Abstract In this work, we consider the topological gradient method to deal with an inverse problem associated with three-dimensional Stokes equations. The problem consists in detecting unknown point forces acting on fluid from measurements on the boundary of the domain. We present an asymptotic expansion of the considered cost function using the topological sensitivity analysis method. A detection algorithm is then presented using the developed formula. Some numerical tests are presented to show the efficiency of the presented algorithm.
{"title":"On the topological gradient method for an inverse problem resolution","authors":"Mohamed Abdelwahed, Nejmeddine Chorfi","doi":"10.1515/anona-2023-0109","DOIUrl":"https://doi.org/10.1515/anona-2023-0109","url":null,"abstract":"Abstract In this work, we consider the topological gradient method to deal with an inverse problem associated with three-dimensional Stokes equations. The problem consists in detecting unknown point forces acting on fluid from measurements on the boundary of the domain. We present an asymptotic expansion of the considered cost function using the topological sensitivity analysis method. A detection algorithm is then presented using the developed formula. Some numerical tests are presented to show the efficiency of the presented algorithm.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136304599","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 We present a survey concerning the convergence, as the viscosity goes to zero, of the solutions to the three-dimensional evolutionary Navier-Stokes equations to solutions of the Euler equations. After considering the Cauchy problem, particular attention is given to the convergence under Navier slip-type boundary conditions. We show that, in the presence of flat boundaries (typically, the half-space case), convergence holds, uniformly in time, with respect to the initial data’s norm. In spite of this result (and of a similar result for arbitrary two-dimensional domains), strong inviscid limit results are proved to be false in general domains, in correspondence to a very large family of smooth initial data. In Section 6, we present a result in this direction.
{"title":"A survey on some vanishing viscosity limit results","authors":"H. Beirão da Veiga, F. Crispo","doi":"10.1515/anona-2022-0309","DOIUrl":"https://doi.org/10.1515/anona-2022-0309","url":null,"abstract":"Abstract We present a survey concerning the convergence, as the viscosity goes to zero, of the solutions to the three-dimensional evolutionary Navier-Stokes equations to solutions of the Euler equations. After considering the Cauchy problem, particular attention is given to the convergence under Navier slip-type boundary conditions. We show that, in the presence of flat boundaries (typically, the half-space case), convergence holds, uniformly in time, with respect to the initial data’s norm. In spite of this result (and of a similar result for arbitrary two-dimensional domains), strong inviscid limit results are proved to be false in general domains, in correspondence to a very large family of smooth initial data. In Section 6, we present a result in this direction.","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":"48168710","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 the present work, we establish a blow-up criterion for viscoelastic wave equations with nonlinear damping, logarithmic source, delay in the velocity, and acoustic boundary conditions. Due to the nonlinear damping term, we cannot apply the concavity method introduced by Levine. Thus, we use the energy method to show that the solution with negative initial energy blows up after finite time. Furthermore, we investigate the upper and lower bounds of the blow-up time.
{"title":"Blow-up for logarithmic viscoelastic equations with delay and acoustic boundary conditions","authors":"Sun‐Hye Park","doi":"10.1515/anona-2022-0310","DOIUrl":"https://doi.org/10.1515/anona-2022-0310","url":null,"abstract":"Abstract In the present work, we establish a blow-up criterion for viscoelastic wave equations with nonlinear damping, logarithmic source, delay in the velocity, and acoustic boundary conditions. Due to the nonlinear damping term, we cannot apply the concavity method introduced by Levine. Thus, we use the energy method to show that the solution with negative initial energy blows up after finite time. Furthermore, we investigate the upper and lower bounds of the blow-up time.","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":"46601371","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 study the asymptotic behaviour at infinity for viscosity solutions to a singular Monge-Ampère equation in half space from affine geometry. In particular, we extend the Liouville theorem for smooth solutions to the case of viscosity solutions by a completely different method from the smooth case.
{"title":"Generalized Liouville theorem for viscosity solutions to a singular Monge-Ampère equation","authors":"H. Jian, Xianduo Wang","doi":"10.1515/anona-2022-0284","DOIUrl":"https://doi.org/10.1515/anona-2022-0284","url":null,"abstract":"Abstract In this article, we study the asymptotic behaviour at infinity for viscosity solutions to a singular Monge-Ampère equation in half space from affine geometry. In particular, we extend the Liouville theorem for smooth solutions to the case of viscosity solutions by a completely different method from the smooth case.","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":"47584886","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 The article investigates a second-order nonlinear difference equation of Emden-Fowler type Δ2u(k)±kαum(k)=0, {Delta }^{2}uleft(k)pm {k}^{alpha }{u}^{m}left(k)=0, where k k is the independent variable with values k=k0,k0+1,… k={k}_{0},{k}_{0}+1,ldots hspace{0.33em} , u:{k0,k0+1,…}→R u:left{{k}_{0},{k}_{0}+1,ldots hspace{0.33em}right}to {mathbb{R}} is the dependent variable, k0 {k}_{0} is a fixed integer, and Δ2u(k) {Delta }^{2}uleft(k) is its second-order forward difference. New conditions with respect to parameters α∈R alpha in {mathbb{R}} and m∈R min {mathbb{R}} , m≠1 mne 1 , are found such that the equation admits a solution asymptotically repre
摘要研究了Emden-Fowler型二阶非线性差分方程Δ 2u (k)±k α u m (k)=0, {Delta ^}2u{}left (k) pm k{^ }{alpha u}{^}m{}left (k)=0，其中k k为自变量，其值为k=k 0,k 0+1，…k={k＿0},{k＿0}+1, {}{}ldots, hspace{0.33em}u:{ k 0,k 0+1，…}→R u。left ｛{k＿0},{k＿0}+1, {}{}ldotshspace{0.33em}right｝ to{mathbb{R}}为因变量，k 0 {k＿0}为固定整数，Δ 2u (k) {}{Delta ^}2u{}left (k)为其二阶正方差。关于参数α∈R alphain{mathbb{R}}和m∈R m in{mathbb{R}}, m≠1 m ne 1的新条件，使得方程的解渐近地表示为一个幂函数，该幂函数渐近地等价于非线性二阶微分Emden-Fowler方程y″(x)±x α ym (x) = 0的精确解。{Y} ^{^{primeprime}}left (x) pm x{^ }{alpha Y}{ ^}m{}left (x)=0。不仅给出了解本身的两项渐近表示，而且给出了解的一阶和二阶正差的两项渐近表示。讨论了以前已知的结果，并考虑了说明性的例子。
{"title":"Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type","authors":"Josef Diblík, Evgeniya Korobko","doi":"10.1515/anona-2023-0105","DOIUrl":"https://doi.org/10.1515/anona-2023-0105","url":null,"abstract":"Abstract The article investigates a second-order nonlinear difference equation of Emden-Fowler type <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"> <m:msup> <m:mrow> <m:mi mathvariant=\"normal\">Δ</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>k</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>±</m:mo> <m:msup> <m:mrow> <m:mi>k</m:mi> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msup> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>m</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>k</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> </m:math> {Delta }^{2}uleft(k)pm {k}^{alpha }{u}^{m}left(k)=0, where <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>k</m:mi> </m:math> k is the independent variable with values <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>k</m:mi> <m:mo>=</m:mo> <m:msub> <m:mrow> <m:mi>k</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>k</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:mn>1</m:mn> <m:mo form=\"prefix\">,</m:mo> <m:mrow> <m:mo>…</m:mo> </m:mrow> <m:mspace width=\"0.33em\" /> </m:math> k={k}_{0},{k}_{0}+1,ldots hspace{0.33em} , <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>u</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>k</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>k</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mrow> <m:mo>…</m:mo> </m:mrow> <m:mspace width=\"0.33em\" /> </m:mrow> <m:mo>}</m:mo> </m:mrow> <m:mo>→</m:mo> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:math> u:left{{k}_{0},{k}_{0}+1,ldots hspace{0.33em}right}to {mathbb{R}} is the dependent variable, <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mi>k</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> </m:math> {k}_{0} is a fixed integer, and <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi mathvariant=\"normal\">Δ</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>k</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {Delta }^{2}uleft(k) is its second-order forward difference. New conditions with respect to parameters <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>α</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:math> alpha in {mathbb{R}} and <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>m</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:math> min {mathbb{R}} , <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>m</m:mi> <m:mo>≠</m:mo> <m:mn>1</m:mn> </m:math> mne 1 , are found such that the equation admits a solution asymptotically repre","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136002854","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 The current article is concerned with the traveling fronts for a class of double degenerate equations with delay. We first show that the traveling fronts decay algebraically at one end, while those may decay exponentially or algebraically at the other end, which depend on the wave speed of traveling fronts. Based on the asymptotical behavior, the uniqueness and stability of traveling fronts are then proved. Of particular interest is the effect of the lower order term and higher order term on the critical speed. We mention that, under the double degenerate case, the nonlinear reaction is less competitive due to the appearance of degeneracy. This yields that the critical speed depends on the lower order term and higher order term, which is different from the nondegenerate case.
{"title":"Front propagation in a double degenerate equation with delay","authors":"Wei-Jian Bo, Shiliang Wu, Li-Jun Du","doi":"10.1515/anona-2022-0313","DOIUrl":"https://doi.org/10.1515/anona-2022-0313","url":null,"abstract":"Abstract The current article is concerned with the traveling fronts for a class of double degenerate equations with delay. We first show that the traveling fronts decay algebraically at one end, while those may decay exponentially or algebraically at the other end, which depend on the wave speed of traveling fronts. Based on the asymptotical behavior, the uniqueness and stability of traveling fronts are then proved. Of particular interest is the effect of the lower order term and higher order term on the critical speed. We mention that, under the double degenerate case, the nonlinear reaction is less competitive due to the appearance of degeneracy. This yields that the critical speed depends on the lower order term and higher order term, which is different from the nondegenerate case.","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":"44528231","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 formulate a model describing the evolution of thickness of a grounded shallow ice sheet. The thickness of the ice sheet is constrained to be nonnegative. This renders the problem under consideration an obstacle problem. A rigorous analysis shows that the model is thus governed by a set of variational inequalities that involve nonlinearities in the time derivative and in the elliptic term, and that it admits solutions, whose existence is established by means of a semi-discrete scheme and the penalty method.
{"title":"On the dynamics of grounded shallow ice sheets: Modeling and analysis","authors":"Paolo Piersanti, R. Temam","doi":"10.1515/anona-2022-0280","DOIUrl":"https://doi.org/10.1515/anona-2022-0280","url":null,"abstract":"Abstract In this article, we formulate a model describing the evolution of thickness of a grounded shallow ice sheet. The thickness of the ice sheet is constrained to be nonnegative. This renders the problem under consideration an obstacle problem. A rigorous analysis shows that the model is thus governed by a set of variational inequalities that involve nonlinearities in the time derivative and in the elliptic term, and that it admits solutions, whose existence is established by means of a semi-discrete scheme and the penalty method.","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":"47943948","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 the present article, we are concerned with the following problem: v t = Δ v + ∣ x ∣ β e v , x ∈ R N , t > 0 , v ( x , 0 ) = v 0 ( x ) , x ∈ R N , left{phantom{rule[-1.25em]{}{0ex}}begin{array}{ll}{v}_{t}=Delta v+| x{| }^{beta }{e}^{v},hspace{1.0em}& xin {{mathbb{R}}}^{N},hspace{0.33em}tgt 0, vleft(x,0)={v}_{0}left(x),hspace{1.0em}& xin {{mathbb{R}}}^{N},end{array}right. where N ≥ 3 Nge 3 , 0 < β < 2 0lt beta lt 2 , and v 0 {v}_{0} is a continuous function in R N {{mathbb{R}}}^{N} . We prove the existence and asymptotic behavior of forward self-similar solutions in the case where v 0 {v}_{0} decays at the rate − ( 2 + β ) log ∣ x ∣ -left(2+beta )log | x| as ∣ x ∣ → ∞ | x| to infty . Particularly, we obtain the optimal decay bound for initial value v 0 {v}_{0} .
摘要本文研究以下问题:v t = Δ v +∣x∣β e v, x∈rn, t > 0, v (x, 0) = v 0 (x)， x∈rn, left ｛phantom{rule[-1.25em]{}{0ex}}begin{array}{ll}{v}_{t}=Delta v+| x{| }^{beta }{e}^{v},hspace{1.0em}& xin {{mathbb{R}}}^{N},hspace{0.33em}tgt 0, vleft(x,0)={v}_{0}left(x),hspace{1.0em}& xin {{mathbb{R}}}^{N},end{array}right。其中N≥3n ge 3, 0 < β < 20 ltbetalt 2, {v0 }v＿0{是R N中的连续函数}{{mathbb{R}}} ^{N}。在v 0 {v＿0}衰减速率为- (2+ β) log∣x∣- {}left (2+ beta) log | x|为∣x∣→∞| x| toinfty的情况下，证明了前向自相似解的存在性和渐近性。特别地，我们得到了初始值{v0 }v＿0{的最优衰减界。}
{"title":"Existence and blow-up of solutions in Hénon-type heat equation with exponential nonlinearity","authors":"Dong-sheng Gao, Jun Wang, Xuan Wang","doi":"10.1515/anona-2022-0290","DOIUrl":"https://doi.org/10.1515/anona-2022-0290","url":null,"abstract":"Abstract In the present article, we are concerned with the following problem: v t = Δ v + ∣ x ∣ β e v , x ∈ R N , t > 0 , v ( x , 0 ) = v 0 ( x ) , x ∈ R N , left{phantom{rule[-1.25em]{}{0ex}}begin{array}{ll}{v}_{t}=Delta v+| x{| }^{beta }{e}^{v},hspace{1.0em}& xin {{mathbb{R}}}^{N},hspace{0.33em}tgt 0, vleft(x,0)={v}_{0}left(x),hspace{1.0em}& xin {{mathbb{R}}}^{N},end{array}right. where N ≥ 3 Nge 3 , 0 < β < 2 0lt beta lt 2 , and v 0 {v}_{0} is a continuous function in R N {{mathbb{R}}}^{N} . We prove the existence and asymptotic behavior of forward self-similar solutions in the case where v 0 {v}_{0} decays at the rate − ( 2 + β ) log ∣ x ∣ -left(2+beta )log | x| as ∣ x ∣ → ∞ | x| to infty . Particularly, we obtain the optimal decay bound for initial value v 0 {v}_{0} .","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":"44574952","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 We consider analytic families of planar vector fields depending analytically on the parameters in Λ Lambda that guarantee the existence of a (may be degenerate and with characteristic directions) monodromic singularity. We characterize the structure of the asymptotic Dulac series of the Poincaré map associated to the singularity when the family possesses a Puiseux inverse integrating factor in terms of its multiplicity and index. This characterization is only valid in a restricted monodromic parameter space Λ Λ ∗ Lambda backslash {Lambda }^{ast } associated to the nonexistence of local curves with zero angular speed. As a byproduct, we are able to study the center-focus problem (under the assumption of the existence of some Cauchy principal values) in very degenerated cases where no other tools are available. We illustrate the theory with several nontrivial examples.
{"title":"The Poincaré map of degenerate monodromic singularities with Puiseux inverse integrating factor","authors":"I. A. García, J. Giné","doi":"10.1515/anona-2022-0314","DOIUrl":"https://doi.org/10.1515/anona-2022-0314","url":null,"abstract":"Abstract We consider analytic families of planar vector fields depending analytically on the parameters in Λ Lambda that guarantee the existence of a (may be degenerate and with characteristic directions) monodromic singularity. We characterize the structure of the asymptotic Dulac series of the Poincaré map associated to the singularity when the family possesses a Puiseux inverse integrating factor in terms of its multiplicity and index. This characterization is only valid in a restricted monodromic parameter space Λ Λ ∗ Lambda backslash {Lambda }^{ast } associated to the nonexistence of local curves with zero angular speed. As a byproduct, we are able to study the center-focus problem (under the assumption of the existence of some Cauchy principal values) in very degenerated cases where no other tools are available. We illustrate the theory with several nontrivial examples.","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":"43206351","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 are concerned with normalized solutions for the p p -Laplacian equation with a trapping potential and L r {L}^{r} -supercritical growth, where r = p r=p or 2 . 2. The solutions correspond to critical points of the underlying energy functional subject to the L r {L}^{r} -norm constraint, namely, ∫ R N ∣ u ∣ r d x = c {int }_{{{mathbb{R}}}^{N}}| u{| }^{r}{rm{d}}x=c for given c > 0 . cgt 0. When r = p , r=p, we show that such problem has a ground state with positive energy for c c small enough. When r = 2 , r=2, we show that such problem has at least two solutions both with positive energy, which one is a ground state and the other one is a high-energy solution.
摘要在本文中，我们讨论了具有俘获势和Lr｛L｝^｛r｝-超临界生长的p-拉普拉斯方程的归一化解，其中r＝p r＝p或2。2.这些解对应于Lr｛L｝^｛r｝-范数约束下的潜在能量泛函主体的临界点，即，对于给定的c>0，当给定c>0时，ξr NÜuÜr d x=c。cgt 0。当r＝p，r＝p时，我们证明了对于c c足够小，这样的问题具有正能量的基态。当r＝2，r＝2时，我们证明了这类问题至少有两个解都是正能量的，一个是基态，另一个是高能解。
{"title":"Normalized solutions for the p-Laplacian equation with a trapping potential","authors":"Chao Wang, Juntao Sun","doi":"10.1515/anona-2022-0291","DOIUrl":"https://doi.org/10.1515/anona-2022-0291","url":null,"abstract":"Abstract In this article, we are concerned with normalized solutions for the p p -Laplacian equation with a trapping potential and L r {L}^{r} -supercritical growth, where r = p r=p or 2 . 2. The solutions correspond to critical points of the underlying energy functional subject to the L r {L}^{r} -norm constraint, namely, ∫ R N ∣ u ∣ r d x = c {int }_{{{mathbb{R}}}^{N}}| u{| }^{r}{rm{d}}x=c for given c > 0 . cgt 0. When r = p , r=p, we show that such problem has a ground state with positive energy for c c small enough. When r = 2 , r=2, we show that such problem has at least two solutions both with positive energy, which one is a ground state and the other one is a high-energy solution.","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":"43212309","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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