SIAM Journal on Mathematical Analysis, Volume 52, Issue 6, Page 6180-6221, January 2020.
We prove the existence and asymptotic expansion of a large class of solutions to nonlinear Helmholtz equations of the form $(\Delta - \lambda^2) u = N[u]$, where $\Delta = -\sum_j \partial^2_j$ is the Laplacian on $\mathbb{R}^n$, $\lambda$ is a positive real number, and $N[u]$ is a nonlinear operator depending polynomially on $u$ and its derivatives of order up to order two. Nonlinear Helmholtz eigenfunctions with $N[u]= \pm |u|^{p-1} u$ were first considered by Gutiérrez [Math. Ann., 328 (2004), pp. 1--25]. We show that for suitable nonlinearities and for every $f \in H^{k+4}(\mathbb{S}^{n-1})$ of sufficiently small norm, there is a nonlinear Helmholtz function taking the form $u(r, \omega) = r^{-(n-1)/2} ( e^{-i\lambda r} f(\omega) + e^{+i\lambda r} b(\omega) + O(r^{-\epsilon})), \text{ as } r \to \infty, \quad \epsilon > 0$, for some $b \in H^{k}(\mathbb{S}^{n-1})$. Moreover, we prove the result in the general setting of asymptotically conic manifolds. The proof uses an elaboration of anisotropic Sobolev spaces defined by Vasy [A minicourse on microlocal analysis for wave propagation, in Asymptotic Analysis in General Relativity, London Math. Soc. Lecture Note Ser. 443, Cambridge University Press, Cambridge, 2018, pp. 219--374], between which the Helmholtz operator $\Delta - \lambda^2$ acts invertibly. These spaces have a variable spatial weight $\mathsf{l}_\pm$, varying in phase space and distinguishing between the two “radial sets” corresponding to incoming oscillations, $e^{-i\lambda r}$, and outgoing oscillations, $e^{+i\lambda r}$. Our spaces have, in addition, module regularity with respect to two different “test modules” and have algebra (or pointwise multiplication) properties that allow us to treat nonlinearities $N[u]$ of the form specified above.

中文翻译：

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非线性亥姆霍兹特征函数的存在性与渐近性

SIAM数学分析杂志，第52卷，第6期，第6180-6221页，2020年1月。
我们证明了形式为$（\ Delta-\ lambda ^ 2）u = N [u] $的非线性Helmholtz方程的一类解的存在性和渐近展开，其中$ \ Delta =-\ sum_j \ partial ^ 2_j $是$ \ mathbb {R} ^ n $上的拉普拉斯算子，$ \ lambda $是一个正实数，而$ N [u] $是一个非线性运算符，其多项式取决于$ u $及其从阶到阶的导数二。Gutiérrez首先考虑了具有$ N [u] = \ pm | u | ^ {p-1} u $的非线性Helmholtz本征函数。Ann。，328（2004），第1--25页]。我们表明，对于适当的非线性和足够小范数的H ^ {k + 4}（\ mathbb {S} ^ {n-1}）$中的每个$ f，存在一个采用$ u形式的非线性亥姆霍兹函数（r，\ omega）= r ^ {-（n-1）/ 2}（e ^ {-i \ lambda r} f（\ omega）+ e ^ {+ i \ lambda r} b（\ omega）+ O（r ^ {-\ epsilon}）），\ text {as} r \ to \ infty，\ quad \ epsilon> 0 $，在H ^ {k}（\ mathbb {S} ^ {n-1}）$中有$ b \。此外，我们证明了渐近锥流形的一般设置的结果。该证明使用了由Vasy定义的各向异性Sobolev空间[在伦敦大学数学学院的《广义相对论渐近分析》中，对波传播进行了微局部分析的微型课程。Soc。讲义系列 443，剑桥大学出版社，剑桥，2018年，第219--374页]，在两者之间，亥姆霍兹算子$ \ Delta-\ lambda ^ 2 $可逆地作用。这些空间具有可变的空间权重$ \ mathsf {l} _ \ pm $，在相空间中变化，并区分与传入振荡$ e ^ {-i \ lambda r} $和传出振荡对应的两个“径向集” $ e ^ {+ i \ lambda r} $。此外，我们的空间还有