Analytic solutions of the difference equation Ψ(z+h)=M(z)Ψ(z)Psi (z+h)=M(z)Psi (z) are explored. Here zz is a complex variable, h>0h>0 is a parameter, and MM is a given SL(2,C)SL(2,mathbb {C})-valued function. It is assumed that MM
The two-dimensional (2D) domain under study is bounded from below by two semi-infinite and, between them, two finite straight lines; on each of the straight lines (segments), a usually individual impedance boundary condition is imposed. An incident surface wave, propagating from infinity along one semi-infinite segment of the polygonal domain, excites outgoing surface waves both on the same segment (a reflected wave) and on the second semi-infinite segment (a transmitted wave); in addition, a circular (cylindrical) outgoing wave will be generated in the far field. The scattered wave field satisfies the Helmholtz equation and the Robin (in other words, impedance) boundary conditions as well as some special integral form of the Sommerfeld radiation conditions. It is shown that a classical solution of the problem is unique. By the use of some known extension of the Sommerfeld–Malyuzhinets technique, the problem is reduced to functional Malyuzhinets equations and then to a system of integral equations of the second kind with integral operator depending on a characteristic parameter. The Fredholm property of the equations is established, which also leads to the existence of the solution for noncharacteristic values of the parameter. From the Sommerfeld integral representation of the solution, the far-field asymptotics is developed. Numerical results for the scattering diagram are also presented.
{"title":"Scattering of a surface wave in a polygonal domain with impedance boundary","authors":"M. Lyalinov, N. Zhu","doi":"10.1090/spmj/1700","DOIUrl":"https://doi.org/10.1090/spmj/1700","url":null,"abstract":"The two-dimensional (2D) domain under study is bounded from below by two semi-infinite and, between them, two finite straight lines; on each of the straight lines (segments), a usually individual impedance boundary condition is imposed. An incident surface wave, propagating from infinity along one semi-infinite segment of the polygonal domain, excites outgoing surface waves both on the same segment (a reflected wave) and on the second semi-infinite segment (a transmitted wave); in addition, a circular (cylindrical) outgoing wave will be generated in the far field. The scattered wave field satisfies the Helmholtz equation and the Robin (in other words, impedance) boundary conditions as well as some special integral form of the Sommerfeld radiation conditions. It is shown that a classical solution of the problem is unique. By the use of some known extension of the Sommerfeld–Malyuzhinets technique, the problem is reduced to functional Malyuzhinets equations and then to a system of integral equations of the second kind with integral operator depending on a characteristic parameter. The Fredholm property of the equations is established, which also leads to the existence of the solution for noncharacteristic values of the parameter. From the Sommerfeld integral representation of the solution, the far-field asymptotics is developed. Numerical results for the scattering diagram are also presented.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43902415","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The dimension reduction for the viscous flows in thin tube structures leads to equations on the graph for the macroscopic pressure with Kirchhoff type junction conditions at the vertices. Nonlinear equations on the graph generated by the non-Newtonian rheology are treated here. The existence and uniqueness of a solution of this problem is proved. This solution describes the leading term of an asymptotic analysis of the stationary non-Newtonian fluid motion in a thin tube structure with no-slip boundary condition on the lateral boundary.
{"title":"Steady state non-Newtonian flow in a thin tube structure: equation on the graph","authors":"G. Panasenko, K. Pileckas, B. Vernescu","doi":"10.1090/spmj/1702","DOIUrl":"https://doi.org/10.1090/spmj/1702","url":null,"abstract":"The dimension reduction for the viscous flows in thin tube structures leads to equations on the graph for the macroscopic pressure with Kirchhoff type junction conditions at the vertices. Nonlinear equations on the graph generated by the non-Newtonian rheology are treated here. The existence and uniqueness of a solution of this problem is proved. This solution describes the leading term of an asymptotic analysis of the stationary non-Newtonian fluid motion in a thin tube structure with no-slip boundary condition on the lateral boundary.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47948184","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The asymptotics is found for eigenvalues of polynomially compact pseudodifferential operators of the zeroth order.
研究了零阶多项式紧化伪微分算子的特征值的渐近性。
{"title":"Eigenvalue asymptotics for polynomially compact pseudodifferential operators","authors":"G. Rozenblum","doi":"10.1090/spmj/1703","DOIUrl":"https://doi.org/10.1090/spmj/1703","url":null,"abstract":"The asymptotics is found for eigenvalues of polynomially compact pseudodifferential operators of the zeroth order.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47204442","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Formulas are constructed for the short-wave asymptotics in the problem of diffraction of a plane wave on a contour with continuous curvature that is smooth everywhere except for one point near which it has a power-like behavior. The wave field is described in the boundary layers surrounding the singular point of the contour and the limit ray. An expression for the diffracted wave is found.
{"title":"Short wave diffraction on a contour with a Hölder singularity of the curvature","authors":"E. Zlobina, A. Kiselev","doi":"10.1090/spmj/1697","DOIUrl":"https://doi.org/10.1090/spmj/1697","url":null,"abstract":"Formulas are constructed for the short-wave asymptotics in the problem of diffraction of a plane wave on a contour with continuous curvature that is smooth everywhere except for one point near which it has a power-like behavior. The wave field is described in the boundary layers surrounding the singular point of the contour and the limit ray. An expression for the diffracted wave is found.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47807434","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The asymptotics is examined for solutions to the spectral problem for the Laplace operator in a dd-dimensional thin, of diameter O(h)O(h), spindle-shaped domain ΩhOmega ^h with the Dirichlet condition on small, of size h≪1hll 1, terminal zones Γ±hGamma ^h_pm and the Neumann condition on the remaining part of the boundary ∂Ωhpartial Omega ^h. In the limit as h
Bilateral pointwise estimates are provided for positive solutions uu to the sublinear integral equation u=G(σuq)+fin Ω,begin{equation*} u = mathbf {G}(sigma u^q) + f quad text {in } Omega , end{equation*} for 0>q>10 > q > 1, where σ≥0sigma ge 0 is a measurable function or a Radon measure, f≥0f ge 0, and G
A. Anikin, S. Dobrokhotov, V. Nazaikinskii, A. Tsvetkova
The spectral problem −⟨∇,D(x)∇ψ⟩=λψ-langle nabla ,D(x)nabla psi rangle = lambda psi in a bounded two-dimensional domain ΩOmega is considered, where D(x)D(x) is a smooth function positive inside the domain and zero on the boundary whose gradient is different from zero on the boundary. This problem arises in the study of long waves trapped by the shore and by bottom irregularities. For its asymptotic solutions as λ→∞lambda rightarrow infty, explicit formulas are given when D(x)
{"title":"Nonstandard Liouville tori and caustics in asymptotics in the form of Airy and Bessel functions for 2D standing coastal waves","authors":"A. Anikin, S. Dobrokhotov, V. Nazaikinskii, A. Tsvetkova","doi":"10.1090/spmj/1696","DOIUrl":"https://doi.org/10.1090/spmj/1696","url":null,"abstract":"<p>The spectral problem <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"minus mathematical left-angle nabla comma upper D left-parenthesis x right-parenthesis nabla psi mathematical right-angle equals lamda psi\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">⟨<!-- ⟨ --></mml:mo>\u0000 <mml:mi mathvariant=\"normal\">∇<!-- ∇ --></mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>D</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mi mathvariant=\"normal\">∇<!-- ∇ --></mml:mi>\u0000 <mml:mi>ψ<!-- ψ --></mml:mi>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">⟩<!-- ⟩ --></mml:mo>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mi>λ<!-- λ --></mml:mi>\u0000 <mml:mi>ψ<!-- ψ --></mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">-langle nabla ,D(x)nabla psi rangle = lambda psi</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> in a bounded two-dimensional domain <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Omega\">\u0000 <mml:semantics>\u0000 <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">Omega</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is considered, where <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D left-parenthesis x right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>D</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">D(x)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is a smooth function positive inside the domain and zero on the boundary whose gradient is different from zero on the boundary. This problem arises in the study of long waves trapped by the shore and by bottom irregularities. For its asymptotic solutions as <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"lamda right-arrow normal infinity\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>λ<!-- λ --></mml:mi>\u0000 <mml:mo stretchy=\"false\">→<!-- → --></mml:mo>\u0000 <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">lambda rightarrow infty</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, explicit formulas are given when <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D left-parenthesis x right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>D</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45196115","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a field FF, a positive integer mm and an integer n≥2ngeq 2, it is proved that the symbol length of classes in Milnor’s KK-groups KnF/2mKnFK_n F/2^m K_n F that are equivalent to single symbols under the embedding into KnF/2m+1KnF
给定一个域F、一个正整数m和一个整数n≥2ngeq2，证明了Milnor的K-群KnF/2mKnFK_nF/2^mK_nF中等价于嵌入到KnF/2中的单个符号的类的符号长度m+1 K n F K_ n F/2^｛m+1｝K_ n F在假定Fμ2 m+1 Fsupseteqμ_｛2^｛m+1｝｝。由于当n=2n=2时，K2 F/2 m K2 FŞ2 m Br（F）K_，这与指数为2m2^m的中心单代数的符号长度的2 2的上界（由Tignol在1983年证明）一致，该中心单代数是Brauer等价于2 m+12^{m+1}度的单符号代数。还考虑了嵌入到K n F/2 m+1 K n F K_n F/2^｛m+1｝K_n F中的符号长度为2 2、3 3和4 4（最后当n=2 n=2时）的情况。本文最后研究了
{"title":"Symbol length of classes in Milnor 𝐾-groups","authors":"Adam Chapman","doi":"10.1090/spmj/1775","DOIUrl":"https://doi.org/10.1090/spmj/1775","url":null,"abstract":"<p>Given a field <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F\">\u0000 <mml:semantics>\u0000 <mml:mi>F</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">F</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, a positive integer <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"m\">\u0000 <mml:semantics>\u0000 <mml:mi>m</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">m</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and an integer <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n greater-than-or-equal-to 2\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mo>≥<!-- ≥ --></mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">ngeq 2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, it is proved that the symbol length of classes in Milnor’s <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\u0000 <mml:semantics>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-groups <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K Subscript n Baseline upper F slash 2 Superscript m Baseline upper K Subscript n Baseline upper F\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msub>\u0000 <mml:mi>F</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>/</mml:mo>\u0000 </mml:mrow>\u0000 <mml:msup>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:mi>m</mml:mi>\u0000 </mml:msup>\u0000 <mml:msub>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msub>\u0000 <mml:mi>F</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">K_n F/2^m K_n F</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> that are equivalent to single symbols under the embedding into <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K Subscript n Baseline upper F slash 2 Superscript m plus 1 Baseline upper K Subscript n Baseline upper F\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msub>\u0000 <mml:mi>F</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>/</mml:mo>\u0000 </mml:mrow>\u0000 <mml:msup>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>m</mml:mi>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 <mml:msub>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msub>\u0000 <mml:mi>F</mml:mi>\u0000 </mml:","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46291720","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Sufficient conditions are provided for the existence of a ground state solution for the problem generated by the fractional Sobolev inequality in Ω∈C2:Omega in C^2:(−Δ)Spsu(x)+u(x)=u2s∗−1(x)(-Delta )_{Sp}^s u(x) + u(x) = u^{2^*_s-1}(x). Here (−Δ)S
给出了Ω∈C2:Omega中的分数阶Sobolev不等式在C^2中产生的问题基态解存在的充分条件：（−Δ）S p S u（x）+u（x=u 2 s*−1（x）（-Δ）_｛Sp｝^s u（x）+u（x）=u ^｛2^*_s-1｝（x）。这里（−Δ）S p S（-Delta）_｛Sp｝^S代表传统Neumann拉普拉斯算子在Ω中的S次幂，s∈（0，1）s在（0，l）中，2s*=2 n/（n−2s）2^*_s=2n/（n-2s）。对于s=1 s=1的局部情况，Neumann-拉普拉斯算子和Neumann-p-拉普拉斯算子的相应结果早些时候得到。
{"title":"Solvability of a critical semilinear problem with the spectral Neumann fractional Laplacian","authors":"N. Ustinov","doi":"10.1090/spmj/1693","DOIUrl":"https://doi.org/10.1090/spmj/1693","url":null,"abstract":"<p>Sufficient conditions are provided for the existence of a ground state solution for the problem generated by the fractional Sobolev inequality in <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Omega element-of upper C squared colon\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi>\u0000 <mml:mo>∈<!-- ∈ --></mml:mo>\u0000 <mml:msup>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msup>\u0000 <mml:mo>:</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">Omega in C^2:</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis negative normal upper Delta right-parenthesis Subscript upper S p Superscript s Baseline u left-parenthesis x right-parenthesis plus u left-parenthesis x right-parenthesis equals u Superscript 2 Super Subscript s Super Superscript asterisk Superscript minus 1 Baseline left-parenthesis x right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mi mathvariant=\"normal\">Δ<!-- Δ --></mml:mi>\u0000 <mml:msubsup>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>S</mml:mi>\u0000 <mml:mi>p</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>s</mml:mi>\u0000 </mml:msubsup>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:msup>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:msubsup>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:mi>s</mml:mi>\u0000 <mml:mo>∗<!-- ∗ --></mml:mo>\u0000 </mml:msubsup>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">(-Delta )_{Sp}^s u(x) + u(x) = u^{2^*_s-1}(x)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. Here <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis negative normal upper Delta right-parenthesis Subscript upper S p Superscript s\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mi mathvariant=\"normal\">Δ<!-- Δ --></mml:mi>\u0000 <mml:msubsup>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>S</mml:mi>\u0000 ","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45556400","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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