| 陈自高.$\mathbb{R}^n$ 上 $p(x)$-Laplace型椭圆问题的无穷多解[J].数学年刊A辑,2014,35(1):45~60 |
| $\mathbb{R}^n$ 上 $p(x)$-Laplace型椭圆问题的无穷多解 |
| Infinitely Many Solutions to $p(x)$-Laplace Type Elliptic Problems in $\mathbb{R}^n$ |
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| DOI： |
| 中文关键词: 变指数 Sobolev 空间, 散度型算子, $p(x)$-Laplace算子, 多重解 |
| 英文关键词:Variable exponent Sobolev space, Divergence type
operator, $p(x)$-Laplacian, Multiple
solutions |
| 基金项目:国家自然科学基金 (No.11101145) |
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| 中文摘要: |
| 讨论了涉及一般散度型椭圆算子($p(x)$-Laplace算子为其特例)
非线性偏微分方程的弱解存在性和多解性问题, 假定非线性项 $f_1,
f_2$ 其中之一是超线性的, 且满足 Ambrosetti-Rabinowitz 条件,
另一项是次线性的. 所采用的方法依赖于变指数 Sobolev 空间
$W^{1,p(x)}(\mathbb{R}^n)$ 理论.
主要结果的证明基于喷泉定理和对偶喷泉定理. |
| 英文摘要: |
| In this paper, the existence and multiplicity of
weak solutions to nonlinear partial differential equations involving a
general elliptic operator in divergence form (in particular, a
$p(x)$-Laplace operator) in $\mathbb{R}^n$ are investigated,
assumed that one of the nonlinear terms $f_1$ and $ f_2$ is superlinear
and satisfies the Ambrosetti-Rabinowitz type condition and another
one is sublinear. Our approach relies on the theory of variable
exponent Sobolev space $W^{1,p(x)}(\mathbb{R}^n)$. The proofs of our
main results are based on the Fountain theorem and the Dual Fountain
theorem. |
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