Multiple solutions for elliptic systems via trapping regions and related nonsmooth potentials
Multiple solutions for elliptic systems via trapping regions and related nonsmooth potentials
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DOI：
10.1080/00036811.2014.940520
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发表时间：
2015-08
影响因子：
1.1
通讯作者：
S. Carl;D. Motreanu
中科院分区：
文献类型：
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作者：
S. Carl;D. Motreanu
本文研究有界域上拟线性椭圆组的Dirichlet边值问题，其中对角线-拉普拉斯算子为前导微分算子，Carathéodory向量场为右端向量场。只有对接近于零的…施加一定的增长条件，我们才能证明多个非平凡解的存在性，并为它们提供符号信息。更确切地说，我们首先证明了正极小解和负极大解的存在性，其中极大和极小的概念是指由序锥引入的向量值函数的偏序。其次，在所考虑的系统是变分结构的情况下，即充分光滑的情况下，可以证明存在更多的非平凡解，特别是变号解。一方面，我们的方法是基于一系列扩展的陷阱区域来获得极值常号解。另一方面，通过引入某些截断算子，我们构造了一个(非光滑)泛函，它的临界点是由极值解形成的陷域内给定系统的解。这种方法的一个特点是，无论位势多么光滑，都不可避免地要处理非光滑势。用来实现我们目标的进一步工具是微分不等式的比较结果，以及非光滑泛函的变分和拓扑工具，例如非光滑临界点理论和非光滑局部Lipschitz泛函的第二变形引理。值得一提的是，在研究系统时，我们本质上使用的是关于相关椭圆型方程的知识。
This paper deals with the Dirichlet boundary value problem for quasilinear elliptic systems in a bounded domain with a diagonal -Laplacian as leading differential operator and a Carathéodory right-hand side vector field . Only by imposing certain growth conditions on , , near zero we are able to prove the existence of multiple, nontrivial solutions, and provide sign information for them. More precisely, first we show the existence of a positive minimal and a negative maximal solution, where the notion maximal and minimal refer to the partial ordering of vector-valued functions introduced by the order cone . Second, in case the considered system is, in addition, of variational structure, i.e. with sufficiently smooth, further nontrivial solutions can be proved to exist, in particular, solutions that change sign. Our approach is based, on the one hand, on a sequences of expanding trapping regions to get extremal constant sign solutions. On the other hand, introducing certain truncation operators we construct a (nonsmooth) functional whose critical points turn out to be solutions of the given system within the trapping region formed by the extremal solutions. A characteristic feature of this approach is that one cannot avoid to deal with nonsmooth potentials no matter how smooth might be. Further tools used to achieve our goals are comparison results for differential inequalities, and variational and topological tools for nonsmooth functionals such as, e.g. nonsmooth critical point theory and second deformation lemma for nonsmooth, locally Lipschitz functionals. It is also worth pointing out that in the study of system we essentially use knowledge on associated elliptic equations.