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In this article, we prove a compact embedding theorem for the weighted Orlicz-Sobolev space of radially symmetric functions. Using the embedding theorem and critical points theory, we prove the existence of multiple radial solutions and radial ground states for the following modified capillary surface equation $$\displaylines{ -\operatorname{div}\Big(\frac{|\nabla u|^{2p-2}\nabla u} {\sqrt{1+|\nabla u|^{2p}}}\Big) +T(|x|)|u|^{\alpha-2}u=K(|x|)|u|^{s-2}u,\quad u>0,\; x\in\mathbb{R}^{N},\cr u(|x|)\to 0,\quad\text{as } |x|\to \infty, }$$ where $N\geq3$, $1<\alpha<p<2p<N$, $s$ satisfies some suitable conditions, $K(|x|)$ and $T(|x|)$ are continuous, nonnegative functions.