We consider a nonlinear Dirichlet problem driven by the -Laplacian and with a reaction consisting of a singular term plus a negative perturbation. Using regularization of the singular term and truncation and comparison techniques, we show that the problem has a unique positive smooth solution.
{"title":"Positive solutions for singular (p,q)-Laplacian equations with negative perturbation","authors":"Nikolaos S. Papageorgiou, C. Vetro, F. Vetro","doi":"10.58997/ejde.2023.25","DOIUrl":"https://doi.org/10.58997/ejde.2023.25","url":null,"abstract":"We consider a nonlinear Dirichlet problem driven by the -Laplacian and with a reaction consisting of a singular term plus a negative perturbation. Using regularization of the singular term and truncation and comparison techniques, we show that the problem has a unique positive smooth solution.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43705759","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}
In this article, we consider blow-up criteria and instability of standing waves for the fractional Schrodinger-Poisson equation. By using the localized virial estimates, we establish the blow-up criteria for non-radial solutions in both mass-critical and mass-supercritical cases. Based on these blow-up criteria and three variational characterizations of the ground state, we prove that the standing waves are strongly unstable. These obtained results extend the corresponding ones presented in the literature.
{"title":"Blow-up criteria and instability of standing waves for the fractional Schrodinger Poisson equation","authors":"Yi-Na Mo, Min Zhu, Binhua Feng","doi":"10.58997/ejde.2023.24","DOIUrl":"https://doi.org/10.58997/ejde.2023.24","url":null,"abstract":"In this article, we consider blow-up criteria and instability of standing waves for the fractional Schrodinger-Poisson equation. By using the localized virial estimates, we establish the blow-up criteria for non-radial solutions in both mass-critical and mass-supercritical cases. Based on these blow-up criteria and three variational characterizations of the ground state, we prove that the standing waves are strongly unstable. These obtained results extend the corresponding ones presented in the literature.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46801266","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}
Y. Il'yasov, E. D. Da Silva, Maxwell Lizete Da Silva
A minimax variational method for finding mountain pass-type solutions with prescribed energy levels is introduced. The method is based on application of the Linking Theorem to the energy-level nonlinear Rayleigh quotients which critical points correspond to the solutions of the equation with prescribed energy. An application of the method to nonlinear indefinite elliptic problems with nonlinearities that does not satisfy the Ambrosetti-Rabinowitz growth conditions is also presented.
{"title":"Prescribed energy saddle-point solutions of nonlinear indefinite problems","authors":"Y. Il'yasov, E. D. Da Silva, Maxwell Lizete Da Silva","doi":"10.58997/ejde.2023.23","DOIUrl":"https://doi.org/10.58997/ejde.2023.23","url":null,"abstract":"A minimax variational method for finding mountain pass-type solutions with prescribed energy levels is introduced. The method is based on application of the Linking Theorem to the energy-level nonlinear Rayleigh quotients which critical points correspond to the solutions of the equation with prescribed energy. An application of the method to nonlinear indefinite elliptic problems with nonlinearities that does not satisfy the Ambrosetti-Rabinowitz growth conditions is also presented.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47662739","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}
In this article, we study the Hyers-Ulam stability of the first-order linear quaternion-valued differential equations. We transfer a linear quaternion-valued differential equation into a real differential system. The Hyers-Ulam stability results for the linear quaternion-valued differential equations are obtained according to the equivalent relationship between the vector 2-norm and the quaternion module.
{"title":"Hyers-Ulam stability of linear quaternion-valued differential equations","authors":"Jiaojiao Lv, Jinrong Wang, R. Liu","doi":"10.58997/ejde.2023.21","DOIUrl":"https://doi.org/10.58997/ejde.2023.21","url":null,"abstract":"In this article, we study the Hyers-Ulam stability of the first-order linear quaternion-valued differential equations. We transfer a linear quaternion-valued differential equation into a real differential system. The Hyers-Ulam stability results for the linear quaternion-valued differential equations are obtained according to the equivalent relationship between the vector 2-norm and the quaternion module.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47572390","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}
This article concerns the dynamical transitions of the stochastic Swift-Hohenberg equation with multiplicative noise on a two-dimensional domain (-L,L) times (-L, L). With α and L regarded as parameters, we show that the approximate reduced system corresponding to the invariant manifold undergoes a stochastic pitchfork bifurcation near the critical points, and the impact of noise on stochastic bifurcation of the Swift-Hohenberg equation. We find the approximation representation of the manifold and the corresponding reduced systems for stochastic Swift-Hohenberg equation when L2 and √2L1 are close together.
{"title":"Stochastic attractor bifurcation for the two-dimensional Swift-Hohenberg equation with multiplicative noise","authors":"Qingkun Xiao, Hongjun Gao","doi":"10.58997/ejde.2023.20","DOIUrl":"https://doi.org/10.58997/ejde.2023.20","url":null,"abstract":"This article concerns the dynamical transitions of the stochastic Swift-Hohenberg equation with multiplicative noise on a two-dimensional domain (-L,L) times (-L, L). With α and L regarded as parameters, we show that the approximate reduced system corresponding to the invariant manifold undergoes a stochastic pitchfork bifurcation near the critical points, and the impact of noise on stochastic bifurcation of the Swift-Hohenberg equation. We find the approximation representation of the manifold and the corresponding reduced systems for stochastic Swift-Hohenberg equation when L2 and √2L1 are close together.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135996395","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}
This article concerns the existence of multiple non-radial positive solutions of the L2-constrained problem $$displaylines{-Delta{u}-Q(varepsilon x)|u|^{p-2}u=lambda{u},quad text{in }mathbb{R}^N, int_{mathbb{R}^N}|u|^2dx=1,}$$ where (Q(x)) is a radially symmetric function, ε>0 is a small parameter, (Ngeq 2), and (p in (2, 2+4/N)) is assumed to be mass sub-critical. We are interested in the symmetry breaking of the normalized solutions and we prove the existence of multiple non-radial positive solutions as local minimizers of the energy functional.
{"title":"Non-radial normalized solutions for a nonlinear Schrodinger equation","authors":"Zhicheng Tong, Jianqing Chen, Zhi-Qiang Wang","doi":"10.58997/ejde.2023.19","DOIUrl":"https://doi.org/10.58997/ejde.2023.19","url":null,"abstract":"This article concerns the existence of multiple non-radial positive solutions of the L2-constrained problem $$displaylines{-Delta{u}-Q(varepsilon x)|u|^{p-2}u=lambda{u},quad text{in }mathbb{R}^N, int_{mathbb{R}^N}|u|^2dx=1,}$$ where (Q(x)) is a radially symmetric function, ε>0 is a small parameter, (Ngeq 2), and (p in (2, 2+4/N)) is assumed to be mass sub-critical. We are interested in the symmetry breaking of the normalized solutions and we prove the existence of multiple non-radial positive solutions as local minimizers of the energy functional.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42587629","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}
We develop a method for estimation of parameters of an elastic plate resting on a Winkler-type elastic foundation solely from data on the vertical displacements of the plate. The method allows one to estimate components of the external body force density field, plate thickness, elastic foundation stiffness parameters, horizontal displacements of the plate, and stresses. The key idea of the method is that multiple plate models are used simultaneously, namely the proposed reduced three-dimensional (R3D) plate model, the Mindlin plate model, and the thin plate model. The three plate models form a hierarchy of elastic plate models based on assumptions imposed on stresses, with the R3D plate model being the most generalized model and the thin plate model being the most constrained one. The hierarchical relationship among the plate models allows one to incorporate prior information into the estimation technique. The applicability of the proposed estimation method is illustrated by a numerical example.
{"title":"Estimation of plate parameters from vertical displacement data using a family of plate models","authors":"L. White, T. Malysheva, L. Karlstrom","doi":"10.58997/ejde.2023.15","DOIUrl":"https://doi.org/10.58997/ejde.2023.15","url":null,"abstract":"We develop a method for estimation of parameters of an elastic plate resting on a Winkler-type elastic foundation solely from data on the vertical displacements of the plate. The method allows one to estimate components of the external body force density field, plate thickness, elastic foundation stiffness parameters, horizontal displacements of the plate, and stresses. The key idea of the method is that multiple plate models are used simultaneously, namely the proposed reduced three-dimensional (R3D) plate model, the Mindlin plate model, and the thin plate model. The three plate models form a hierarchy of elastic plate models based on assumptions imposed on stresses, with the R3D plate model being the most generalized model and the thin plate model being the most constrained one. The hierarchical relationship among the plate models allows one to incorporate prior information into the estimation technique. The applicability of the proposed estimation method is illustrated by a numerical example.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48236761","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}
In this article we prove that there are no nontrivial solutions tothe Dirichlet problem for the fractional Laplacian$$ displaylines{(-Delta)^s u =f(u) quad text{in }Omega, u=0 quad text{in } mathbb{R}^N backslash Omega,}$$ where (Omega subset mathbb{R}^N) ((Ngeq 1)) is a bounded domain, and f is locally Lipschitz with non-positive primitive (F(t)= int_0^t f(tau)dtau).
{"title":"Nonexitence of nontrivial solutions to Dirichlet problems for the fractional Laplacian","authors":"José Carmona, A. Molino","doi":"10.58997/ejde.2023.16","DOIUrl":"https://doi.org/10.58997/ejde.2023.16","url":null,"abstract":"In this article we prove that there are no nontrivial solutions tothe Dirichlet problem for the fractional Laplacian$$ displaylines{(-Delta)^s u =f(u) quad text{in }Omega, u=0 quad text{in } mathbb{R}^N backslash Omega,}$$ where (Omega subset mathbb{R}^N) ((Ngeq 1)) is a bounded domain, and f is locally Lipschitz with non-positive primitive (F(t)= int_0^t f(tau)dtau).","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45981136","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}
We study the existence and multiplicity of solutions for the nonlocalperturbed Kirchhoff problem$$displaylines{-Big(a+bint_Omega |nabla u|^2,dxBig)Delta u=lambda g(x,u)+f(x,u), quad text{in } Omega, u=0, quadtext{on }partialOmega,}$$ where Ω is a bounded smooth domain in (mathbb{R}^N), (N>4), (a,b, lambda > 0), and (f,g:Omegatimes mathbb{R}to mathbb{R}) are Caratheodory functions, with (f) subcritical, and (g) of arbitrary growth. This paper is motivated by a recent results by Faraci and Silva [4] where existence and multiplicity results were obtained when g is subcritical and f is a power-type function withcritical exponent.
我们研究了非局部扰动Kirchhoff问题$$displaylines｛-Big（a+bint_Omega|nabla u|^2，dxBig）Delta u=lambda g（x，u）+f（x，u），quadtext｛in｝Omega，u=0，quad text{on｝partial Omega，和（f，g:Omegatimesmathbb｛R｝ to mathbb｛R}）是Caratheodory函数，具有（f）次临界和（g）任意增长。本文的动机是Faraci和Silva[4]最近的一个结果，其中当g是亚临界的，f是具有临界指数的幂型函数时，得到了存在性和多重性的结果。
{"title":"Existence and multiplicity results for supercritical nonlocal Kirchhoff problem","authors":"G. Anello","doi":"10.58997/ejde.2023.14","DOIUrl":"https://doi.org/10.58997/ejde.2023.14","url":null,"abstract":"We study the existence and multiplicity of solutions for the nonlocalperturbed Kirchhoff problem$$displaylines{-Big(a+bint_Omega |nabla u|^2,dxBig)Delta u=lambda g(x,u)+f(x,u), quad text{in } Omega, u=0, quadtext{on }partialOmega,}$$ where Ω is a bounded smooth domain in (mathbb{R}^N), (N>4), (a,b, lambda > 0), and (f,g:Omegatimes mathbb{R}to mathbb{R}) are Caratheodory functions, with (f) subcritical, and (g) of arbitrary growth. This paper is motivated by a recent results by Faraci and Silva [4] where existence and multiplicity results were obtained when g is subcritical and f is a power-type function withcritical exponent.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48445767","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}
We consider a class of Schrodinger-Kirchhoff equations in R3 with a general nonlinearity g and coercive sign-changing potential V so that the Schrodinger operator -aΔ +V is indefinite. The nonlinearity considered here satisfies the Ambrosetti-Rabinowitz type condition g(t)t≥μ G(t)>0 with μ>3. We obtain the existence of nontrivial solutions for this problem via Morse theory.
{"title":"Existence of nontrivial solutions for Schrodinger-Kirchhoff equations with indefinite potentials","authors":"Shuai Jiang, Li Yin","doi":"10.58997/ejde.2023.13","DOIUrl":"https://doi.org/10.58997/ejde.2023.13","url":null,"abstract":"We consider a class of Schrodinger-Kirchhoff equations in R3 with a general nonlinearity g and coercive sign-changing potential V so that the Schrodinger operator -aΔ +V is indefinite. The nonlinearity considered here satisfies the Ambrosetti-Rabinowitz type condition g(t)t≥μ G(t)>0 with μ>3. We obtain the existence of nontrivial solutions for this problem via Morse theory.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43120922","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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