In this paper we prove two new abstract compactness criteria in normed spaces. To this end we first introduce the notion of an equinormed set using a suitable family of semi-norms on the given normed space satisfying some natural conditions. Those conditions, roughly speaking, state that the norm can be approximated (on the equinormed sets even uniformly) by the elements of this family. As we are given some freedom of choice of the underlying semi-normed structure that is used to define equinormed sets, our approach opens a new perspective for building compactness criteria in specific normed spaces. As an example we show that natural selections of families of semi-norms in spaces $C(X,R)$ and $l^p$ for $pin[1,+infty)$ lead to the well-known compactness criteria (including the Arzel`a-Ascoli theorem). In the second part of the paper, applying the abstract theorems, we construct a simple compactness criterion in the space of functions of bounded Schramm variation.
{"title":"Compactness in normed spaces: a unified approach through semi-norms","authors":"Jacek Gulgowski, Piotr Kasprzak, Piotr Maćkowiak","doi":"10.12775/tmna.2022.064","DOIUrl":"https://doi.org/10.12775/tmna.2022.064","url":null,"abstract":"In this paper we prove two new abstract compactness criteria in normed spaces. To this end we first introduce the notion of an equinormed set using a suitable family of semi-norms on the given normed space satisfying some natural conditions. Those conditions, roughly speaking, state that the norm can be approximated (on the equinormed sets even uniformly) by the elements of this family. As we are given some freedom of choice of the underlying semi-normed structure that is used to define equinormed sets, our approach opens a new perspective for building compactness criteria in specific normed spaces. As an example we show that natural selections of families of semi-norms in spaces $C(X,R)$ and $l^p$ for $pin[1,+infty)$ lead to the well-known compactness criteria (including the Arzel`a-Ascoli theorem). In the second part of the paper, applying the abstract theorems, we construct a simple compactness criterion in the space of functions of bounded Schramm variation.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":"2014 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135959832","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}
Thi Thu Huong Nguyen, Dao Trong Quyet, Thi Hien Anh Vu
In this paper, we study the Kirchhoff elliptic equations of the form $$ -M(|nabla_lambda u|^2)Delta_lambda u=w(x)f(u) quad mbox{in }mathbb R^{N}, $$ where $M$ is a smooth monotone function, $w$ is a weight function and $f(u)$ is of the form $u^p, e^u$ or $-u^{-p}$. The operator $Delta_lambda$ is strongly degenerate and given by $$ Delta_lambda=sum_{j=1}^N frac{partial}{partial x_j}bigg(lambda_j^2(x)frac{partial }{partial x_j}bigg). $$ We shall prove some classifications of stable solutions to the equation above under general assumptions on $M$ and $lambda_j$, $j=1,ldots,N$.
{"title":"Liouville type theorems for Kirchhoff sub-elliptic equations involving $Delta_lambda$-operators","authors":"Thi Thu Huong Nguyen, Dao Trong Quyet, Thi Hien Anh Vu","doi":"10.12775/tmna.2022.071","DOIUrl":"https://doi.org/10.12775/tmna.2022.071","url":null,"abstract":"In this paper, we study the Kirchhoff elliptic equations of the form $$ -M(|nabla_lambda u|^2)Delta_lambda u=w(x)f(u) quad mbox{in }mathbb R^{N}, $$ where $M$ is a smooth monotone function, $w$ is a weight function and $f(u)$ is of the form $u^p, e^u$ or $-u^{-p}$. The operator $Delta_lambda$ is strongly degenerate and given by $$ Delta_lambda=sum_{j=1}^N frac{partial}{partial x_j}bigg(lambda_j^2(x)frac{partial }{partial x_j}bigg). $$ We shall prove some classifications of stable solutions to the equation above under general assumptions on $M$ and $lambda_j$, $j=1,ldots,N$.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135966916","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 paper we study the existence of solutions to the following generalized nonlinear two-parameter problem begin{equation*} a(u, v) = lambda b(u, v) + mu m(u, v) + varepsilon F(u, v), end{equation*} for a triple $(a, b, m)$ of continuous, symmetric bilinear forms on a real separable Hilbert space $V$ and nonlinear form $F$. This problem is a natural abstraction of nonlinear problems that occur for a large class of differential operators, various elliptic pde's with nonlinearities in either the differential equation and/or the boundary conditions being a special subclass. First, a Fredholm alternative for the associated linear two-parameter eigenvalue problem is developed, and then this is used to construct a nonlinear version of the Fredholm alternative. Lastly, the Steklov-Robin Fredholm equation is used to exemplify the abstract results.
{"title":"A Fredholm alternative for elliptic equations with interior and boundary nonlinear reactions","authors":"Daniel Maroncelli, Mauricio A. Rivas","doi":"10.12775/tmna.2022.054","DOIUrl":"https://doi.org/10.12775/tmna.2022.054","url":null,"abstract":"In this paper we study the existence of solutions to the following generalized nonlinear two-parameter problem begin{equation*} a(u, v) = lambda b(u, v) + mu m(u, v) + varepsilon F(u, v), end{equation*} for a triple $(a, b, m)$ of continuous, symmetric bilinear forms on a real separable Hilbert space $V$ and nonlinear form $F$. This problem is a natural abstraction of nonlinear problems that occur for a large class of differential operators, various elliptic pde's with nonlinearities in either the differential equation and/or the boundary conditions being a special subclass. First, a Fredholm alternative for the associated linear two-parameter eigenvalue problem is developed, and then this is used to construct a nonlinear version of the Fredholm alternative. Lastly, the Steklov-Robin Fredholm equation is used to exemplify the abstract results.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":"2014 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135966917","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 work, we consider an activator-inhibitor system, known as the Gierer-Meinhardt model. Using the linear stability analysis at the unique positive equilibrium, we derive the conditions of the Hopf bifurcation. We compute the normal form of this bifurcation up to the third degree and obtain the direction of the Hopf bifurcation. Finally, we provide numerical simulations to illustrate the theoretical results of this paper. In this study, we will use the technique of normal form and center manifold theorem.
{"title":"Analysis of the Hopf bifurcation in a Diffusive Gierer-Meinhardt Model","authors":"Rasoul Asheghi","doi":"10.12775/tmna.2022.050","DOIUrl":"https://doi.org/10.12775/tmna.2022.050","url":null,"abstract":"In this work, we consider an activator-inhibitor system, known as the Gierer-Meinhardt model. Using the linear stability analysis at the unique positive equilibrium, we derive the conditions of the Hopf bifurcation. We compute the normal form of this bifurcation up to the third degree and obtain the direction of the Hopf bifurcation. Finally, we provide numerical simulations to illustrate the theoretical results of this paper. In this study, we will use the technique of normal form and center manifold theorem.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135966918","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 2010, M. Sakai and the first author showed that the topological complexity of a space $X$ coincides with the fibrewise unpointed L-S category of a pointed fibrewise space $proj_{1} colon X times X to X$ with the diagonal map $Delta colon X to X times X$ as its section. In this paper, we describe our algorithm how to determine the fibrewise L-S category or the Topological Complexity of a topological spherical space form. Especially, for $S^3/Q_8$ where $Q_8$ is the quaternion group, we write a python code to realise the algorithm to determine its Topological Complexity.
{"title":"Topological complexity of $S^3/Q_8$ as fibrewise L-S category","authors":"Norio Iwase, Yuya Miyata","doi":"10.12775/tmna.2022.068","DOIUrl":"https://doi.org/10.12775/tmna.2022.068","url":null,"abstract":"In 2010, M. Sakai and the first author showed that the topological complexity of a space $X$ coincides with the fibrewise unpointed L-S category of a pointed fibrewise space $proj_{1} colon X times X to X$ with the diagonal map $Delta colon X to X times X$ as its section. In this paper, we describe our algorithm how to determine the fibrewise L-S category or the Topological Complexity of a topological spherical space form. Especially, for $S^3/Q_8$ where $Q_8$ is the quaternion group, we write a python code to realise the algorithm to determine its Topological Complexity.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135966915","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 paper, a class of degenerative quasilinear Schrödinger equations with vanishing potentials and critical Sobolev exponents is considered. The main operator involved in these equations is not strictly elliptic. Under suitable conditions, the existence of nontrivial solutions to the equations is obtained by employing variational methods and the decay rate of the solutions is established.
{"title":"Solutions to degenerative generalized quasilinear Schrödinger equations involving vanishing potentials and critical exponent","authors":"Yong Huang, Zhouxin Li, X. Yuan","doi":"10.12775/tmna.2022.045","DOIUrl":"https://doi.org/10.12775/tmna.2022.045","url":null,"abstract":"In this paper, a class of degenerative quasilinear Schrödinger equations with\u0000 vanishing potentials and critical Sobolev exponents is considered.\u0000The main operator involved in these equations is not strictly elliptic. Under suitable conditions, the existence of nontrivial solutions to the equations is obtained\u0000by employing variational methods and the decay rate of the solutions is established.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47537449","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 study of the boundary value problem with arbitrary measurable data originated in the dissertation of Luzin where he investigated the Dirichlet problem for harmonic functions in the unit disk. Recently, in cite{R7}, we studied the Hilbert, Poincaré and Neumann boundary value problems with arbitrary measurable data for generalized analytic and generalized harmonic functions and provided applications to relevant problems in mathematical physics. The present paper is devoted to the study of the boundary value problem with arbitrary measurable boundary data in a domain with rectifiable boundary corresponding to semi-linear equation with suitable nonlinear source. We construct a completely continuous operator and generate nonclassical solutions to the Hilbert and Poincaré boundary value problems with arbitrary measurable data for Vekua type and Poisson equations, respectively. Based on that, we prove the existence of solutions of the Hilbert boundary value problem for the nonlinear Vekua type equation with arbitrary measurable data in a domain with rectifiable boundary. It is necessary to point out that our approach differs from the classical variational approach in PDE as it is based on the geometric interpretation of boundary values as angular (along non-tangential paths) limits. The latter makes it possible to also obtain a theorem on the boundary value problem for directional derivatives, and, in particular, of the Neumann problem with arbitrary measurable data for the Poisson equation with nonlinear sources in any Jordan domain with rectifiable boundary. As a result we arrive at applications to some problems of mathematical physics.
{"title":"Hilbert and Poincaré problems for semi-linear equations in rectifiable domains","authors":"V. Ryazanov","doi":"10.12775/tmna.2022.044","DOIUrl":"https://doi.org/10.12775/tmna.2022.044","url":null,"abstract":"The study of the boundary value problem with arbitrary\u0000measurable data originated in the dissertation of Luzin where\u0000he investigated the Dirichlet problem for harmonic functions in the unit\u0000disk.\u0000Recently, in cite{R7}, we studied the Hilbert, Poincaré and Neumann\u0000boundary value problems with arbitrary measurable data for\u0000generalized analytic and generalized harmonic functions and provided\u0000applications to relevant problems in mathematical physics.\u0000The present paper is devoted to the study of the boundary value\u0000problem with arbitrary measurable boundary data in a domain with\u0000rectifiable boundary corresponding to semi-linear equation with\u0000suitable nonlinear source. We construct a completely continuous\u0000operator and generate nonclassical solutions to the Hilbert and\u0000Poincaré boundary value problems with arbitrary measurable data for\u0000Vekua type and Poisson equations, respectively. Based on that, we\u0000prove the existence of solutions of the Hilbert boundary value\u0000problem for the nonlinear Vekua type equation with arbitrary\u0000measurable data in a domain with rectifiable boundary.\u0000It is necessary to point out that our approach differs from the\u0000classical variational approach in PDE as it is based on the\u0000geometric interpretation of boundary values as angular (along\u0000non-tangential paths) limits.\u0000The latter makes it possible to also obtain a theorem on the\u0000boundary value problem for directional derivatives,\u0000 and, in\u0000particular, of the Neumann problem with arbitrary measurable\u0000data for the Poisson equation with nonlinear sources in any Jordan\u0000domain with rectifiable boundary.\u0000As a result we arrive at applications to some problems of\u0000mathematical physics.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47244046","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 analyse an $alpha$-th order, $1 < alpha leq 2$, nabla fractional three-point boundary value problem (BVP). We construct the Green's function associated to this problem and derive a few of its important properties. We then establish sufficient conditions on existence and uniqueness of solutions for the corresponding nonlinear BVP using the modern ideas of continuation methods for contractive maps. Our results extend recent results on nabla fractional BVPs. Finally, we provide an example to illustrate the applicability of main results.
{"title":"Existence theory for nabla fractional three-point boundary value problems via continuation methods for contractive maps","authors":"J. Jonnalagadda","doi":"10.12775/tmna.2022.043","DOIUrl":"https://doi.org/10.12775/tmna.2022.043","url":null,"abstract":"In this article, we analyse an $alpha$-th order, $1 < alpha leq 2$, nabla fractional\u0000 three-point boundary value problem (BVP). We construct the Green's function\u0000 associated to this problem and derive a few of its important properties.\u0000We then establish sufficient conditions on existence and uniqueness of solutions\u0000for the corresponding nonlinear BVP using the modern ideas of continuation methods\u0000 for contractive maps. Our results extend recent results on nabla fractional BVPs. Finally, we provide an example to illustrate the applicability of main results.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46207326","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 main goal of the present paper is to prove the existence of saddle-type solutions for the following class of quasilinear problems $$ -Delta_{Phi}u + V'(u)=0quad text{in }mathbb{R}^2, $$% where $$ Delta_{Phi}u=text{div}(phi(|nabla u|)nabla u), $$% $Phicolon mathbb{R}rightarrow [0,+infty)$ is an N-function and the potential $V$ satisfies some technical condition and we have as an example $ V(t)=Phi(|t^2-1|)$.
{"title":"Existence of saddle-type solutions for a class of quasilinear problems in R^2","authors":"C. O. Alves, Renan J. S. Isneri, P. Montecchiari","doi":"10.12775/tmna.2022.039","DOIUrl":"https://doi.org/10.12775/tmna.2022.039","url":null,"abstract":"The main goal of the present paper is to prove the existence of saddle-type solutions for the following class of quasilinear problems\u0000$$\u0000-Delta_{Phi}u + V'(u)=0quad text{in }mathbb{R}^2,\u0000$$%\u0000where\u0000$$\u0000Delta_{Phi}u=text{div}(phi(|nabla u|)nabla u),\u0000$$%\u0000$Phicolon mathbb{R}rightarrow [0,+infty)$ is an N-function\u0000and the potential $V$ satisfies some technical condition and we have\u0000as an example $ V(t)=Phi(|t^2-1|)$.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43840733","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 work, we are concerned with the existence of a ground state solution for a Kirchhoff weighted problem under boundary Dirichlet condition in the unit ball of $mathbb{R}^{4}$. The nonlinearities have critical growth in view of Adams' inequalities. To prove the existence result, we use Pass Mountain Theorem. The main difficulty is the loss of compactness due to the critical exponential growth of the nonlinear term $f$. The associated energy function does not satisfy the condition of compactness. We provide a new condition for growth and we stress its importance to check the min-max compactness level.
{"title":"Weighted fourth order equation of Kirchhoff type in dimension 4 with non-linear exponential growth","authors":"Rached Jaidane","doi":"10.12775/tmna.2023.005","DOIUrl":"https://doi.org/10.12775/tmna.2023.005","url":null,"abstract":"In this work, we are concerned with the existence of a ground state solution\u0000 for a Kirchhoff weighted problem under boundary Dirichlet condition\u0000 in the unit ball of $mathbb{R}^{4}$.\u0000 The nonlinearities have critical growth in view of Adams'\u0000 inequalities. To prove the existence result, we use Pass Mountain Theorem.\u0000The main difficulty is\u0000the loss of compactness due to the critical exponential growth of the nonlinear\u0000term $f$. The associated energy function does not satisfy\u0000 the condition of compactness. We provide a new condition for growth and we stress its importance\u0000 to check the min-max compactness level.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46910351","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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