W01,p(x) versus C1 Local Minimizers for a Functional with Critical Growth (Journal of Partial Differential Equations, 2014)
Let $Ω⊂\mathbb{R}^N$, $(N ≥ 2)$ be a bounded smooth domain, p is Hölder continuous on $\overline{Ω}$, $$1 ‹ p^–:=inf_Ωp(x)≤p^+=sup_Ωp(x)›∞,$$ and $f : \overline{Ω}×\mathbb{R}→\mathbb{R}$ be a C¹ function with $f (x,s) ≥ 0, ∀(x,s)∈Ω×\mathbb{R}^+$ and $sup_x∈Ω f (x,s)≤ C(1+s)^{q(x)}$, $∀s∈\mathbb{R}^+, ∀x∈Ω$ for some $0‹q(x)∈C(\overline{Ω})$ satisfying $1›p(x)›q(x)≥p^∗(x)-1, ∀x∈\overline{Ω}$ and $1‹p^-≤p^+‹q^-≤q^+$. As usual, $p^∗(x)= \frac{Np(x)}{N-p(x)}$ if $p(x)‹N and p^∗(x)=∞$ if $p(x)≥N$. Consider the functional $I :W^{1,p(x)}_0 (Ω)→\mathbb{R}$ defined as $$I(u)^{def} = \int_Ω\frac{1}{p(x)}|∇u|^{p(x)}dx-\int_ΩF(x,u^+)dx, ∀u∈W^{1,p(x)}_0 (Ω),$$ where $F(x,u)=\int^s_0 f (x,s)ds$. Theorem1.1 proves that if $u_0∈C¹(\overline{Ω})$ is a local minimum of I in the $C¹(\overline{Ω})∩C_0(\overline{Ω})$ 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.
Keywords: p(x)-Laplacian equation; variational methods; local minimizer