z dxnv – Ω ∫ h dxv ≥ Ω ∫ z dxnv – Ω ∫ ∈ − ≥   sup ( , ) ξ ξ R g x dxv 0 , tak kak un ( x ) < 0 y zn ( x ) ≥ g x u xn−( ), ( ) = lim ( , )( )η η→u xn g x ≥ inf ( , ) ξ ξ ∈ −R g x ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1 TEOREMA SUWESTVOVANYQ DLQ ODNOHO KLASSA SYL|NO … 109   zn ( x ) ≤ g x u xn+( ), ( ) = lim ( , )( )η η→u xn g x ≤ sup ( , ) ξ ξ ∈ −  R g x , çto protyvoreçyt (19) ((20)). Takym obrazom, m¥ pryxodym k v¥vodu o tom, çto posledovatel\nost\ ( un ) obobwenn¥x reßenyj approksymyrugwej zadaçy (10), (11) ohranyçena v C1 Ω( ). V sylu (13) A un = fn , hde fn = ( λ1 + δn ) un – zn + h ∈ L q ( Ω ). Uçyt¥vaq ohranyçen- nost\ ( un ) v C1 Ω( ), zaklgçaem o suwestvovanyy konstant¥ β > 0, dlq koto- roj fn Lq ( )Ω ≤ β. Otsgda y yz teorem ob ocenke syl\n¥x reßenyj πllypty- çeskyx kraev¥x zadaç [2] sleduet ohranyçennost\ ( un ) v Wq 2( )Ω . Yz reflek- syvnosty Wq 2( )Ω zaklgçaem o suwestvovanyy podposledovatel\nosty unk( ) posledovatel\nosty ( un ), slabo sxodqwejsq v Wq 2( )Ω k u. Sledovatel\no, unk → u v C1 Ω( ), Aunk ⇀ Au y λ δ1 +( )nk unk → λ1 u v L q ( Ω ). Yz (13) poluça- em, çto znk ⇀ z v L q ( Ω ) y z ( x ) = – A u ( x ) + λ1 un ( x ) + h ( x ). V sylu slaboj zamknutosty S G [13] zaklgçaem, çto z ∈ S G u , yz çeho sleduet, çto z ( x ) ∈ g x u x g x u x− +( ) ( )[ ], ( ) , , ( ) dlq poçty vsex x ∈ Ω. Sledovatel\no, u — obobwennoe reßenye zadaçy (1), (2). Teorema dokazana. Zameçanyq. 3. Ynteresno otmetyt\, çto esly dlq funkcyy h ∈ L q ( Ω ) v¥- polneno (i)-uslovye, to ono v¥polneno y dlq h + h1 , hde h1 ∈ L q ( Ω ) — pro- yzvol\naq funkcyq, ortohonal\naq Ker ( L ) v L 2 ( Ω ). Takym obrazom, teorema daet dostatoçn¥e uslovyq suwestvovanyq reßenyq zadaçy (1), (2) srazu dlq celoho klassa funkcyj    h + h1 : h udovletvorqet (i)-uslovyg, a Ω ∫ =     h dx1 0ϕ . 4. Kak pokaz¥vaet sledugwyj kontrprymer, (i)-uslovye nel\zq oslabyt\, za- menyv eho suwestvovanyem poloΩytel\noho çysla r, dlq kotoroho v¥polnqet- sq lybo neravenstvo Ω ∫ > sup ( , ) ( ) ξ ξ ϕ r g x x dx ≤ Ω ∫ h x x dx( ) ( )ϕ ≤ Ω ∫ <− inf ( , ) ( ) ξ ξ ϕ r g x x dx , (21) lybo neravenstvo Ω ∫ <− sup ( , ) ( ) ξ ξ ϕ r g x x dx ≤ Ω ∫ h x x dx( ) ( )ϕ ≤ Ω ∫ > inf ( , ) ( ) ξ ξ ϕ r g x x dx . (22) Dejstvytel\no, v ohranyçennoj oblasty Ω ⊂ R m s dostatoçno hladkoj hranycej Γ rassmotrym zadaçu – ∆ u ( x ) – λ1 u ( x ) + g ( u ( x ) ) = 0, x ∈ Ω, (23) u Γ = 0, (24) ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1 110 V. N. PAVLENKO, E. A. ÇYÛ hde λ1 — naymen\ßee sobstvennoe znaçenye operatora – ∆ s hranyçn¥m uslo- vyem (24), a nelynejnost\ g ( ξ ) = 0 1 1 1 1 1 , , , , , , . ξ ξ ∈ −∞ −( ] + ∞[ ) ∈ −( )    ∪ Oçevydno, çto dlq funkcyj g y h v¥polnen¥ uslovyq (21), (22). Predpolo- Ωym, çto suwestvuet u ∈ Wq 2( )Ω ∩ Wq 1� ( )Ω — reßenye zadaçy (23), (24). Tohda, umnoΩyv obe çasty uravnenyq (23) na sobstvennug funkcyg ϕ operatora – ∆ s hranyçn¥m uslovyem (24), sootvetstvugwug λ1 , y proyntehryrovav po çastqm, poluçym Ω ∫ ( )g u x x dx( ) ( )ϕ = 0. Poskol\ku funkcyq g neotrycatel\na, a ϕ lybo poloΩytel\na, lybo otryca- tel\na na Ω, to g u x( )( ) = 0 ∀ x ∈ Ω. Sledovatel\no, u x( ) > 1 dlq lgboho x ∈ Ω, çto nevozmoΩno v sylu neprer¥vnosty u v Ω y hranyçnoho uslo- vyqK(24). 1. Krasnosel\skyj M. A., Pokrovskyj A. V. System¥ s hysterezysom. – M.: Nauka, 1983. – 272 s. 2. Hylbarh D., Trudynher N. ∏llyptyçeskye dyfferencyal\n¥e uravnenyq s çastn¥my proyz- vodn¥my vtoroho porqdka. – M.: Nauka, 1989. – 496 s. 3. Landesman E., Lazer A. Nonlinear perturbations of linear elliptic boundary value problems at resonance // J. Math. and Mech. – 1970. – 19, # 7. – P. 609 – 623. 4. Basile N., Mininni M. Some solvability results for elliptic boundary value problems in resonance at the first eigenvalue with discontinuous nonlinearities // Boll. Unione math. ital. – 1980. – 17-B, # 3. – P. 1023 – 1033. 5. Massabo I. Elliptic boundary value problems at resonance with discontinuous nonlinearities // Ibid. Ser 5. – 1980. – 17-B, # 3. – P. 1302 – 1320. 6. Chang K.-C. Variational methods for nondifferentiable function and their applications to partial differential equations // J. Math. Anal. and Appl. – 1981. – 80, # 1. – P. 102 – 129. 7. Pavlenko V. N., Vynokur V. V. Rezonansn¥e kraev¥e zadaçy dlq uravnenyj πllyptyçeskoho typa s razr¥vnoj nelynejnost\g // Yzv. vuzov. Matematyka. – 2001. – # 5. – S. 43 – 58. 8. Pavlenko V. N., Vynokur V. V. Teorem¥ suwestvovanyq dlq uravnenyj s nekoπrcytyvn¥my razr¥vn¥my operatoramy // Ukr. mat. Ωurn. – 2002. – 54, # 3. – S. 349 – 363. 9. Bartolo P., Benci V., Fortunato D. Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity // Nonlinear Anal. – 1983. – 7, # 9. – P. 981 – 1012. 10. Iannacci R., Nkashama M. N., Ward J. R. Nonlinear second order elliptic partial differential equations at resonance // Proc. Amer. Math. Soc. – 1989. – 311, # 2. – P. 711 – 725. 11. Pavlenko V. N., ÇyΩ E. A. Zadaça Dyryxle dlq uravnenyq Laplasa s razr¥vnoj nelynej- nost\g bez uslovyq Landesmana – Lazera // Vestn. Çelqbyn. un-ta. Ser. 3. Matematyka. Me- xanyka. Ynformatyka. – 2002. – # 1(6). – S. 120 – 126. 12. Lad¥Ωenskaq O. A., Ural\ceva N. N, Lynejn¥e y kvazylynejn¥e uravnenyq πllyptyçesko- ho typa. – M.: Nauka, 1964. – 540 s. 13. Pavlenko V. N. Upravlenye synhulqrn¥my raspredelenn¥my systemamy parabolyçeskoho typa s razr¥vn¥my nelynejnostqmy // Ukr. mat. Ωurn. – 1994. – 46, # 6. – S. 729 – 736. 14. Ma T. W. Topological degree for set valued compact vector fields in locally convex spaces // Rozprawy Mat. – 1972. – 92. – P. 3 – 47. Poluçeno 17.08.2003 ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 1
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spelling umjimathkievua-article-35772020-03-18T19:59:02Z Existence Theorem for One Class of Strongly Resonance Boundary-Value Problems of Elliptic Type with Discontinuous Nonlinearities Теорема существования для одного класса сильно резонансных краевых задач эллиптического типа с разрывными нелинейностями Pavlenko, V. N. Chizh, E. A. Павленко, В. Н. Чиж, Е. А. Павленко, В. Н. Чиж, Е. А. We consider the Dirichlet problem for an equation of the elliptic type with a nonlinearity discontinuous with respect to the phase variable in the resonance case; it is not required that the nonlinearity satisfy the Landesman-Lazer condition. Using the regularization of the original equation, we establish the existence of a generalized solution of the problem indicated. Розглянуто задачу Діріхле для рівняння еліптичного типу з розривною за фазовою змінною нелінійністю в резонансному випадку, причому нелінійність може не задовольняти умову Ландесмана - Лазера. За допомогою регуляризації початкового рівняння встановлено існування узагальненого рівняння вказаної задачі. Institute of Mathematics, NAS of Ukraine 2005-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3577 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 1 (2005); 102–110 Український математичний журнал; Том 57 № 1 (2005); 102–110 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3577/3887 https://umj.imath.kiev.ua/index.php/umj/article/view/3577/3888 Copyright (c) 2005 Pavlenko V. N.; Chizh E. A.
spellingShingle Pavlenko, V. N.
Chizh, E. A.
Павленко, В. Н.
Чиж, Е. А.
Павленко, В. Н.
Чиж, Е. А.
Existence Theorem for One Class of Strongly Resonance Boundary-Value Problems of Elliptic Type with Discontinuous Nonlinearities
title Existence Theorem for One Class of Strongly Resonance Boundary-Value Problems of Elliptic Type with Discontinuous Nonlinearities
title_alt Теорема существования для одного класса сильно резонансных краевых задач эллиптического типа с разрывными нелинейностями
title_full Existence Theorem for One Class of Strongly Resonance Boundary-Value Problems of Elliptic Type with Discontinuous Nonlinearities
title_fullStr Existence Theorem for One Class of Strongly Resonance Boundary-Value Problems of Elliptic Type with Discontinuous Nonlinearities
title_full_unstemmed Existence Theorem for One Class of Strongly Resonance Boundary-Value Problems of Elliptic Type with Discontinuous Nonlinearities
title_short Existence Theorem for One Class of Strongly Resonance Boundary-Value Problems of Elliptic Type with Discontinuous Nonlinearities
title_sort existence theorem for one class of strongly resonance boundary-value problems of elliptic type with discontinuous nonlinearities
url https://umj.imath.kiev.ua/index.php/umj/article/view/3577
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