论文标题
某些奇异随机椭圆形PDE的变分方法
Variational methods for some singular stochastic elliptic PDEs
论文作者
论文摘要
我们使用非线性分析中的一些工具来研究两个奇异随机椭圆PDE的示例，这些示例无法通过收缩原理或Schauder固定点定理来解决。让$ξ$站在封闭的Riemannian Surface $ S $上的空间白噪声。我们证明存在方程$$（-Δ + a）u = f（u） +ξu$$，具有潜在的$ a \ in l^p（s）$和$ p> 1 $，以及受增长条件的$ f $。在$ f $的额外奇偶校赛条件下 - 例如，当$ f（u）= u \ vert u \ vert^\ ell $（$ \ ell $ a in innteger）中时，我们进一步证明，这个方程式具有无限的解决方案，与迄今为止在近似奇异的现有pdes pdeption中所证明的有良好的结果形成了鲜明的对比。通过将方程式视为表征基于安德森操作员$ h =δ+δ$的能量函数的临界点，并求助于山间通过定理的变体来获得这种结果。但是，有一些有趣的方程式无法将其表征为能量功能的临界点。在$ \ mathbb {t}^2 $ $ $（-δ + a）u = big（w \ star f（u） + big） + big） +ξu$ $上的单一奇异choquard pecard方程就是这种情况。在系数上$ a，w，f $和$ g $。
We use some tools from nonlinear analysis to study two examples of singular stochastic elliptic PDEs that cannot be solved by the contraction principle or the Schauder fixed point theorem. Let $ξ$ stand for a spatial white noise on a closed Riemannian surface $S$. We prove the existence of a solution to the equation $$ (-Δ+ a)u = f(u) + ξu $$ with a potential $a\in L^p(S)$ and $p>1$, and $f$ subject to growth conditions. Under an additional parity condition on $f$ -- met for instance when $f(u) = u\vert u\vert^\ell$, with $\ell$ an even innteger, we further prove that this equation has infinitely many solutions, in stark contrast with all the well-posedness results that have been proved so far for singular stochastic PDEs under a small parameter assumption. This kind of results is obtained by seeing the equation as characterizing the critical points of an energy functional based on the Anderson operator $H=Δ+ ξ$ and by resorting to variants of the mountain pass theorem. There are however some interesting equations that cannot be characterized as the critical points of an energy functional. Such is the case of the singular Choquard-Pecard equation on $\mathbb{T}^2$ $$ (-Δ+ a)u = \big(w\star f(u)\big)g(u) + ξu $$ One can use Ghoussoub's machinery of self-dual functionals to prove the existence of a solution to that equation as the minimum of a self-dual strongly coercive functional under proper assumptions on the coefficients $a,w,f$ and $g$.
