Compact embeddings for radial Sobolev spaces of two variable exponents and nonlinear elliptic problems with variable critical exponent

In this article, we establish the compact embedding for radial Sobolev spaces $W_{{\rm rad}}^{1,\Phi}(\mathbb R^N)$ of two variable exponents, where $\Phi(x,t)=t^{p(x)}(\log(e+t))^{q(x)}$. Here, $p(\cdot)$ and $q(\cdot)$ are variable exponents satisfying the log-Hölder and log log-Hölder conditions, respectively. The result is new even for the case $q(\cdot)=0$. As an application, we study the existence of a weak solution to $$ -{\rm div} (|\nabla u|^{p-2} \nabla u) + u^{p-1} = u^{r(x)-1} \,\,\text{ in }\,\, \mathbb R^N, $$ $ u \in W_{\rm rad}^{1, p} (\mathbb R^N),$ $ u \ge 0 $ in $ \mathbb R^N $ when $p=\text{ess}\,\inf_{x \in \mathbb R^N} r(x)$.