In this article, our main aim is to investigate the existence of radial kk-convex solutions for the following Dirichlet system with kk-Hessian operators: Sk(D2u
在本文中，我们的主要目的是研究以下具有 k k 个黑森算子的狄利克特系统的径向 k k -凸解的存在性： S k ( D 2 u ) = λ 1 ν 1 ( ∣ x ∣ ) ( - u ) p 1 ( - v ) q
{"title":"k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities","authors":"Xingyue He, Chenghua Gao, Jingjing Wang","doi":"10.1515/anona-2023-0136","DOIUrl":"https://doi.org/10.1515/anona-2023-0136","url":null,"abstract":"\u0000 <jats:p>In this article, our main aim is to investigate the existence of radial <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0136_eq_001.png\" />\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mi>k</m:mi>\u0000 </m:math>\u0000 <jats:tex-math>k</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula>-convex solutions for the following Dirichlet system with <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0136_eq_002.png\" />\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mi>k</m:mi>\u0000 </m:math>\u0000 <jats:tex-math>k</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula>-Hessian operators: <jats:disp-formula id=\"j_anona-2023-0136_eq_001\">\u0000 <jats:alternatives>\u0000 <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0136_eq_003.png\" />\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\">\u0000 <m:mfenced open=\"{\" close=\"\">\u0000 <m:mrow>\u0000 <m:mtable displaystyle=\"true\">\u0000 <m:mtr>\u0000 <m:mtd columnalign=\"left\">\u0000 <m:msub>\u0000 <m:mrow>\u0000 <m:mi>S</m:mi>\u0000 </m:mrow>\u0000 <m:mrow>\u0000 <m:mi>k</m:mi>\u0000 </m:mrow>\u0000 </m:msub>\u0000 <m:mrow>\u0000 <m:mo>(</m:mo>\u0000 <m:mrow>\u0000 <m:msup>\u0000 <m:mrow>\u0000 <m:mi>D</m:mi>\u0000 </m:mrow>\u0000 <m:mrow>\u0000 <m:mn>2</m:mn>\u0000 </m:mrow>\u0000 </m:msup>\u0000 <m:mi>u</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":"140519518","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}
In this article, we are concerned about the existence, uniqueness, and nonexistence of the positive solution for: −Δu−12(x⋅∇u)=μh(x)u
本文关注以下正解的存在性、唯一性和不存在性： - Δ u - 1 2 ( x ⋅∇ u ) = μ h ( x ) u q - 1 + λ u - u p , x∈ R N , u ( x ) → 0 , as ∣ x ∣ → + ∞ , left{begin{array}{l}-Delta u-frac{1}{2}left(xcdot nabla u)=mu hleft(x){u}^{q-
{"title":"Existence and uniqueness of solution for a singular elliptic differential equation","authors":"Shanshan Gu, Bianxia Yang, Wenrui Shao","doi":"10.1515/anona-2023-0126","DOIUrl":"https://doi.org/10.1515/anona-2023-0126","url":null,"abstract":"\u0000 <jats:p>In this article, we are concerned about the existence, uniqueness, and nonexistence of the positive solution for: <jats:disp-formula id=\"j_anona-2023-0126_eq_001\">\u0000 <jats:alternatives>\u0000 <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0126_eq_001.png\" />\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\">\u0000 <m:mfenced open=\"{\" close=\"\">\u0000 <m:mrow>\u0000 <m:mtable displaystyle=\"true\">\u0000 <m:mtr>\u0000 <m:mtd columnalign=\"left\">\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:mfrac>\u0000 <m:mrow>\u0000 <m:mn>1</m:mn>\u0000 </m:mrow>\u0000 <m:mrow>\u0000 <m:mn>2</m:mn>\u0000 </m:mrow>\u0000 </m:mfrac>\u0000 <m:mrow>\u0000 <m:mo>(</m:mo>\u0000 <m:mrow>\u0000 <m:mi>x</m:mi>\u0000 <m:mo>⋅</m:mo>\u0000 <m:mrow>\u0000 <m:mo>∇</m:mo>\u0000 </m:mrow>\u0000 <m:mi>u</m:mi>\u0000 </m:mrow>\u0000 <m:mo>)</m:mo>\u0000 </m:mrow>\u0000 <m:mo>=</m:mo>\u0000 <m:mi>μ</m:mi>\u0000 <m:mi>h</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:mi>u</m:mi>\u0000 </m:mrow>\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":"140524673","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}
The compressible Navier-Stokes-Smoluchowski equations under investigation concern the behavior of the mixture of fluid and particles at a macroscopic scale. We devote to the existence of the global classical solution near the stationary solution based on the energy method under weaker conditions imposed on the external potential compared with Chen et al. (Global existence and time–decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations, Discrete Contin. Dyn. Syst. 36 (2016), no. 10, 5287–5307). Under further assumptions that the stationary solution (ρs(x),0,0)T{left({rho }_{s}left(x),0,0)}^{T} is in a small neighborhood of the constant state
所研究的可压缩纳维-斯托克斯-斯莫卢霍夫斯基方程涉及流体和粒子在宏观尺度上的混合行为。与 Chen 等人（Global existence and time-decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations, Discrete Contin.Dyn.Syst.36 (2016)，第 10 期，5287-5307）。进一步假设静止解 ( ρ s ( x ) , 0 , 0 ) T {left({rho }_{s}left(x),0,0)}^{T} 在恒定状态 ( ρ ¯ , 0 , 0 ) 的一个小邻域内。 T {left(bar{rho},0,0)}^{T}在无穷远处，我们还可以通过能量法和线性 L p {L}^{p} 的结合得到解的时间衰减率。 - L q {L}^{q} 衰减估计。
{"title":"Global existence and decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in ℝ3","authors":"Leilei Tong","doi":"10.1515/anona-2023-0131","DOIUrl":"https://doi.org/10.1515/anona-2023-0131","url":null,"abstract":"\u0000 <jats:p>The compressible Navier-Stokes-Smoluchowski equations under investigation concern the behavior of the mixture of fluid and particles at a macroscopic scale. We devote to the existence of the global classical solution near the stationary solution based on the energy method under weaker conditions imposed on the external potential compared with Chen et al. (Global existence and time–decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations, Discrete Contin. Dyn. Syst. 36 (2016), no. 10, 5287–5307). Under further assumptions that the stationary solution <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0131_eq_001.png\" />\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:msup>\u0000 <m:mrow>\u0000 <m:mrow>\u0000 <m:mo>(</m:mo>\u0000 <m:mrow>\u0000 <m:msub>\u0000 <m:mrow>\u0000 <m:mi>ρ</m:mi>\u0000 </m:mrow>\u0000 <m:mrow>\u0000 <m:mi>s</m:mi>\u0000 </m:mrow>\u0000 </m:msub>\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:mo>,</m:mo>\u0000 <m:mn>0</m:mn>\u0000 <m:mo>,</m:mo>\u0000 <m:mn>0</m:mn>\u0000 </m:mrow>\u0000 <m:mo>)</m:mo>\u0000 </m:mrow>\u0000 </m:mrow>\u0000 <m:mrow>\u0000 <m:mi>T</m:mi>\u0000 </m:mrow>\u0000 </m:msup>\u0000 </m:math>\u0000 <jats:tex-math>{left({rho }_{s}left(x),0,0)}^{T}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula> is in a small neighborhood of the constant state <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0131_eq_002.png\" />\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\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":"140519507","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}
In this article, we consider the bounded variation capacity (BV capacity) and characterize the Sobolev-type inequalities related to BV functions in a general framework of strictly local Dirichlet spaces with a doubling measure via the BV capacity. Under a weak Bakry-Émery curvature-type condition, we give the connection between the Hausdorff measure and the Hausdorff capacity, and discover some capacitary inequalities and Maz’ya-Sobolev inequalities for BV functions. The De Giorgi characterization for total variation is also obtained with a quasi-Bakry-Émery curvature condition. It should be noted that the results in this article are proved if the Dirichlet space supports the weak ( 1 , 2 ) left(1,2) -Poincaré inequality instead of the weak ( 1 , 1 ) left(1,1) -Poincaré inequality compared with the results in the previous references.
{"title":"The bounded variation capacity and Sobolev-type inequalities on Dirichlet spaces","authors":"Xiangyun Xie, Yu Liu, Pengtao Li, Jizheng Huang","doi":"10.1515/anona-2023-0119","DOIUrl":"https://doi.org/10.1515/anona-2023-0119","url":null,"abstract":"\u0000 In this article, we consider the bounded variation capacity (BV capacity) and characterize the Sobolev-type inequalities related to BV functions in a general framework of strictly local Dirichlet spaces with a doubling measure via the BV capacity. Under a weak Bakry-Émery curvature-type condition, we give the connection between the Hausdorff measure and the Hausdorff capacity, and discover some capacitary inequalities and Maz’ya-Sobolev inequalities for BV functions. The De Giorgi characterization for total variation is also obtained with a quasi-Bakry-Émery curvature condition. It should be noted that the results in this article are proved if the Dirichlet space supports the weak \u0000 \u0000 \u0000 \u0000 \u0000 (\u0000 \u0000 1\u0000 ,\u0000 2\u0000 \u0000 )\u0000 \u0000 \u0000 left(1,2)\u0000 \u0000 -Poincaré inequality instead of the weak \u0000 \u0000 \u0000 \u0000 \u0000 (\u0000 \u0000 1\u0000 ,\u0000 1\u0000 \u0000 )\u0000 \u0000 \u0000 left(1,1)\u0000 \u0000 -Poincaré inequality compared with the results in the previous references.","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":"140526306","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}
In this article, we study the following quasilinear equation with nonlocal nonlinearity −Δu−κuΔ(u2)+λu=(∣x∣−μ*F(u))f(u
在本文中，我们将研究以下具有非局部非线性的准线性方程 - Δ u - κ u Δ ( u 2 ) +λ u = ( ∣ x ∣ - μ * F ( u ) ) f ( u ) , in R N , -Delta u-kappa uDelta left({u}^{2})+lambda u=left({| x| }^{mu }* Fleft(u))fleft(u),hspace{1em}hspace{0.1em}text{in}hspace{0.1em}hspace{0.
{"title":"Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities","authors":"Yue Jia, Xianyong Yang","doi":"10.1515/anona-2023-0130","DOIUrl":"https://doi.org/10.1515/anona-2023-0130","url":null,"abstract":"\u0000 <jats:p>In this article, we study the following quasilinear equation with nonlocal nonlinearity <jats:disp-formula id=\"j_anona-2023-0130_eq_001\">\u0000 <jats:alternatives>\u0000 <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0130_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:mi>κ</m:mi>\u0000 <m:mi>u</m:mi>\u0000 <m:mi>Δ</m:mi>\u0000 <m:mrow>\u0000 <m:mo>(</m:mo>\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:mrow>\u0000 <m:mo>)</m:mo>\u0000 </m:mrow>\u0000 <m:mo>+</m:mo>\u0000 <m:mi>λ</m:mi>\u0000 <m:mi>u</m:mi>\u0000 <m:mo>=</m:mo>\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>x</m:mi>\u0000 <m:mo>∣</m:mo>\u0000 </m:mrow>\u0000 <m:mrow>\u0000 <m:mo>−</m:mo>\u0000 <m:mi>μ</m:mi>\u0000 </m:mrow>\u0000 </m:msup>\u0000 <m:mo>*</m:mo>\u0000 <m:mi>F</m:mi>\u0000 <m:mrow>\u0000 <m:mo>(</m:mo>\u0000 <m:mrow>\u0000 <m:mi>u</m:mi>\u0000 </m:mrow>\u0000 <m:mo>)</m:mo>\u0000 </m:mrow>\u0000 </m:mrow>\u0000 <m:mo>)</m:mo>\u0000 </m:mrow>\u0000 <m:mi>f</m:mi>\u0000 <m:mrow>\u0000 <m:mo>(</m:mo>\u0000 <m:mrow>\u0000 <m:mi>u</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":"140522614","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}
Abstract In this article, we consider anisotropic parabolic systems of p p -Laplace type. The model case is the parabolic p i {p}_{i} -Laplace system u t − ∑ i = 1 n ∂ ∂ x i ( ∣ D i u ∣ p i − 2 D i u ) = 0 {u}_{t}-mathop{sum }limits_{i=1}^{n}frac{partial }{partial {x}_{i}}({| {D}_{i}u| }^{{p}_{i}-2}{D}_{i}u)=0 with exponents p i ≥ 2 {p}_{i}ge 2 . Under the assumption that the exponents are not too far apart, i.e., the difference p max − p min {p}_{max }-{p}_{min } is sufficiently small, we establish a higher integrability result for weak solutions. This extends a result, which was only known for the elliptic setting, to the parabolic setting.
{"title":"Higher integrability for anisotropic parabolic systems of p-Laplace type","authors":"Leon Mons","doi":"10.1515/anona-2022-0308","DOIUrl":"https://doi.org/10.1515/anona-2022-0308","url":null,"abstract":"Abstract In this article, we consider anisotropic parabolic systems of p p -Laplace type. The model case is the parabolic p i {p}_{i} -Laplace system u t − ∑ i = 1 n ∂ ∂ x i ( ∣ D i u ∣ p i − 2 D i u ) = 0 {u}_{t}-mathop{sum }limits_{i=1}^{n}frac{partial }{partial {x}_{i}}({| {D}_{i}u| }^{{p}_{i}-2}{D}_{i}u)=0 with exponents p i ≥ 2 {p}_{i}ge 2 . Under the assumption that the exponents are not too far apart, i.e., the difference p max − p min {p}_{max }-{p}_{min } is sufficiently small, we establish a higher integrability result for weak solutions. This extends a result, which was only known for the elliptic setting, to the parabolic setting.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45131011","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}
Abstract Our aim of this article is to study qualitative properties of Dirichlet problems involving the Hardy-Leray operator ℒ V ≔ − Δ + V {{mathcal{ {mathcal L} }}}_{V}:= -Delta +V , where V ( x ) = ∑ i = 1 m μ i ∣ x − A i ∣ 2 Vleft(x)={sum }_{i=1}^{m}frac{{mu }_{i}}{{| x-{A}_{i}| }^{2}} , with μ i ≥ − ( N − 2 ) 2 4 {mu }_{i}ge -frac{{left(N-2)}^{2}}{4} being the Hardy-Leray potential containing the polars’ set A m = { A i : i = 1 , … , m } {{mathcal{A}}}_{m}=left{{A}_{i}:i=1,ldots ,mright} in R N {{mathbb{R}}}^{N} ( N ≥ 2 Nge 2 ). Since the inverse-square potentials are critical with respect to the Laplacian operator, the coefficients { μ i } i = 1 m {left{{mu }_{i}right}}_{i=1}^{m} and the locations of polars { A i } left{{A}_{i}right} play an important role in the properties of solutions to the related Poisson problems subject to zero Dirichlet boundary conditions. Let Ω Omega be a bounded domain containing A m {{mathcal{A}}}_{m} . First, we obtain increasing Dirichlet eigenvalues: ℒ V u = λ u in Ω , u = 0 on ∂ Ω , {{mathcal{ {mathcal L} }}}_{V}u=lambda uhspace{1.0em}{rm{in}}hspace{0.33em}Omega ,hspace{1.0em}u=0hspace{1.0em}{rm{on}}hspace{0.33em}partial Omega , and the positivity of the principle eigenvalue depends on the strength μ i {mu }_{i} and polars’ setting. When the spectral does not contain the origin, we then consider the weak solutions of the Poisson problem ( E ) ℒ V u = ν in Ω , u = 0 on ∂ Ω , left(E)hspace{1.0em}hspace{1.0em}{{mathcal{ {mathcal L} }}}_{V}u=nu hspace{1em}{rm{in}}hspace{0.33em}Omega ,hspace{1.0em}u=0hspace{1em}{rm{on}}hspace{0.33em}partial Omega , when ν nu belongs to L p ( Ω ) {L}^{p}left(Omega ) , with p > 2 N N + 2 pgt frac{2N}{N+2} in the variational framework, and we obtain a global weighted L ∞ {L}^{infty } estimate when p > N 2 pgt frac{N}{2} . When the principle eigenvalue is positive and ν nu is a Radon measure, we build a weighted distributional framework to show the existence of weak solutions of problem ( E ) left(E) . Moreover, via this weighted distributional framework, we can obtain a sharp assumption of ν ∈ C γ ( Ω ¯ A m ) nu in {{mathcal{C}}}^{gamma }left(bar{Omega }setminus {{mathcal{A}}}_{m}) for the existence of isolated singular solutions for problem ( E ) left(E) .
摘要本文的目的是研究涉及Hardy-Leray算子的Dirichlet问题的定性性质ℒ V−Δ+V-{A}_{i} |｝^｛2｝｝，其中μi≥−（N−2）2 4｛mu｝_{{A}_{i} ：i=1，ldots，mright｝在R N｛mathbb｛R｝｝^｛N｝中（N≥2Nge2）。由于平方反比势对于拉普拉斯算子是关键的，因此系数｛μi｝i＝1m｛lang1033｛mu｝_｛i｝right｝｝_{{A}_{i} 在零Dirichlet边界条件下相关Poisson问题解的性质中起着重要作用。设ΩOmega是一个包含a m｛mathcal｛a｝｝_｛m｝的有界域。首先，我们获得了增加的狄利克雷特征值：ℒ V u=λu，单位为Ω，u=0，在¦ΒΩ上，｛mathcal｛math L｝｝_{V}u=lambda uhspace｛1.0em｝｛rm｛in｝｝space｛0.33em｝Omega，space{1.0em}u=0hspace｛1.0em｝｛rm｛on｝｝space｛0.33em｝partial Omega，并且主特征值的正性取决于强度μi｛mu｝_｛i｝和polar的设置。当谱不包含原点时，我们考虑泊松问题（E）的弱解ℒ V u=¦Α，u=¦ΒΩ上的0，left（E）hspace｛1.0em｝space｛1.0em｝｛mathcal｛L｝｝_{V}u=nuhspace｛1em｝｛rm｛in｝｝space｛0.33em｝Omega，space{1.0em}u=0hspace｛1em｝｛rm｛on｝｝space｛0.33em｝partialOmega，当Γnu属于Lp（Ω）｛L｝^｛p｝left（Omega）时，在变分框架中p＞2N＋2p｝｛｛N＋2｝，并且当p＞N2｝｛。当主特征值为正，且Γnu为Radon测度时，我们建立了一个加权分布框架来证明问题（E）left（E）弱解的存在性。此外，通过这个加权分布框架，我们可以得到一个关于问题（E）left（E）存在孤立奇异解的尖锐假设，即｛mathcal｛C｝｝｝^｛gamma｝left（bar｛Omega｝setminus｛math cal｛a｝｝｝}_{m｝）中的Γ∈Cγ（Ωam）nu。
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