Dedicated to the memory of LOTHAR COLLATZ Die von STUMMEL und GRIGORIEFF begriindete und spater von zahlreichen Autoren auf nichtlineare Probleme erweiterte allgemeine Theorie der Konvergenz von Diskretisierungsalgorithmen geniigt nicht vollstandig der numerischen Behandlung hearer oder nichtlinearer, schwach formulierter Probleme. Eine derartige Erweiterung wird hier zusammen mit Beispielen fur nichtlineare elliptische und hyperbolische partielle Differentialgleichungen angegeben. Oft fiihrt der Ubergang von klassischen zu schwachen Losungen zu einem Verlust der Eindeutigkeit, so daJ Eindeutigkeit nur durch zusatzliche Bedingungen wie etwa geeignete Ungleichungsrestriktionen garantiert werden kann. Wie solche Restriktionen im allgemeinen diskretisiert werden miissen, um eine Konvergenz der numerischen Losung gegen die entsprechende eindeutige schwache Losung zu erreichen, wird ebenfalls diskutiert, und die Ergebnisse werden insbesondere mit den Entropie-Bedingungen von Erhaltungssatzen verglichen. Die Beispiele werden nur behandelt, um die allgemeine Theorie zu motivieren und deren Anwendbarkeit zu demonstrieren. The general theory of the convergence of discretization algorithms founded by STUMMEL and GRIGORIEFF and later extended to nonlinear problems by numerous authors does not completely fit the numerical treatment of linear or nonlinear weakly formulated problems. Such an extension is presented together with examples of nonlinear elliptic and hyperbolic PDEs. Often the transition from classsical to weak solutions leads to a loss of uniqueness such that uniqueness can only be guaranteed by additional conditions, e.g. suitable inequality constraints. How such constraints must generally be discretized in order to make the numerical solution converge to the corresponding unique weak solution is also discussed, and the results are particularly compared with entropy conditions of conservation laws. The examples are only treated in order to motivate the general theory and to demonstrate its applicability. OcHosaHHan CTYMMEJIEM u ~PMI-OPMEBOM u pacutupartnasl n o 3~e MnozorlucneHHHbiuu asmopaMu Ha cnyrlaii nenuHeiiHbix 3adar o6u4an meopusl cxoduuocmu aflropumMo8 dyc~pemu3ayuu necoseputenHo coomsemcmsyem rlucnenftoii o6pa6om~e nuHeiiHbrx UAU cna60 @opuynuposanHbix Henuneiimix 3adav. Jaemcsl mauoe pacutupewe sMecme c npuuepauu dnsl HefluHeiinbix 3 n f l~n m~r l e c~~x u zunep6onuuec~ux ypasnenuii 6 VacmHbix npou36odnbix. Liacmo nepexod om KnaccurlecKux Ha cfla6bie peutewisl sedem Ha nomepm eduttcmsemocmu maK, rlmo M O X H O 06ecnerlusamb eduncmsennocmb monbKo npu dononnumenbitbix ycnosuslx, Ha npuuep npu nodxodnupx ycilosuslx s aude HepaseHcms. M3yrlaemcsl u sonpoc, Kau 8006tye mame ycnosun Hado duc~pemu3uposam~ rlrno6bi 06ecnevusamb cxoduuocmb Y U C A~H~O Z O peuteHus K coomsemcmsymtqeuy eduncmsennouy cna6ouy peweHum u cpasnslmmcn pe3ynbmambr 8 o c o 6 e~~o c m u u ycilosusluu snmponuu 3a~onos coxpaHeHun. IIpuMepbr monbuo u3yrlammcn 6 uavecmse uomusayuu o6u4eli meopuu u no~a3amenbcmsa ezo npuueltuwocmu. MSC (1980): ...
We study the existence of ground state solutions of a Schrödinger–Poisson–Slater‐type equation with critical growth. By using the Nehari–Pohozaev manifold, we obtain the existence of ground state solutions of this system.
This paper is concerned with the theory of the convergence of discretizations f o r set-valued operator equations. It is a continuation of investigations in 121, 171, [5], some results for one-valued operators have been generalrzed herein. As an application, two examples for set-valued differential equations are discussed.Define set convergence by S, + S, if any &-neighborhood of S contains S, for all n sufficiently large. Theorem 2.1: In equations (l), (2) assume that y , -+ y, [{A,}, A] is d-closed, and {S,} is d-compact. Then {S,}* c S and S' ,z + S. If S, # 0 for n E N', then S # 0. P r o o f : Let z E {SrL}*, then there are N' and x, E S,, n E N', such that x, ---f 2 . Since [{A,,}, A] is d-closed and y, + y, it follows that x E D(A) and y E Ax, that is z E S and {Sn}* c S. The rest follows from 111.Now we consider the weak formulation of problem (1). Let Z be a Banach space, J be an index set, {y(@), @ E J } c Z be a given set, and { A ( @ ) , @ E J } be a set of operators that map D = n D(A(@)) c X into 22, here D(A(@)) is the domain of A ( @ ) . @ E J
This paper deals with the following Kirchhoff–Schrödinger–Newton system with critical growth { − M ( ∫ Ω | ∇ u | 2 d x ) Δ u = ϕ | u | 2 ∗ − 3 u + λ | u | p − 2 u , i n Ω , − Δ ϕ = | u | 2 ∗ − 1 , i n Ω , u = ϕ = 0 , o n ∂ Ω , where Ω ⊂ R N ( N ≥ 3 ) is a smooth bounded domain, M ( t ) = 1 + b t θ − 1 with t > 0 , 1 < θ < N + 2 N − 2 , b > 0 , 1 < p < 2 , λ > 0 is a parameter, 2 ∗ = 2 N N − 2 is the critical Sobolev exponent. By using the variational method and the Brézis–Lieb lemma, the existence and multiplicity of positive solutions are established.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.