(untitled) (Journal of Partial Differential Equations, 2014)
Let Ω⊂R^N, (N ≥ 2) be a bounded smooth domain, p is Hölder continuous on Ω, 1 ‹ p^–:=inf_Ωp(x)≤p^+=sup_Ωp(x)›∞, and f : Ω×R→R be a C¹ function with f (x,s) ≥ 0, ∀(x,s)∈Ω×R^+ and sup_x∈Ω f (x,s)≤ C(1+s)^q(x), ∀s∈R^+, ∀x∈Ω for some 0‹q(x)∈C(Ω) satisfying 1›p(x)›q(x)≥p^∗(x)-1, ∀x∈Ω and 1‹p^-≤p^+‹q^-≤q^+. As usual, p^∗(x)= Np(x)N-p(x) if p(x)‹N and p^∗(x)=∞ if p(x)≥N. Consider the functional I :W^1,p(x)_0 (Ω)→R defined as I(u)^def = _Ω1p(x)|∇u|^p(x)dx-_ΩF(x,u^+)dx, ∀u∈W^1,p(x)_0 (Ω), where F(x,u)=^s_0 f (x,s)ds. Theorem1.1 proves that if u_0∈C¹(Ω) is a local minimum of I in the C¹(Ω)∩C_0(Ω) topology, then it is also a local minimum in W^1,p(x)_0 (Ω) topology. This result is useful for proving multiple solutions to the associated Euler-lagrange equation (P) defined below.