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Multi-bump solutions for quasilinear elliptic equations with variable exponents and critical growth in ℝN
Communications in Contemporary Mathematics ( IF 1.6 ) Pub Date : 2020-02-26 , DOI: 10.1142/s0219199720500133 Chao Ji 1 , Vicenţiu D. Rădulescu 2, 3
Communications in Contemporary Mathematics ( IF 1.6 ) Pub Date : 2020-02-26 , DOI: 10.1142/s0219199720500133 Chao Ji 1 , Vicenţiu D. Rădulescu 2, 3
Affiliation
In this paper, we are concerned with the existence of multi-bump solutions for the following class of p ( x ) -Laplacian equations:
− div ( | ∇ u | p ( x ) − 2 ∇ u ) + ( λ V ( x ) + Z ( x ) ) | u | p ( x ) − 2 u = α f ( x , u ) + u q ( x ) − 1 , in ℝ N , u ∈ W 1 , p ( x ) ( ℝ N ) , u > 0 ,
where α > 0 and λ ≥ 1 are two real parameters, the nonlinearity f : ℝ N × ℝ → ℝ is a continuous function with subcritical growth, N > p + = sup x ∈ ℝ N p ( x ) , the exponent q ( x ) can be equal to the critical exponent p ∗ ( x ) = N p ( x ) N − p ( x ) at some points of ℝ N including at infinity and the potentials V , Z : ℝ N → ℝ are continuous functions verifying some conditions. We show that if the zero set of V has several isolated connected components Ω 1 , … , Ω k such that the interior of Ω i is not empty and ∂ Ω i is smooth, then for λ > 0 large enough there exists, for any non-empty subset Γ ⊂ { 1 , … , k } , a bump solution trapped in a neighborhood of ⋃ j ∈ Γ Ω j . The proofs are based on variational and topological methods.
中文翻译:
ℝN 中具有可变指数和临界增长的拟线性椭圆方程的多凹凸解
在本文中,我们关注以下类别的多凸点解决方案的存在p ( X ) -拉普拉斯方程:
- div ( | ∇ 你 | p ( X ) - 2 ∇ 你 ) + ( λ 五 ( X ) + Z ( X ) ) | 你 | p ( X ) - 2 你 = α F ( X , 你 ) + 你 q ( X ) - 1 , 在 ℝ ñ , 你 ∈ W 1 , p ( X ) ( ℝ ñ ) , 你 > 0 ,
在哪里α > 0 和λ ≥ 1 是两个实参数,非线性F : ℝ ñ × ℝ → ℝ 是具有亚临界增长的连续函数,ñ > p + = 支持 X ∈ ℝ ñ p ( X ) , 指数q ( X ) 可以等于临界指数p * ( X ) = ñ p ( X ) ñ - p ( X ) 在某些时候ℝ ñ 包括无穷大和势能五 ,Z : ℝ ñ → ℝ 是验证某些条件的连续函数。我们证明,如果零集五 有几个孤立的连接组件Ω 1 , … , Ω ķ 这样的内部Ω 一世 不是空的并且∂ Ω 一世 是光滑的,那么对于λ > 0 存在足够大,对于任何非空子集Γ ⊂ { 1 , … , ķ } ,困在附近的凹凸解决方案⋃ j ∈ Γ Ω j . 证明基于变分和拓扑方法。
更新日期:2020-02-26
中文翻译:
ℝN 中具有可变指数和临界增长的拟线性椭圆方程的多凹凸解
在本文中,我们关注以下类别的多凸点解决方案的存在