arXiv subjects
Publications and source records attributed to A. Nekvinda.
We prove the solvability of the Dirichlet problem for the variable exponent $p$-Laplacian with boundary data in $W^{1,p(x)}(Ω)$ on a bounded, smooth domain $Ω\subset {\mathbb R}^n$. Our main focus will be on an a.e. finite variable exponent $p(\cdot)$ with $n < \inf\limits_{x\in Ω}p(x)$ and $\sup\limits_{x\in Ω}p(x) = \infty$ under the sole assumption that $p\in C(Ω)$.
Let $Ω$ be an open subset of $\mathbb{R}^{N}$, and let $p,\, q:Ω\rightarrow \left[ 1,\infty \right] $ be measurable functions. We give a necessary and sufficient condition for the embedding of the variable exponent space $L^{p(\cdot )}\left( Ω\right) $ in $L^{q(\cdot )}\left( Ω\right) $ to be almost compact. This leads to a condition on $Ω, \, p$ and $q$ sufficient to ensure that the Sobolev space $W^{1,p(\cdot )}\left( Ω\right) $ based on $L^{p(\cdot )}\left( Ω\right) $ is compactly embedded in $L^{q(\cdot )}\left( Ω\right) ;$ compact embedding results of this type already in the literature are included as special cases.