| Abstract: |
| We present new existence and multiplicity results for the mixed fractional $p$-Laplacian equation
\[
{\mathcal A}(u) := (- \Delta)_{p_1}^{s_1}\, u + (- \Delta)_{p_2}^{s_2}\, u = f(|x|,u)
\]
with
\[
u \in {\mathcal D}_\text{rad}^{s_1,p_1}({\mathbb R}^N) \cap {\mathcal D}_\text{rad}^{s_2,p_2}({\mathbb R}^N),
\]
where $s_i \in (0,1)$ and $1 < p_i < N/s_i$ for $i = 1,2$ and $f$ is a Caratheodory 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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