In this article, we study the generalized Kadomtsev-Petviashvili equation witha potential $$ (-u_{xx}+D_{x}^{-2}u_{yy}+V(varepsilon x,varepsilon y)u-f(u))_{x}=0quad text{in }mathbb{R}^2, $$ where (D_{x}^{-2}h(x,y)=int_{-infty }^{x}int_{-infty }^{t}h(s,y),ds,dt ), (f) is a nonlinearity, (varepsilon) is a small positive parameter, and the potential (V) satisfies a local condition. We prove the existence of nontrivial solitary waves for the modified problem by applying penalization techniques. The relationship between the number of positive solutions and the topology of the set where (V) attains its minimum is obtained by using Ljusternik-Schnirelmann theory. With the help of Moser's iteration method, we verify that the solutions of the modified problem are indeed solutions of the original roblem for (varepsilon>0) small enough.
{"title":"Multiplicity of solutions for a generalized Kadomtsev-Petviashvili equation with potential in R^2","authors":"Zheng Xie, Jing Chen","doi":"10.58997/ejde.2023.48","DOIUrl":"https://doi.org/10.58997/ejde.2023.48","url":null,"abstract":"In this article, we study the generalized Kadomtsev-Petviashvili equation witha potential $$ (-u_{xx}+D_{x}^{-2}u_{yy}+V(varepsilon x,varepsilon y)u-f(u))_{x}=0quad text{in }mathbb{R}^2, $$ where (D_{x}^{-2}h(x,y)=int_{-infty }^{x}int_{-infty }^{t}h(s,y),ds,dt ), (f) is a nonlinearity, (varepsilon) is a small positive parameter, and the potential (V) satisfies a local condition. We prove the existence of nontrivial solitary waves for the modified problem by applying penalization techniques. The relationship between the number of positive solutions and the topology of the set where (V) attains its minimum is obtained by using Ljusternik-Schnirelmann theory. With the help of Moser's iteration method, we verify that the solutions of the modified problem are indeed solutions of the original roblem for (varepsilon>0) small enough.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45875334","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}
by fractional noise. Existence and uniqueness of a mild solution is given bya fixed point argument. Then, we explore Holder regularity of the mildsolution in (C([0,T_{*}];L^p(Omega;dot{H}^{gamma}))) for some stoppingtime (T_{*}).
通过分数噪声。一个不动点论证给出了一个温和解的存在唯一性。然后，我们在一些停止时间（T_。
{"title":"Stochastic Burgers equations with fractional derivative driven by fractional noise","authors":"Yubo Duan, Yiming Jiang, Yang Tian, Yawei Wei","doi":"10.58997/ejde.2023.49","DOIUrl":"https://doi.org/10.58997/ejde.2023.49","url":null,"abstract":"by fractional noise. Existence and uniqueness of a mild solution is given bya fixed point argument. Then, we explore Holder regularity of the mildsolution in (C([0,T_{*}];L^p(Omega;dot{H}^{gamma}))) for some stoppingtime (T_{*}).","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47524717","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}
Francisco Odair de Paiva, Sandra Machado de Souza Lima, O. Miyagaki
We consider the elliptic problem $$ - Delta_A u + u = a_{lambda}(x) |u|^{q-2}u+b_{mu}(x) |u|^{p-2}u , $$ for (x in mathbb{R}^N), ( 1 < q < 2 < p < 2^*= 2N/(N-2)), (a_{lambda}(x)) is a sign-changing weight function, (b_{mu}(x)) satisfies some additional conditions, (u in H^1_A(mathbb{R}^N)) and (A:mathbb{R}^N to mathbb{R}^N) is a magnetic potential. Exploring the Bahri-Li argument and some preliminary results we will discuss the existence of a four nontrivial solutions to the problem in question.
{"title":"Existence of at least four solutions for Schrodinger equations with magnetic potential involving and sign-changing weight function","authors":"Francisco Odair de Paiva, Sandra Machado de Souza Lima, O. Miyagaki","doi":"10.58997/ejde.2023.47","DOIUrl":"https://doi.org/10.58997/ejde.2023.47","url":null,"abstract":"We consider the elliptic problem $$ - Delta_A u + u = a_{lambda}(x) |u|^{q-2}u+b_{mu}(x) |u|^{p-2}u , $$ for (x in mathbb{R}^N), ( 1 < q < 2 < p < 2^*= 2N/(N-2)), (a_{lambda}(x)) is a sign-changing weight function, (b_{mu}(x)) satisfies some additional conditions, (u in H^1_A(mathbb{R}^N)) and (A:mathbb{R}^N to mathbb{R}^N) is a magnetic potential. Exploring the Bahri-Li argument and some preliminary results we will discuss the existence of a four nontrivial solutions to the problem in question.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46765733","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 blow up behavior for the Henon type parabolic equation withexponential nonlinearity, $$ u_t=Delta u+|x|^{sigma}e^uquad text{in } B_Rtimes mathbb{R}_+, $$ where (sigmageq 0) and (B_R={xinmathbb{R}^N: |x|10+4sigma) and (N=10+4sigma), the asymptotic expansions of stationary solutions have different forms, so two cases are discussed separately. Moreover, different inner region widths in two cases are also obtained.
{"title":"Asymptotic behavior of blowup solutions for Henon type parabolic equations with exponential nonlinearity","authors":"Caihong Chang, Zhengce Zhang","doi":"10.58997/ejde.2022.42","DOIUrl":"https://doi.org/10.58997/ejde.2022.42","url":null,"abstract":"This article concerns the blow up behavior for the Henon type parabolic equation withexponential nonlinearity, $$ u_t=Delta u+|x|^{sigma}e^uquad text{in } B_Rtimes mathbb{R}_+, $$ where (sigmageq 0) and (B_R={xinmathbb{R}^N: |x|<R}).We consider all cases in which blowup of solutions occurs, i.e. (Ngeq 10+4sigma).Grow up rates are established by a certain matching of different asymptotic behaviorsin the inner region (near the singularity) and the outer region (close to the boundary).For the cases (N>10+4sigma) and (N=10+4sigma), the asymptotic expansions of stationary solutions have different forms, so two cases are discussed separately. Moreover, different inner region widths in two cases are also obtained.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48470912","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}
Fariba Gharehgazlouei, J. Graef, S. Heidarkhani, L. Kong
In this article, the authors consider a fractional p-Laplacian elliptic Dirichlet problem. Using critical point theory and the variational method, they investigate the existence of at least one, two, and three solutions to the problem. Examples illustrating the results are interspaced in the paper.
{"title":"Existence and multiplicity of solutions to a fractional p-Laplacian elliptic Dirichlet problem","authors":"Fariba Gharehgazlouei, J. Graef, S. Heidarkhani, L. Kong","doi":"10.58997/ejde.2023.46","DOIUrl":"https://doi.org/10.58997/ejde.2023.46","url":null,"abstract":"In this article, the authors consider a fractional p-Laplacian elliptic Dirichlet problem. Using critical point theory and the variational method, they investigate the existence of at least one, two, and three solutions to the problem. Examples illustrating the results are interspaced in the paper.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47168386","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}
K. Vidhyaa, E. Thandapani, J. Alzabut, Abdullah Ozbekler
We obtain oscillation conditions for non-canonical second-order nonlinear delay difference equations with a superlinear neutral term. To cope with non-canonical types of equations, we propose new oscillation criteria for the main equation when the neutral coefficient does not satisfy any of the conditions that call it to either converge to (0) or (infty). Our approach differs from others in that we first turn into the non-canonical equation to a canonical form and as a result, we only require one condition to weed out non-oscillatory solutions in order to induce oscillation. The conclusions made here are new and have been condensed significantly from those found in the literature. For the sake of confirmation, we provide examplesthat cannot be included in earlier works.
{"title":"Oscillation criteria for non-canonical second-order nonlinear delay difference equations with a superlinear neutral term","authors":"K. Vidhyaa, E. Thandapani, J. Alzabut, Abdullah Ozbekler","doi":"10.58997/ejde.2023.45","DOIUrl":"https://doi.org/10.58997/ejde.2023.45","url":null,"abstract":"We obtain oscillation conditions for non-canonical second-order nonlinear delay difference equations with a superlinear neutral term. To cope with non-canonical types of equations, we propose new oscillation criteria for the main equation when the neutral coefficient does not satisfy any of the conditions that call it to either converge to (0) or (infty). Our approach differs from others in that we first turn into the non-canonical equation to a canonical form and as a result, we only require one condition to weed out non-oscillatory solutions in order to induce oscillation. The conclusions made here are new and have been condensed significantly from those found in the literature. For the sake of confirmation, we provide examplesthat cannot be included in earlier works.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45381927","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 stability of a porous thermoelastic system with Gurtin-Pipkin flux. Under suitable assumptions for the derivative of the heat flux relaxation kernel, we establish the existence and uniqueness of solution by applying the semigroup theory, and prove the exponential stability of system without considering the wave velocity by the means of estimates of the resolvent
{"title":"Exponential stability for porous thermoelastic systems with Gurtin-Pipkin flux","authors":"Jianghao Hao, Jing Yang","doi":"10.58997/ejde.2023.44","DOIUrl":"https://doi.org/10.58997/ejde.2023.44","url":null,"abstract":"In this article, we study the stability of a porous thermoelastic system with Gurtin-Pipkin flux. Under suitable assumptions for the derivative of the heat flux relaxation kernel, we establish the existence and uniqueness of solution by applying the semigroup theory, and prove the exponential stability of system without considering the wave velocity by the means of estimates of the resolvent","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49104401","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 is devoted to exploring the existence and the form of finite order transcendental entire solutions of Fermat-type second order partial differential-difference equations $$ Big(frac{partial^2 f}{partial z_1^2}+deltafrac{partial^2 f}{partial z_2^2} +etafrac{partial^2 f}{partial z_1partial z_2}Big)^2 +f(z_1+c_1,z_2+c_2)^2=e^{g(z_1,z_2)} $$ and $$ Big(frac{partial^2 f}{partial z_1^2}+deltafrac{partial^2 f}{partial z_2^2} +etafrac{partial^2 f}{partial z_1partial z_2}Big)^2+(f(z_1+c_1,z_2+c_2) -f(z_1,z_2))^2=e^{g(z)}, $$ where (delta,etainmathbb{C}) and (g(z_1,z_2)) is a polynomial in (mathbb{C}^2). Our results improve the results of Liu and Dong [23] Liu et al. [24] and Liu and Yang [25] Several examples confirm that the form of tr
本文探讨了费马型二阶偏微分-差分方程$$ Big(frac{partial^2 f}{partial z_1^2}+deltafrac{partial^2 f}{partial z_2^2} +etafrac{partial^2 f}{partial z_1partial z_2}Big)^2 +f(z_1+c_1,z_2+c_2)^2=e^{g(z_1,z_2)} $$和$$ Big(frac{partial^2 f}{partial z_1^2}+deltafrac{partial^2 f}{partial z_2^2} +etafrac{partial^2 f}{partial z_1partial z_2}Big)^2+(f(z_1+c_1,z_2+c_2) -f(z_1,z_2))^2=e^{g(z)}, $$的有限阶超越全解的存在性和形式，其中(delta,etainmathbb{C})和(g(z_1,z_2))是(mathbb{C}^2)中的一个多项式。我们的结果改进了Liu and Dong [23] Liu et al.[24]和Liu and Yang[25]的结果
{"title":"Solutions of complex nonlinear functional equations including second order partial differential and difference in C^2","authors":"H. Xu, Goutam Haldar","doi":"10.58997/ejde.2023.43","DOIUrl":"https://doi.org/10.58997/ejde.2023.43","url":null,"abstract":"This article is devoted to exploring the existence and the form of finite order transcendental entire solutions of Fermat-type second order partial differential-difference equations $$ Big(frac{partial^2 f}{partial z_1^2}+deltafrac{partial^2 f}{partial z_2^2} +etafrac{partial^2 f}{partial z_1partial z_2}Big)^2 +f(z_1+c_1,z_2+c_2)^2=e^{g(z_1,z_2)} $$ and $$ Big(frac{partial^2 f}{partial z_1^2}+deltafrac{partial^2 f}{partial z_2^2} +etafrac{partial^2 f}{partial z_1partial z_2}Big)^2+(f(z_1+c_1,z_2+c_2) -f(z_1,z_2))^2=e^{g(z)}, $$ where (delta,etainmathbb{C}) and (g(z_1,z_2)) is a polynomial in (mathbb{C}^2). Our results improve the results of Liu and Dong [23] Liu et al. [24] and Liu and Yang [25] Several examples confirm that the form of tr","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41530039","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 establish the existence and uniqueness of viscosity solutions tothe Dirichlet problem $$displaylines{ Delta_infty^h u=f(x,u), quad hbox{in } Omega,cr u=q, quadhbox{on }partialOmega,}$$ where (qin C(partialOmega)), (h>1), (Delta_infty^h u=|Du|^{h-3}Delta_infty u). The operator (Delta_infty u=langle D^2uDu,Du rangle) is the infinity Laplacian which is strongly degenerate, quasilinear and it is associated with the absolutely minimizing Lipschitz extension. When the nonhomogeneous term (f(x,t)) is non-decreasing in (t), we prove the existence of the viscosity solution via Perron's method. We also establish a uniqueness result based on the perturbation analysis of the viscosity solutions. If the function (f(x,t)) is nonpositive (nonnegative) and non-increasing in (t), we also give the existence of viscosity solutions by an iteration technique under the condition that the domain has small diameter. Furthermore, we investigate the existence and uniqueness of viscosity solutions to the boundary-value problem with singularity $$displaylines{ Delta_infty^h u=-b(x)g(u), quad hbox{in } Omega, cr u>0, quad hbox{in } Omega, cr u=0, quad hbox{on }partialOmega, }$$ when the domain satisfies some regular condition. We analyze asymptotic estimates for the viscosity solution near the boundary.
{"title":"Viscosity solutions to the infinity Laplacian equation with lower terms","authors":"Cuicui Li, Fang Liu","doi":"10.58997/ejde.2023.42","DOIUrl":"https://doi.org/10.58997/ejde.2023.42","url":null,"abstract":"We establish the existence and uniqueness of viscosity solutions tothe Dirichlet problem $$displaylines{ Delta_infty^h u=f(x,u), quad hbox{in } Omega,cr u=q, quadhbox{on }partialOmega,}$$ where (qin C(partialOmega)), (h>1), (Delta_infty^h u=|Du|^{h-3}Delta_infty u). The operator (Delta_infty u=langle D^2uDu,Du rangle) is the infinity Laplacian which is strongly degenerate, quasilinear and it is associated with the absolutely minimizing Lipschitz extension. When the nonhomogeneous term (f(x,t)) is non-decreasing in (t), we prove the existence of the viscosity solution via Perron's method. We also establish a uniqueness result based on the perturbation analysis of the viscosity solutions. If the function (f(x,t)) is nonpositive (nonnegative) and non-increasing in (t), we also give the existence of viscosity solutions by an iteration technique under the condition that the domain has small diameter. Furthermore, we investigate the existence and uniqueness of viscosity solutions to the boundary-value problem with singularity $$displaylines{ Delta_infty^h u=-b(x)g(u), quad hbox{in } Omega, cr u>0, quad hbox{in } Omega, cr u=0, quad hbox{on }partialOmega, }$$ when the domain satisfies some regular condition. We analyze asymptotic estimates for the viscosity solution near the boundary.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42021349","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 investigate the space-time decay rates of strong solution to a two-phase flow model with magnetic field in the whole space (mathbb{R}^3 ). Based on the temporal decay results by Xiao [24] we show that for any integer (ellgeq 3), the space-time decay rate of (k(0leq k leq ell))-order spatial derivative of the strong solution in the weighted Lebesgue space ( L_gamma^2 ) is (t^{-frac{3}{4}-frac{k}{2}+gamma}). Moreover, we prove that the space-time decay rate of (k(0leq k leq ell-2))-order spatial derivative of the difference between two velocities of the fluid in the weighted Lebesgue space ( L_gamma^2 ) is (t^{-frac{5}{4}-frac{k}{2}+gamma}), which is faster than ones of the two velocities themselves.
研究了具有磁场的两相流模型在全空间内强溶液的时空衰减率(mathbb{R}^3 )。基于Xiao[24]的时间衰减结果，我们证明了对于任意整数(ellgeq 3)，在加权勒贝格空间( L_gamma^2 )中强解的(k(0leq k leq ell))阶空间导数的时空衰减率为(t^{-frac{3}{4}-frac{k}{2}+gamma})。此外，我们还证明了在加权勒贝格空间( L_gamma^2 )中流体两种速度之差的(k(0leq k leq ell-2)) -阶空间导数的时空衰减速率为(t^{-frac{5}{4}-frac{k}{2}+gamma})，比两种速度本身的衰减速率更快。
{"title":"Space-time decay rates of a two-phase flow model with magnetic field in R^3","authors":"Qin Ye, Yinghui Zhang","doi":"10.58997/ejde.2023.41","DOIUrl":"https://doi.org/10.58997/ejde.2023.41","url":null,"abstract":"We investigate the space-time decay rates of strong solution to a two-phase flow model with magnetic field in the whole space (mathbb{R}^3 ). Based on the temporal decay results by Xiao [24] we show that for any integer (ellgeq 3), the space-time decay rate of (k(0leq k leq ell))-order spatial derivative of the strong solution in the weighted Lebesgue space ( L_gamma^2 ) is (t^{-frac{3}{4}-frac{k}{2}+gamma}). Moreover, we prove that the space-time decay rate of (k(0leq k leq ell-2))-order spatial derivative of the difference between two velocities of the fluid in the weighted Lebesgue space ( L_gamma^2 ) is (t^{-frac{5}{4}-frac{k}{2}+gamma}), which is faster than ones of the two velocities themselves.","PeriodicalId":49213,"journal":{"name":"Electronic Journal of Differential Equations","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41782775","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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