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Multiple solutions of double phase variational problems with variable exponent
Advances in Calculus of Variations ( IF 1.7 ) Pub Date : 2020-10-01 , DOI: 10.1515/acv-2018-0003 Xiayang Shi 1 , Vicenţiu D. Rădulescu 2 , Dušan D. Repovš 3 , Qihu Zhang 4
Advances in Calculus of Variations ( IF 1.7 ) Pub Date : 2020-10-01 , DOI: 10.1515/acv-2018-0003 Xiayang Shi 1 , Vicenţiu D. Rădulescu 2 , Dušan D. Repovš 3 , Qihu Zhang 4
Affiliation
Abstract This paper deals with the existence of multiple solutions for the quasilinear equation - div 𝐀 ( x , ∇ u ) + | u | α ( x ) - 2 u = f ( x , u ) in ℝ N , {-\operatorname{div}\mathbf{A}(x,\nabla u)+|u|^{\alpha(x)-2}u=f(x,u)\quad\text% {in ${\mathbb{R}^{N}}$,}} which involves a general variable exponent elliptic operator 𝐀 {\mathbf{A}} in divergence form. The problem corresponds to double phase anisotropic phenomena, in the sense that the differential operator has various types of behavior like | ξ | q ( x ) - 2 ξ {|\xi|^{q(x)-2}\xi} for small | ξ | {|\xi|} and like | ξ | p ( x ) - 2 ξ {|\xi|^{p(x)-2}\xi} for large | ξ | {|\xi|} , where 1 < α ( ⋅ ) ≤ p ( ⋅ ) < q ( ⋅ ) < N {1<\alpha(\,\cdot\,)\leq p(\,\cdot\,)
中文翻译：
变指数双相变分问题的多重解
摘要 本文讨论拟线性方程的多重解的存在性——div 𝐀 ( x , ∇ u ) + | 你| α ( x ) - 2 u = f ( x , u ) in ℝ N , {-\operatorname{div}\mathbf{A}(x,\nabla u)+|u|^{\alpha(x )-2}u=f(x,u)\quad\text% {in ${\mathbb{R}^{N}}$,}} 涉及一般可变指数椭圆算子 𝐀 {\mathbf{A} } 发散形式。该问题对应于双相各向异性现象，因为微分算子具有各种类型的行为，例如 | ξ | q ( x ) - 2 ξ {|\xi|^{q(x)-2}\xi} 小 | ξ | {|\xi|} 和喜欢 | ξ | p ( x ) - 2 ξ {|\xi|^{p(x)-2}\xi} 对于大 | ξ | {|\xi|} , 其中 1 < α ( ⋅ ) ≤ p ( ⋅ ) < q ( ⋅ ) < N {1<\alpha(\,\cdot\,)\leq p(\,\cdot \,)
更新日期：2020-10-01
中文翻译：
变指数双相变分问题的多重解
摘要 本文讨论拟线性方程的多重解的存在性——div 𝐀 ( x , ∇ u ) + | 你| α ( x ) - 2 u = f ( x , u ) in ℝ N , {-\operatorname{div}\mathbf{A}(x,\nabla u)+|u|^{\alpha(x )-2}u=f(x,u)\quad\text% {in ${\mathbb{R}^{N}}$,}} 涉及一般可变指数椭圆算子 𝐀 {\mathbf{A} } 发散形式。该问题对应于双相各向异性现象，因为微分算子具有各种类型的行为，例如 | ξ | q ( x ) - 2 ξ {|\xi|^{q(x)-2}\xi} 小 | ξ | {|\xi|} 和喜欢 | ξ | p ( x ) - 2 ξ {|\xi|^{p(x)-2}\xi} 对于大 | ξ | {|\xi|} , 其中 1 < α ( ⋅ ) ≤ p ( ⋅ ) < q ( ⋅ ) < N {1<\alpha(\,\cdot\,)\leq p(\,\cdot \,)
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