论文标题
具有可变指数的双期变异问题的多个解决方案
Multiple solutions of double phase variational problems with variable exponent
论文作者
论文摘要
本文介绍了quasilinear方程的多个解决方案$ - \ mathrm {div} \，\ mathbf {a}（x，x，\ nabla u）| u | ^{α（x）-2} u = f（x，u）$ in $ \ mathbb {r} ^{n} $，它涉及divergence形式的一般变量指数椭圆算子$ \ mathbf {a} $。该问题对应于双期各向异性现象，从某种意义上说，差异操作员具有$ | | ξ| ^{q（x）-2}ξ$用于小$ | ξ| $和喜欢$ | ξ| ^{p（x）-2}ξ$对于大$ | ξ| $，其中$ 1 <α（\ cdot）\ leq p（\ cdot）<q（\ cdot）<n $。我们的目的是通过使用具有可变指数的广义orlicz-sobolev空间中的临界点理论的工具来变异问题。我们的结果将先前的作品Azzollini，D'Avenia和Pomponio（2014）和Chorfi和Rădulescu（2016）扩展到$ p $ p $和$ q $的情况下，当$ p $ p $和$ q $是持续的情况下，当$ p（\ cdot）$和$％q（\ cdot）$时，情况是功能。我们还实质上削弱了他们的某些假设，通过使用加权方法克服了缺乏紧凑性。
This paper deals with the existence of multiple solutions for the quasilinear equation $-\mathrm{div}\,\mathbf{A}(x,\nabla u)| u| ^{α(x)-2}u=f(x,u)$ in $ \mathbb{R} ^{N}$, which involves a general variable exponent elliptic operator $\mathbf{ A}$ in divergence form. The problem corresponds to double phase anisotropic phenomena, in the sense that the differential operator has behaviors like $ | ξ| ^{q(x)-2}ξ$ for small $| ξ| $ and like $| ξ| ^{p(x)-2}ξ$ for large $ | ξ| $, where $1<α(\cdot )\leq p(\cdot )<q(\cdot )<N$. Our aim is to approach variationally the problem by using the tools of critical points theory in generalized Orlicz-Sobolev spaces with variable exponent. Our results extend the previous works Azzollini, d'Avenia, and Pomponio (2014) and Chorfi and Rădulescu (2016), from the case when exponents $p$ and $q$ are constant, to the case when $p(\cdot )$ and $% q(\cdot )$ are functions. We also substantially weaken some of their hypotheses overcome the lack of compactness by using the weighting method.
