Volume 27, Issue 2
W01,p(x) versus C1 Local Minimizers for a Functional with Critical Growth

K. Saoudi

10.4208/jpde.v27.n2.2

J. Part. Diff. Eq., 27 (2014), pp. 115-124.

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• Abstract

Let $Ω⊂\mathbb{R}^N$, $(N ≥ 2)$ be a bounded smooth domain, p is Hölder continuous on $\overline{\Omega}$, $$1 ‹ p^–:=inf_Ωp(x)≤p^+=sup_Ωp(x)›∞,$$ and $f : \overline{\Omega}×\mathbb{R}→\mathbb{R}$ 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 $0‹q(x)∈C(\overline{\Omega})$ satisfying $1›p(x)›q(x)≥p^∗(x)-1, ∀x∈\overline{\Omega}$ 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 $u_0∈C¹(\overline{\Omega})$ is a local minimum of I in the $C¹(\overline{\Omega})∩C_0(\overline{\Omega})$ 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