| Abstract: |
| \usepackage{amsmath,amssymb}
We present new existence and multiplicity results for the double phase problem
\[
{\mathcal A}(u) := - \text{ div} \left(|\nabla u|^{p-2}\, \nabla u + |x|^{-b}\, |\nabla u|^{q-2}\, \nabla u\right) = f(|x|,u)
\]
with
\[
u \in {\mathcal D}_\text{rad}^{1,p}({\mathbb R}^N) \cap {\mathcal D}_\text{rad}^{1,q}({\mathbb R}^N;|x|^{-b}),
\]
where $1 < p < q < N$, $0 \le b < N - q$, and $f$ is a Carath\`{e}odory function on $[0,\infty) \times {\mathbb R}$ that satisfies a suitable growth condition. Our results are based on certain scaling properties of the operator ${\mathcal A}$ and recently developed variational methods for scaled functionals. |
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