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Two non-trivial solutions for a non-homogeneous Neumann problem: An Orlicz-Sobolev space setting. (English) Zbl 1172.35026

Let Ω N (N3) be a smooth bounded domain, ν be the outer normal of Ω and λ>0 is a parameter. Suppose that the function a:(0,) is such that

φ(t)=a(|t|)tift0andφ(t)=0ift=0

is an odd, strictly increasing homeomorphism from to . Then the authors of the paper proved that the Neumann problem

-div(a(|u(x)|)u(x))+a(|u(x)|)u(x)=λf(x,u(x)),forxΩ,u(x)/ν=0,forxΩ,

has at least two non-trivial weak solutions for any λ in some open interval Λ and the norms of the solutions are bouned above by a constant, if the following further conditions are satisfied:

1<lim inf t tφ(t) Φ(t)sup t>0 tφ(t) Φ(t)<andN<p 0 <lim inf t log(Φ(t)) log(t), whereΦ(t)= 0 t φ(s)dsandp 0 =inf t>0 tφ(t) Φ(t)·

c 0 >0ands(0,p 0 -1)suchthat|f(x,t)|c 0 (1+|t| s )forevery(x,t)Ω×·

bsuchthatB F = Ω F(x,b)dx>0withF(x,t)= 0 t f(x,s)dsfort·

δ>0suchthatf(x,t)t0foreveryxΩandt[-δ,δ]·

MSC:
35J65Nonlinear boundary value problems for linear elliptic equations
35J20Second order elliptic equations, variational methods
35J70Degenerate elliptic equations
35D05Existence of generalized solutions of PDE (MSC2000)
35B45A priori estimates for solutions of PDE