We investigate the following elliptic equations: $$\left\{ {\matrix{ { - M\left( {\int_{{\mathbb{R}^N}} {\phi ({{\left| {\nabla u} \right|}^2}){\rm{d}}x} } \right){\rm{div(}}\phi \prime ({{\left| {\nabla u} \right|}^2})\nabla u{\rm{) + }}{{\left| u \right|}^{\alpha - 2}}u = \lambda h(x,u),} \hfill \cr {u(x) \to 0,\;\;\;\;\;{\rm{as}}\left| x \right| \to \infty ,} \hfill \cr } } \right.\;\;\;\;\;\;\;\;{\rm{in}}\;\;\;\;{\mathbb{R}^N},$$ where N ≥ 2, 1 < p < q < N, α < q, 1 < α < p*q†/p† with $${p^ * } = {\textstyle{{Np} \over {N - p}}},\;\;\phi (t)$$ behaves like tq/2 for small t and tp/2 for large t, and p′ and q′ are the conjugate exponents of p and q, respectively. We study the existence of nontrivial radially symmetric solutions for the problem above by applying the mountain pass theorem and the fountain theorem. Moreover, taking into account the dual fountain theorem, we show that the problem admits a sequence of small-energy, radially symmetric solutions.

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Orlicz-Sobolev 空间设置中包含非齐次算子的拟线性椭圆方程的径向对称解

我们研究以下椭圆方程： $$\left\{ {\matrix{ { - M\left( {\int_{{\mathbb{R}^N}} {\phi ({{\left| {\nabla u } \right|}^2}){\rm{d}}x} } \right){\rm{div(}}\phi \prime ({{\left| {\nabla u} \right|}^ 2})\nabla u{\rm{) + }}{{\left| u \right|}^{\alpha - 2}}u = \lambda h(x,u),} \hfill \cr {u(x) \to 0,\;\;\;\;\;{\ rm{as}}\left| x \右| \to \infty ,} \hfill \cr } } \right.\;\;\;\;\;\;\;\;{\rm{in}}\;\;\;\;{\mathbb{ R}^N},$$ 其中 N ≥ 2, 1 < p < q < N, α < q, 1 < α < p*q†/p† with $${p^ * } = {\textstyle{{ Np} \over {N - p}}},\;\;\phi (t)$$ 表现为小 t 的 tq/2 和大 t 的 tp/2，并且 p' 和 q' 是共轭指数分别为 p 和 q。我们通过应用山口定理和喷泉定理来研究上述问题的非平凡径向对称解的存在性。而且，