<em>W</em><sub>0</sub><sup>1,<em>p(x) </em></sup>versus <em>C</em><sup>1</sup> Local Minimizers for a Functional with Critical Growth
Year: 2014
Author: K. Saoudi
Journal of Partial Differential Equations, Vol. 27 (2014), Iss. 2 : pp. 115–124
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.
You do not have full access to this article.
Already a Subscriber? Sign in as an individual or via your institution
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jpde.v27.n2.2
Journal of Partial Differential Equations, Vol. 27 (2014), Iss. 2 : pp. 115–124
Published online: 2014-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 10
Keywords: p(x)-Laplacian equation