Multiple solutions for ${\bm p}({\bm x})$-Laplacian problems in ${\bf R}^{\bm N}$
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摘要: 在扰动项\,$f_1(x,u),\, f_2(x,u)$~中, 其中一项是超线性并且满足\,Ambrosetti-Rabinowitz\,条件, 另一项为次线性的情形下, 分别利用``喷泉定理''和``对偶喷泉定理'' 研究了无界区域\,$\mathbf{R}^{N}$\,上的\,$p(x)$-Laplace\,方程解的存在性和多解性问题. 此问题是基于变指数\,Lebesgue\,和\,Sobolev\,空间进行讨论的.
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关键词:
- 变指数\,Sobolev\,空间 /
- $p(x)$-Laplacian /
- (PS)$_c^\ast$条件 /
- 喷泉定理 /
- 对偶喷泉定理
Abstract: By using the fountain theorem and the dual fountain theorem, respectively, the existence and multiplicity of solutions for $p(x$)-Laplacian equations in $\mathbf{R}^{N}$ were studied, assumed that one of the perturbation terms $f_1(x,u),\, f_2(x,u)$ is superlinear and satisfies the Ambrosetti-Rabinowitz type condition and the other one is sublinear. The discussion was based on variable exponent Lebesgue and Sobolev spaces. -
[1] {1} ACERBI E, MINGIONE G. Regularity results for a class offunctionals with nonstandard growth [J]. Arch Rational Mech Anal,2001, 156: 121-140.{2} R\r{U}\v{Z}I\v{C}KA M. Electrorheological Fluids: Modeling andMathematical Theory [M]. Berlin: Springer-Verlag, 2000.{3} WINSLOW W M. Induced fibration of suspensions [J]. JAppl Phys, 1949, 20(12\underline{}): 1137-1140.
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