In this article, we consider the multiplicity of positive solutions for a static Schrödinger-Poisson-Slater equation of the type −Δu+u2∗1∣4πx∣u=μf(x)∣u∣p−2u+g
在本文中，我们将考虑静态薛定谔-泊松-斯莱特方程正解的多重性，该方程的类型为 - Δ u + u 2 ∗ 1 ∣ 4 π x ∣ u = μ f ( x ) ∣ u ∣ p - 2 u + g ( x ) ∣ u ∣ 4 u in R 3 , -Delta u+left({u}^{2}ast frac{1}{| 4pi x| }right)u=mu fleft(x){| u| }^{p-2}u+gleft(x){| u| }^{4}uhspace{1em}hspace{0.1em}text{in}hspace{0.1em}hspace{0.
{"title":"Multiple positive solutions for a class of concave-convex Schrödinger-Poisson-Slater equations with critical exponent","authors":"Tian-Tian Zheng, Chun-Yu Lei, Jia-Feng Liao","doi":"10.1515/anona-2023-0129","DOIUrl":"https://doi.org/10.1515/anona-2023-0129","url":null,"abstract":"\u0000 <jats:p>In this article, we consider the multiplicity of positive solutions for a static Schrödinger-Poisson-Slater equation of the type <jats:disp-formula id=\"j_anona-2023-0129_eq_001\">\u0000 <jats:alternatives>\u0000 <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0129_eq_001.png\" />\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\">\u0000 <m:mo>−</m:mo>\u0000 <m:mi mathvariant=\"normal\">Δ</m:mi>\u0000 <m:mi>u</m:mi>\u0000 <m:mo>+</m:mo>\u0000 <m:mfenced open=\"(\" close=\")\">\u0000 <m:mrow>\u0000 <m:msup>\u0000 <m:mrow>\u0000 <m:mi>u</m:mi>\u0000 </m:mrow>\u0000 <m:mrow>\u0000 <m:mn>2</m:mn>\u0000 </m:mrow>\u0000 </m:msup>\u0000 <m:mo>∗</m:mo>\u0000 <m:mfrac>\u0000 <m:mrow>\u0000 <m:mn>1</m:mn>\u0000 </m:mrow>\u0000 <m:mrow>\u0000 <m:mo>∣</m:mo>\u0000 <m:mn>4</m:mn>\u0000 <m:mi>π</m:mi>\u0000 <m:mi>x</m:mi>\u0000 <m:mo>∣</m:mo>\u0000 </m:mrow>\u0000 </m:mfrac>\u0000 </m:mrow>\u0000 </m:mfenced>\u0000 <m:mi>u</m:mi>\u0000 <m:mo>=</m:mo>\u0000 <m:mi>μ</m:mi>\u0000 <m:mi>f</m:mi>\u0000 <m:mrow>\u0000 <m:mo>(</m:mo>\u0000 <m:mrow>\u0000 <m:mi>x</m:mi>\u0000 </m:mrow>\u0000 <m:mo>)</m:mo>\u0000 </m:mrow>\u0000 <m:msup>\u0000 <m:mrow>\u0000 <m:mo>∣</m:mo>\u0000 <m:mi>u</m:mi>\u0000 <m:mo>∣</m:mo>\u0000 </m:mrow>\u0000 <m:mrow>\u0000 <m:mi>p</m:mi>\u0000 <m:mo>−</m:mo>\u0000 <m:mn>2</m:mn>\u0000 </m:mrow>\u0000 </m:msup>\u0000 <m:mi>u</m:mi>\u0000 <m:mo>+</m:mo>\u0000 <m:mi>g</m:mi>\u0000 <m:mrow>\u0000 ","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":4.2,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140522278","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let Ω⊂RnOmega subset {{bf{R}}}^{n} be a smooth bounded domain. In this article, we prove a result of which the following is a by-product: Let q∈]0,1[qin ]0,1{[}, α∈L∞(Ω)
让 Ω ⊂ R nOmega 子集 {{bf{R}}}^{n} 是一个光滑有界域。本文将证明一个结果，下面是其副产品：设 q∈ ] 0 , 1 [ qin ]0,1{[} , α ∈ L ∞ ( Ω ) α > 0 alpha gt 0 , and k ∈ N kin {bf{N}}. .
{"title":"Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions","authors":"B. Ricceri","doi":"10.1515/anona-2023-0104","DOIUrl":"https://doi.org/10.1515/anona-2023-0104","url":null,"abstract":"\u0000 <jats:p>Let <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0104_eq_001.png\" />\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mi mathvariant=\"normal\">Ω</m:mi>\u0000 <m:mo>⊂</m:mo>\u0000 <m:msup>\u0000 <m:mrow>\u0000 <m:mi mathvariant=\"bold\">R</m:mi>\u0000 </m:mrow>\u0000 <m:mrow>\u0000 <m:mi>n</m:mi>\u0000 </m:mrow>\u0000 </m:msup>\u0000 </m:math>\u0000 <jats:tex-math>Omega subset {{bf{R}}}^{n}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula> be a smooth bounded domain. In this article, we prove a result of which the following is a by-product: Let <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0104_eq_002.png\" />\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mi>q</m:mi>\u0000 <m:mo>∈</m:mo>\u0000 <m:mrow>\u0000 <m:mo stretchy=\"false\">]</m:mo>\u0000 <m:mrow>\u0000 <m:mn>0</m:mn>\u0000 <m:mo>,</m:mo>\u0000 <m:mn>1</m:mn>\u0000 </m:mrow>\u0000 <m:mo stretchy=\"false\">[</m:mo>\u0000 </m:mrow>\u0000 </m:math>\u0000 <jats:tex-math>qin ]0,1{[}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula>, <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0104_eq_003.png\" />\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mi>α</m:mi>\u0000 <m:mo>∈</m:mo>\u0000 <m:msup>\u0000 <m:mrow>\u0000 <m:mi>L</m:mi>\u0000 </m:mrow>\u0000 <m:mrow>\u0000 <m:mi>∞</m:mi>\u0000 </m:mrow>\u0000 </m:msup>\u0000 <m:mrow>\u0000 <m:mo>(</m:mo>\u0000 <m:mrow>\u0000 <m:mi mathvariant=\"normal\">Ω</m:mi>\u0000 </m:mrow>\u0000 <m:mo>)</m","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":4.2,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140519417","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article is devoted to a global Calderón-Zygmund estimate in the framework of Lorentz spaces for the mm-order gradients of weak solution to a higher-order elliptic equation with pp-growth. We prove the main result based on a proper power decay estimation of the upper-level set by the principle of layer cake representation for the Lγ,q{L}^{gamma ,q}-estimate of Dmu{D}^{m}u
本文致力于在洛伦兹空间框架内对具有 p p 增长的高阶椭圆方程弱解的 m m 阶梯度进行全局卡尔德隆-齐格蒙估计。我们通过层蛋糕表示原理对上层集进行适当的功率衰减估计，证明了 L γ , q {L}^{gamma ,q} 的主要结果。 -D m u {D}^{m}u 的估计，同时系数满足小 BMO 半规范，底层域的边界在 Reifenberg 意义上是平的。特别是，一个棘手的问题是确定高导数的法向分量受任意边界点邻域解的高导数水平分量控制，这可以通过将所考虑的解与一些参考问题的解进行比较来实现。
{"title":"Gradient estimates for a class of higher-order elliptic equations of p-growth over a nonsmooth domain","authors":"H. Tian, Shenzhou Zheng","doi":"10.1515/anona-2023-0132","DOIUrl":"https://doi.org/10.1515/anona-2023-0132","url":null,"abstract":"\u0000 <jats:p>This article is devoted to a global Calderón-Zygmund estimate in the framework of Lorentz spaces for the <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0132_eq_001.png\" />\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mi>m</m:mi>\u0000 </m:math>\u0000 <jats:tex-math>m</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula>-order gradients of weak solution to a higher-order elliptic equation with <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0132_eq_002.png\" />\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mi>p</m:mi>\u0000 </m:math>\u0000 <jats:tex-math>p</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula>-growth. We prove the main result based on a proper power decay estimation of the upper-level set by the principle of layer cake representation for the <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0132_eq_003.png\" />\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:msup>\u0000 <m:mrow>\u0000 <m:mi>L</m:mi>\u0000 </m:mrow>\u0000 <m:mrow>\u0000 <m:mi>γ</m:mi>\u0000 <m:mo>,</m:mo>\u0000 <m:mi>q</m:mi>\u0000 </m:mrow>\u0000 </m:msup>\u0000 </m:math>\u0000 <jats:tex-math>{L}^{gamma ,q}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula>-estimate of <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0132_eq_004.png\" />\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:msup>\u0000 <m:mrow>\u0000 <m:mi>D</m:mi>\u0000 </m:mrow>\u0000 <m:mrow>\u0000 <m:mi>m</m:mi>\u0000 </m:mrow>\u0000 </m:msup>\u0000 <m:mi>u</m:mi>\u0000 </m:math>\u0000 <jats:tex-math>{D}^{m}u</jats:tex-mat","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":4.2,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140522805","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article is devoted to the study of the initial boundary value problem for a mixed pseudo-parabolic r ( x ) rleft(x) -Laplacian-type equation. First, by employing the imbedding theorems, the theory of potential wells, and the Galerkin method, we establish the existence and uniqueness of global solutions with subcritical initial energy, critical initial energy, and supercritical initial energy, respectively. Then, we obtain the decay estimate of global solutions with sub-sharp-critical initial energy, sharp-critical initial energy, and supercritical initial energy, respectively. For supercritical initial energy, we also need to analyze the properties of ω omega -limits of solutions. Finally, we discuss the finite-time blowup of solutions with sub-sharp-critical initial energy and sharp-critical initial energy, respectively.
本文主要研究混合伪抛物线 r ( x ) rleft(x) -拉普拉斯型方程的初边界值问题。首先，利用嵌入定理、势井理论和 Galerkin 方法，分别建立了亚临界初值能量、临界初值能量和超临界初值能量全局解的存在性和唯一性。然后，我们分别得到了亚临界初能、锐临界初能和超临界初能全局解的衰减估计值。对于超临界初能，我们还需要分析解的ω omega -极限的性质。最后，我们将分别讨论亚锐临界初能和锐临界初能的解的有限时间炸毁问题。
{"title":"Global existence and finite-time blowup for a mixed pseudo-parabolic r(x)-Laplacian equation","authors":"Jiazhuo Cheng, Qiru Wang","doi":"10.1515/anona-2023-0133","DOIUrl":"https://doi.org/10.1515/anona-2023-0133","url":null,"abstract":"\u0000 This article is devoted to the study of the initial boundary value problem for a mixed pseudo-parabolic \u0000 \u0000 \u0000 \u0000 r\u0000 \u0000 (\u0000 \u0000 x\u0000 \u0000 )\u0000 \u0000 \u0000 rleft(x)\u0000 \u0000 -Laplacian-type equation. First, by employing the imbedding theorems, the theory of potential wells, and the Galerkin method, we establish the existence and uniqueness of global solutions with subcritical initial energy, critical initial energy, and supercritical initial energy, respectively. Then, we obtain the decay estimate of global solutions with sub-sharp-critical initial energy, sharp-critical initial energy, and supercritical initial energy, respectively. For supercritical initial energy, we also need to analyze the properties of \u0000 \u0000 \u0000 \u0000 ω\u0000 \u0000 omega \u0000 \u0000 -limits of solutions. Finally, we discuss the finite-time blowup of solutions with sub-sharp-critical initial energy and sharp-critical initial energy, respectively.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":4.2,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140525963","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that the classical strong maximum principle, concerning positive supersolutions of linear elliptic equations vanishing on the boundary of the domain can be extended, under suitable conditions, to the case in which the forcing term is sign-changing. In addition, for the case of solutions, the normal derivative on the boundary may also vanish on the boundary (definition of flat solution). This leads to examples in which the unique continuation property fails. As a first application, we show the existence of positive solutions for a sublinear semilinear elliptic problem of indefinite sign. A second application, concerning the positivity of solutions of the linear heat equation, for some large values of time, with forcing and/or initial datum changing sign, is also given.
{"title":"Beyond the classical strong maximum principle: Sign-changing forcing term and flat solutions","authors":"Jesús Ildefonso Díaz, J. Hernandez","doi":"10.1515/anona-2023-0128","DOIUrl":"https://doi.org/10.1515/anona-2023-0128","url":null,"abstract":"\u0000 We show that the classical strong maximum principle, concerning positive supersolutions of linear elliptic equations vanishing on the boundary of the domain can be extended, under suitable conditions, to the case in which the forcing term is sign-changing. In addition, for the case of solutions, the normal derivative on the boundary may also vanish on the boundary (definition of flat solution). This leads to examples in which the unique continuation property fails. As a first application, we show the existence of positive solutions for a sublinear semilinear elliptic problem of indefinite sign. A second application, concerning the positivity of solutions of the linear heat equation, for some large values of time, with forcing and/or initial datum changing sign, is also given.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":4.2,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140519792","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jinxia Cen, Stanisław Migórski, Jen-Chih Yao, Shengda Zeng
This work is devoted to study the convection–reaction–diffusion behavior of contaminant in the recovered fracturing fluid which flows in the wellbore from shale gas reservoir. First, we apply various constitutive laws for generalized non-Newtonian fluids, diffusion principles, and friction relations to formulate the recovered fracturing fluid model. The latter is a partial differential system composed of a nonlinear and nonsmooth stationary incompressible Navier-Stokes equation with a multivalued friction boundary condition, and a nonlinear convection–reaction–diffusion equation with mixed Neumann boundary conditions. Then, we provide the weak formulation of the fluid model which is a hemivariational inequality driven by a nonlinear variational equation. We establish existence of solutions to the recovered fracturing fluid model via a surjectivity theorem for multivalued operators combined with an alternative iterative method and elements of nonsmooth analysis.
{"title":"Variational–hemivariational system for contaminant convection–reaction–diffusion model of recovered fracturing fluid","authors":"Jinxia Cen, Stanisław Migórski, Jen-Chih Yao, Shengda Zeng","doi":"10.1515/anona-2023-0141","DOIUrl":"https://doi.org/10.1515/anona-2023-0141","url":null,"abstract":"\u0000 This work is devoted to study the convection–reaction–diffusion behavior of contaminant in the recovered fracturing fluid which flows in the wellbore from shale gas reservoir. First, we apply various constitutive laws for generalized non-Newtonian fluids, diffusion principles, and friction relations to formulate the recovered fracturing fluid model. The latter is a partial differential system composed of a nonlinear and nonsmooth stationary incompressible Navier-Stokes equation with a multivalued friction boundary condition, and a nonlinear convection–reaction–diffusion equation with mixed Neumann boundary conditions. Then, we provide the weak formulation of the fluid model which is a hemivariational inequality driven by a nonlinear variational equation. We establish existence of solutions to the recovered fracturing fluid model via a surjectivity theorem for multivalued operators combined with an alternative iterative method and elements of nonsmooth analysis.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":4.2,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140518775","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Consideration herein is the stability issue of peaked solitary wave solution for the modified Camassa-Holm-Novikov equation, which is derived from the shallow water theory. This wave configuration accommodates the ordered trains of the modified Camassa-Holm-Novikov-peaked solitary solution. With the application of conservation laws and the monotonicity property of the localized energy functionals, we prove the orbital stability of this wave profile in the H 1 ( R ) {H}^{1}left({mathbb{R}}) energy space according to the modulation argument.
本文考虑的是修正的卡马萨-霍尔姆-诺维科夫方程的峰状孤波解的稳定性问题，该方程源于浅水理论。这种波形构造可容纳修正的卡马萨-霍尔姆-诺维科夫峰孤波解的有序波列。通过应用守恒定律和局部能量函数的单调性，我们根据调制论证证明了该波谱在 H 1 ( R ) {H}^{1}left({mathbb{R}}) 能量空间中的轨道稳定性。
{"title":"Orbital stability of the trains of peaked solitary waves for the modified Camassa-Holm-Novikov equation","authors":"Ting Luo","doi":"10.1515/anona-2023-0124","DOIUrl":"https://doi.org/10.1515/anona-2023-0124","url":null,"abstract":"\u0000 Consideration herein is the stability issue of peaked solitary wave solution for the modified Camassa-Holm-Novikov equation, which is derived from the shallow water theory. This wave configuration accommodates the ordered trains of the modified Camassa-Holm-Novikov-peaked solitary solution. With the application of conservation laws and the monotonicity property of the localized energy functionals, we prove the orbital stability of this wave profile in the \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 H\u0000 \u0000 \u0000 1\u0000 \u0000 \u0000 \u0000 (\u0000 \u0000 R\u0000 \u0000 )\u0000 \u0000 \u0000 {H}^{1}left({mathbb{R}})\u0000 \u0000 energy space according to the modulation argument.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":4.2,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140521511","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A singular nonlinear differential equation zσdwdz=aw+zwf(z,w),{z}^{sigma }frac{{rm{d}}w}{{rm{d}}z}=aw+zwfleft(z,w), where σ>1sigma gt 1, is considered in a neighbourhood of the point
奇异非线性微分方程 z σ d w d z = a w + z w f ( z , w ) , {z}^{sigma }frac{rm{d}}w}{rm{d}}z}=aw+zwfleft(z,w), 其中 σ > 1 sigma gt 1 , 在点 z = 0 z=0 的邻域中考虑，如果 σ sigma 是自然数，则该点位于复平面 C {mathbb{C}} 中；如果 σ sigma 是有理数，则该点位于有理函数的黎曼曲面中；如果 σ sigma 是无理数，则该点位于对数函数的黎曼曲面中。假定 w = w ( z ) w=wleft(z) , a ∈ C ⧹ { 0 } ain {mathbb{C}}setminus left{0right} , 并且函数 f f 是有理数。 并且函数 f f 在 C × C 的原点邻域中是解析的 {mathbb{C}}times {mathbb{C}}. .考虑到 σ sigma 是整数、有理数或无理数，对于上述每一种情况，都证明了解析解 w = w ( z) 的存在性。
{"title":"Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point","authors":"J. Diblík, M. Ruzicková","doi":"10.1515/anona-2023-0120","DOIUrl":"https://doi.org/10.1515/anona-2023-0120","url":null,"abstract":"\u0000 <jats:p>A singular nonlinear differential equation <jats:disp-formula id=\"j_anona-2023-0120_eq_001\">\u0000 <jats:alternatives>\u0000 <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0120_eq_001.png\" />\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\">\u0000 <m:msup>\u0000 <m:mrow>\u0000 <m:mi>z</m:mi>\u0000 </m:mrow>\u0000 <m:mrow>\u0000 <m:mi>σ</m:mi>\u0000 </m:mrow>\u0000 </m:msup>\u0000 <m:mfrac>\u0000 <m:mrow>\u0000 <m:mi mathvariant=\"normal\">d</m:mi>\u0000 <m:mi>w</m:mi>\u0000 </m:mrow>\u0000 <m:mrow>\u0000 <m:mi mathvariant=\"normal\">d</m:mi>\u0000 <m:mi>z</m:mi>\u0000 </m:mrow>\u0000 </m:mfrac>\u0000 <m:mo>=</m:mo>\u0000 <m:mi>a</m:mi>\u0000 <m:mi>w</m:mi>\u0000 <m:mo>+</m:mo>\u0000 <m:mi>z</m:mi>\u0000 <m:mi>w</m:mi>\u0000 <m:mi>f</m:mi>\u0000 <m:mrow>\u0000 <m:mo>(</m:mo>\u0000 <m:mrow>\u0000 <m:mi>z</m:mi>\u0000 <m:mo>,</m:mo>\u0000 <m:mi>w</m:mi>\u0000 </m:mrow>\u0000 <m:mo>)</m:mo>\u0000 </m:mrow>\u0000 <m:mo>,</m:mo>\u0000 </m:math>\u0000 <jats:tex-math>{z}^{sigma }frac{{rm{d}}w}{{rm{d}}z}=aw+zwfleft(z,w),</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:disp-formula> where <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0120_eq_002.png\" />\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mi>σ</m:mi>\u0000 <m:mo>></m:mo>\u0000 <m:mn>1</m:mn>\u0000 </m:math>\u0000 <jats:tex-math>sigma gt 1</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula>, is considered in a neighbourhood of the point <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0120_eq_003.png\" />\u0000 <m","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":4.2,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140525677","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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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