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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
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)2u) + (λV (x) + Z(x))|u|p(x)2u = αf(x,u) + uq(x)1,in N,u W1,p(x)(N),u > 0, where α > 0 and λ 1 are two real parameters, the nonlinearity f : N × is a continuous function with subcritical growth, N > p+ =supxNp(x), the exponent q(x) can be equal to the critical exponent p(x) = Np(x) Np(x) at some points of N including at infinity and the potentials V, Z : N are continuous functions verifying some conditions. We show that if the zero set of V has several isolated connected components Ω1,, Ωk such that the interior of Ωi is not empty and Ωi 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 jΓΩj. 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(||p(X)-2) + (λ (X) + Z(X))||p(X)-2 = αF(X,) + q(X)-1, ñ, W1,p(X)(ñ), > 0, 在哪里α > 0λ 1是两个实参数,非线性F ñ × 是具有亚临界增长的连续函数,ñ > p+ =支持Xñp(X), 指数q(X)可以等于临界指数p*(X) = ñp(X) ñ-p(X)在某些时候ñ包括无穷大和势能,Z ñ 是验证某些条件的连续函数。我们证明,如果零集有几个孤立的连接组件Ω1,, Ωķ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jΓΩj. 证明基于变分和拓扑方法。
更新日期:2020-02-26
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