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On heteroclinic solutions for BVPs involving ϕ ‐Laplacian operators without asymptotic or growth assumptions
Author(s) -
Minhós Feliz
Publication year - 2019
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201700470
Subject(s) - mathematics , homeomorphism (graph theory) , laplace operator , function (biology) , order (exchange) , mathematical analysis , pure mathematics , discrete mathematics , finance , evolutionary biology , economics , biology
In this paper we consider the second order discontinuous equation in the real line,( ϕ ( a ( t ) u ′ ( t ) ) ) ′=f ( t , u ( t ) , u ′ ( t ) ) , a.e. t ∈ R ,u ( − ∞ )=A ,u ( + ∞ ) = B ,with ϕ an increasing homeomorphism such that ϕ ( 0 ) = 0 and ϕ ( R ) = R , a ∈ C ( R ) with a ( t ) > 0 , for t ∈ R , f : R 3 → R a L 1 ‐Carathéodory function and A , B ∈ R verifying an adequate relation. We remark that the existence of heteroclinic solutions is obtained without asymptotic or growth assumptions on the nonlinearities ϕ and f . Moreover, as far as we know, our main result is even new when ϕ ( y ) = y , that is, for the equation( a ( t ) u ′ ( t ) ) ′ = f ( t , u ( t ) , u ′ ( t ) ) , a.e. t ∈ R .