Positive solutions for a class of equations with a p-Laplace like operator and weights Original Research Article

In this paper we will study the problem of existence of positive solutions to the problem View the MathML source(D){(a(r)ϕ(u′))′+b(r)g(u)=0, a.e. in (0,R),limr→0a(r)ϕ(u′(r))=0,u(R)=0, Turn MathJax on where ϕϕ is an odd increasing homeomorphism of RR and g∈C(R)g∈C(R) is such that g(z)>0g(z)>0 for all z>0z>0 with g(0)=0g(0)=0. The functions aa and bb, that we will refer to as weight functions, satisfy a(r)>0a(r)>0, b(r)>0b(r)>0 for all r∈(0,R]r∈(0,R] and are such that View the MathML sourcea,b∈C1(0,R]∩L1(0,R). If ϕϕ has the form ϕ(z)=zm(|z|)ϕ(z)=zm(|z|), and View the MathML sourcea(r)=rN−1ã(r),b(r)=rN−1b̃(r),N≥2, Turn MathJax on then solutions of problem (D)(D) provide solutions with radial symmetry for the problem View the MathML source(P){div(ã(|x|)m(|∇u|)∇u)+b̃(|x|)g(u)=0,x∈Ω,u=0,x∈∂Ω, Turn MathJax on where Ω=B(0,R)Ω=B(0,R) denotes the ball with center 0 and radius R>0R>0 in RNRN.