In this paper we are concerned with questions of multiplicity and concentration behavior of positive solutions of the elliptic problemwhere ε is a small positive parameter, f : R → R is a continuous function, Lε is a nonlocal operator defined byM : IR+ → IR+ and V : IR 3 → IR are continuous functions which verify some hypotheses.
ResumoEsta dissertação tem como objetivo principal entender mais profundamente as mais variadas nuances do aparecimento de caos em sistemas de teoria de campos, além de um estudo sobre o comportamento das soluções solitônicas desses sistemas. Apresentamos um método, baseado na estrutura de mínimos de potenciais, a fim de obter informações concretas sobre o comportamento das soluções solitônicas destes potenciais. Verificamos ainda a existência de novas soluções topológicas para um modelo queé aplicado na descrição dos chamados twistons. Essas soluções possuem a particularidade de serem degeneradas, assim, para quebrar essa degenerescência, adicionamos ao potencial inicial um termo perturbativo. Determinamos também, novas soluções topológicas para uma lagrangiana contendo um campo escalar complexo, estudada por Trullinger e Subbaswamy em (Trullinger, S. E.; Subbaswamy, K. R.
AbstractThe main objective of this work is a deeper comprehension about the different kinds of the appearing of chaos in field theory´s systems, besides the study about the behavior of these soliton solutions systems. We present a method, based on the structure of potential's minima, to get information about the behavior of the solitons solutions. We calculate new topological solutions for a model that was applied to the description of the so called twistons. These solutions are degenerated and in order to break this degeneracy, we added a perturbative term into the initial potential. We found new topological solutions for a specific Lagrangian that has a complex scalar field, studied by Trullinger and Subbaswamy in (Trullinger, S. E.; Subbaswamy, K. R. Physical Review A 19 (1979) 1340.), and applied the method of Poincaré section in this model checking its chaotic regions.
We study the existence, nonexistence and multiplicity of positive solutions for a general class of degenerate nonlocal problems. The nonexistence results are provided in two different situations: under monotonicity assumptions involving the nonlinearity and without monotonicity conditions. A multiplicity result of positive solutions with ordered L p -norms complement some recent results obtained in the Literature. Examples are provided in order to illustrate the applicability of our theorems.
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