Compact Imbedding Theorems with Symmetry of Strauss–Lions Type for the Space W1,()(Ω)

Nonlinear diffusion equations driven by the p(·)-Laplacian

“…However, both definitions are equivalent under (2.1) given below (see [20] and also [39] for an unusual phenomenon of discontinuous exponents). In this paper, we use the notation of Let us exhibit the Poincaré and Sobolev inequalities (see [24,26,35] and references therein for more details). To do so, we introduce the log-Hölder condition:…”

Compact Sobolev embedding theorems involving symmetry and its application

“…4, we need some theories on the space W 1,p(x) (Ω), which we call variable exponent Sobolev spaces. Firstly, we state some basic properties which will be used later (for details, see [14,16]). …”

“…Sobolev embedding theorems for the variable exponent Sobolev space W 1,p(x) (Ω) have been studied by many authors (see e.g. [10][11][12][13][14][15][16]). In fact, for 1 < p(x) < N, if Ω is bounded and p(x) ∈ C(Ω), there is a compact embedding from W 1,p(x) (Ω) to…”

“…is radially symmetric and p (x) q(x) p * (x), ∀x ∈ R N (see [16]). In this paper, we will extend Wang's embedding results (see [4]) from…”

Functions of Bounded (&lt;i&gt;p&lt;/i&gt;(&amp;sdot;), 2)-Variation in De la Vall&#233;e Poussin-Wiener’s Sense with Variable Exponent

“…These and related problems are the subject of active research to this day. These problems are interesting in applications (see [10] [11] [12] [13]) and gave rise to a revival of the interest in Lebesgue and Sobolev spaces with variable exponent, the origins of which can be traced back to the work of Orlicz [14] in the 1930's. In the 1950's, this study was carried on by Nakano [15] [16] who made the first systematic study of spaces with variable exponent.…”