Symmetry via antisymmetric maximum principles in nonlocal problems of variable order
Symmetry via antisymmetric maximum principles in nonlocal problems of variable order
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DOI：
10.1007/s10231-014-0462-y
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发表时间：
2014-06
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影响因子：
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通讯作者：
Sven Jarohs;T. Weth
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文献类型：
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作者：
Sven Jarohs;T. Weth
我们考虑非线性问题(P)qquad左{u=f(x，u)&quad in Omega，u=0&quad on R^N∖Ω，(P)Iu=f(x，u)inΩ，u=0 in RNΩinΩ⊂R^NΩ⊂RN，其中i是一个非局部算子，它可以是各向异性的，且阶数可以是变化的.我们假设i，Ωi，Ω上的弱对称性和单调性，以及f f关于固定方向的非线性，例如x_1x_1，我们证明了(P)(P)的任何非负弱解uu在x_1x_1中是对称的。此外，我们还有如下选择：u≡0 u≡0 inΩΩ，或者u u在|x_1||x1|中严格递减。证明依赖于一类相关线性问题反对称超解的新的最大值原理。
We consider the nonlinear problem (P)\qquad\left {I u= f (x, u) &\quad in\Omega,\u= 0 &\quad on\mathbb R^ N ∖ Ω\.(P) I u= f (x, u) in Ω, u= 0 on RN\Ω in an open bounded set Ω ⊂ R^ N Ω⊂ RN, where I I is a nonlocal operator, which may be anisotropic and may have varying order. We assume mild symmetry and monotonicity assumptions on I, Ω I, Ω and the nonlinearity f f with respect to a fixed direction, say x_1 x 1, and we show that any nonnegative weak solution u u of (P)(P) is symmetric in x_1 x 1. Moreover, we have the following alternative: Either u ≡ 0 u≡ 0 in Ω Ω, or u u is strictly decreasing in| x_1|| x 1|. The proof relies on new maximum principles for antisymmetric supersolutions of an associated class of linear problems.