Three Solutions to a Neumann Problem for Elliptic Equations with Variable Exponent

摘要

In this article, we study the following nonlinear Neumann boundary value problem {~-div(|▽u|~(p(x)-2)▽u)+a(x)|u|~(p(x)-2)u=λf(x,u) in x ∈ Ω/ partial deriv u/partial deriv v=0 on x ∈partial derivΩ where Ω, is contained in R~N (N ≥ 3), Ω is a bounded smooth domain and, p ∈ C (Ω) with inf_(x∈Ω) p(x) > N, f : R →R is a continuous function, and v is the unit normal exterior vector on partial deriv Ω and a ∈ L~∞(Ω), with ess infa(x) = a_0 > 0 and λ > 0 is a real number. We first deal with the case that f(t) = b|t|~(q-2) t - d|t|~(s-2) t, t∈ R, where b and d are positive constants. Then we deal with the case that f(x, t) = |t|~(q(x)-2)t, x ∈ Ω, t ∈ R, where q,s ∈ C (Ω). Using the direct Ricceri variational principle, we establish the existence of at least three weak solutions of this problem in weighted-variable-exponent Sobolev space W_a~(1,p(x))(Ω).