Abstract We consider the asymptotic behavior of solutions to the Monge-Ampère equations with slow convergence rate at infinity and fulfill previous results under faster convergence rate by Bao et al. [Monge-Ampère equation on exterior domains, Calc. Var PDE. 52 (2015), 39–63]. Different from known results, we obtain the limit of Hessian and/or gradient of solution at infinity relying on the convergence rate. The basic idea is to use a revised level set method, the spherical harmonic expansion, and the iteration method.
{"title":"Asymptotic behavior of solutions to the Monge-Ampère equations with slow convergence rate at infinity","authors":"Zixiao Liu, J. Bao","doi":"10.1515/ans-2022-0052","DOIUrl":"https://doi.org/10.1515/ans-2022-0052","url":null,"abstract":"Abstract We consider the asymptotic behavior of solutions to the Monge-Ampère equations with slow convergence rate at infinity and fulfill previous results under faster convergence rate by Bao et al. [Monge-Ampère equation on exterior domains, Calc. Var PDE. 52 (2015), 39–63]. Different from known results, we obtain the limit of Hessian and/or gradient of solution at infinity relying on the convergence rate. The basic idea is to use a revised level set method, the spherical harmonic expansion, and the iteration method.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"23 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42303461","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Existence of two solutions to a parametric singular quasi-linear elliptic problem is proved. The equation is driven by the Φ Phi -Laplacian operator, and the reaction term can be nonmonotone. The main tools employed are the local minimum theorem and the Mountain pass theorem, together with the truncation technique. Global C 1 , τ {C}^{1,tau } regularity of solutions is also investigated, chiefly via a priori estimates and perturbation techniques.
{"title":"Existence of two solutions for singular Φ-Laplacian problems","authors":"P. Candito, U. Guarnotta, R. Livrea","doi":"10.1515/ans-2022-0037","DOIUrl":"https://doi.org/10.1515/ans-2022-0037","url":null,"abstract":"Abstract Existence of two solutions to a parametric singular quasi-linear elliptic problem is proved. The equation is driven by the Φ Phi -Laplacian operator, and the reaction term can be nonmonotone. The main tools employed are the local minimum theorem and the Mountain pass theorem, together with the truncation technique. Global C 1 , τ {C}^{1,tau } regularity of solutions is also investigated, chiefly via a priori estimates and perturbation techniques.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"22 1","pages":"659 - 683"},"PeriodicalIF":1.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48268822","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This article studies point concentration phenomena of nonlinear Schrödinger equations with magnetic potentials and constant electric potentials. The existing results show that a common magnetic field has no effect on the locations of point concentrations, as long as the electric potential is not a constant. This article finds out the role of the magnetic fields in the locations of point concentrations when the electric potential is a constant.
{"title":"Concentrations for nonlinear Schrödinger equations with magnetic potentials and constant electric potentials","authors":"Liping Wang, Chunyi Zhao","doi":"10.1515/ans-2022-0026","DOIUrl":"https://doi.org/10.1515/ans-2022-0026","url":null,"abstract":"Abstract This article studies point concentration phenomena of nonlinear Schrödinger equations with magnetic potentials and constant electric potentials. The existing results show that a common magnetic field has no effect on the locations of point concentrations, as long as the electric potential is not a constant. This article finds out the role of the magnetic fields in the locations of point concentrations when the electric potential is a constant.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"22 1","pages":"574 - 593"},"PeriodicalIF":1.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46112724","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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 the existence of weak C 1 , 1 {C}^{1,1} solution of the k k -Hessian equation on a closed Hermitian manifold under the optimal assumption of the function in the right-hand side of the equation. The key points are to show the weak C 1 , 1 {C}^{1,1} estimates. We prove a Cherrier-type inequality to obtain the C 0 {C}^{0} estimate, and the complex Hessian estimate is proved by using an auxiliary function, which was motivated by Hou et al. and Tosatti and Weinkove. Our result generalizes the Kähler case proved by Dinew et al.
摘要本文讨论了k k -Hessian方程在封闭厄米流形上，在方程右侧函数的最优假设下，弱c1,1 {C}^{1,1}解的存在性。关键是要显示弱c1,1 {C}^{1,1}估计。我们证明了cherrier型不等式，得到了c0 {C}^{0}估计，并利用辅助函数证明了复Hessian估计，该辅助函数由Hou等人以及Tosatti和Weinkove提出。我们的结果推广了Dinew等人证明的Kähler情况。
{"title":"Regularity of degenerate k-Hessian equations on closed Hermitian manifolds","authors":"Dekai Zhang","doi":"10.1515/ans-2022-0025","DOIUrl":"https://doi.org/10.1515/ans-2022-0025","url":null,"abstract":"Abstract In this article, we are concerned with the existence of weak C 1 , 1 {C}^{1,1} solution of the k k -Hessian equation on a closed Hermitian manifold under the optimal assumption of the function in the right-hand side of the equation. The key points are to show the weak C 1 , 1 {C}^{1,1} estimates. We prove a Cherrier-type inequality to obtain the C 0 {C}^{0} estimate, and the complex Hessian estimate is proved by using an auxiliary function, which was motivated by Hou et al. and Tosatti and Weinkove. Our result generalizes the Kähler case proved by Dinew et al.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"22 1","pages":"534 - 547"},"PeriodicalIF":1.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43775925","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The purpose of this article is to establish sharp conditions for the existence of normalized solutions to a class of scalar field equations involving mixed fractional Laplacians with different orders. This study includes the case when one operator is local and the other one is non-local. This type of equation arises in various fields ranging from biophysics to population dynamics. Due to the importance of these applications, this topic has very recently received an increasing interest. In this article, we provide a complete description of the existence/non-existence of ground state solutions using constrained variational approaches. This study addresses the mass subcritical, critical and supercritical cases. Our model presents some difficulties due to the “conflict” between the different orders and requires a novel analysis, especially in the mass supercritical case. We believe that our results will open the door to other valuable contributions in this important field.
{"title":"Normalized solutions for a class of scalar field equations involving mixed fractional Laplacians","authors":"Tingjian Luo, H. Hajaiej","doi":"10.1515/ans-2022-0013","DOIUrl":"https://doi.org/10.1515/ans-2022-0013","url":null,"abstract":"Abstract The purpose of this article is to establish sharp conditions for the existence of normalized solutions to a class of scalar field equations involving mixed fractional Laplacians with different orders. This study includes the case when one operator is local and the other one is non-local. This type of equation arises in various fields ranging from biophysics to population dynamics. Due to the importance of these applications, this topic has very recently received an increasing interest. In this article, we provide a complete description of the existence/non-existence of ground state solutions using constrained variational approaches. This study addresses the mass subcritical, critical and supercritical cases. Our model presents some difficulties due to the “conflict” between the different orders and requires a novel analysis, especially in the mass supercritical case. We believe that our results will open the door to other valuable contributions in this important field.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"22 1","pages":"228 - 247"},"PeriodicalIF":1.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42123780","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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 investigate the existence of multiple positive solutions to the following multi-critical Schrödinger equation: (0.1) − Δ u + λ V ( x ) u = μ ∣ u ∣ p − 2 u + ∑ i = 1 k ( ∣ x ∣ − ( N − α i ) ∗ ∣ u ∣ 2 i ∗ ) ∣ u ∣ 2 i ∗ − 2 u in R N , u ∈ H 1 ( R N ) , left{begin{array}{l}-Delta u+lambda Vleft(x)u=mu | u{| }^{p-2}u+mathop{displaystyle sum }limits_{i=1}^{k}left(| x{| }^{-left(N-{alpha }_{i})}ast | u{| }^{{2}_{i}^{ast }})| u{| }^{{2}_{i}^{ast }-2}uhspace{1.0em}hspace{0.1em}text{in}hspace{0.1em}hspace{0.33em}{{mathbb{R}}}^{N},hspace{1.0em} uhspace{0.33em}in {H}^{1}left({{mathbb{R}}}^{N}),hspace{1.0em}end{array}right. where λ , μ ∈ R + , N ≥ 4 lambda ,mu in {{mathbb{R}}}^{+},Nge 4 , and 2 i ∗ = N + α i N − 2 {2}_{i}^{ast }=frac{N+{alpha }_{i}}{N-2} with N − 4 < α i < N N-4lt {alpha }_{i}lt N , i = 1 , 2 , … , k i=1,2,ldots ,k are critical exponents and 2 < p < 2 min ∗ = min { 2 i ∗ : i = 1 , 2 , … , k } 2lt plt {2}_{min }^{ast }={rm{min }}left{{2}_{i}^{ast }:i=1,2,ldots ,kright} . Suppose that Ω = int V − 1 ( 0 ) ⊂ R N Omega ={rm{int}}hspace{0.33em}{V}^{-1}left(0)subset {{mathbb{R}}}^{N} is a bounded domain, we show that for λ lambda large, problem (0.1) possesses at least cat Ω ( Ω ) {{rm{cat}}}_{Omega }left(Omega ) positive solutions.
在这篇文章中，我们正在研究应用多学科薛定谔方程的存在:(0。1)−Δu + V (x) u =λμ∣u p∣−2 u +∑x i = 1 k(∣∣−(N−αi)∗∣u∣2我我∗∗)∣u∣2−2 in R N u, u H∈R 1 (N),向左拐{开始{}{}- l阵lambda 三角洲u + V向左拐(x) u = mu | u ^ {p - 2}{|的u + mathop { displaystyle sum的 limits_ {i = 1} k ^{} 向左拐(| x{|} ^{向左拐(N - {{i})}的阿尔法的在| u的{|}^ {{2}{i) ^{在}})|的u {|} ^ {{2} {i} ^{在的u - 2的hspace {1 . 0em} hspace = 0。1em的文本{在} hspace{0。1em} hspace {0 . 33em} {{ mathbb {R}}} N ^ {}, {1 . 0em} hspace u hspace{0。33em} H在{}^{1}向左拐({{R mathbb {}}} ^ {N}),伦敦hspace {1 . 0em} {阵列望远镜的吧。λ,哪里μ+∈R, N≥4你在{{ lambda, mathbb {R}}} + ^ {}, N ge 4和2我∗= N +αi N−2{2}_我{}^{在}= frac {N + { {i}}{已经开始了的阿尔法的N和N−4 <αi < N-4它{阿尔法}{ N,这是我的k = 1, 2, ... i = 1.2, ldots, k是连接exponents和< p < 2 min i∗∗= min {2: i = 1, 2, ... k的中尉p {2}{敏}^{在}的= min{罗{}}的左派 {{2}{i} ^{在}:i = 1.2, ldots, k对)。想那Ω= int V−1(0 - 9)⊂R N ω={罗{int)}} hspace {0 V . 33em}{} ^{- 1}左(0 - 9)子集{{R mathbb {}}} ^ {N}是一个bounded域名,我们为λ lambda大秀那,问题(0。1)possesses至少油漆Ω(Ω)的油漆{{罗{}}}{ Omega欧米茄的左边()积极解决方案。
{"title":"Multiple solutions to multi-critical Schrödinger equations","authors":"Ziyi Xu, Jianfu Yang","doi":"10.1515/ans-2022-0014","DOIUrl":"https://doi.org/10.1515/ans-2022-0014","url":null,"abstract":"Abstract In this article, we investigate the existence of multiple positive solutions to the following multi-critical Schrödinger equation: (0.1) − Δ u + λ V ( x ) u = μ ∣ u ∣ p − 2 u + ∑ i = 1 k ( ∣ x ∣ − ( N − α i ) ∗ ∣ u ∣ 2 i ∗ ) ∣ u ∣ 2 i ∗ − 2 u in R N , u ∈ H 1 ( R N ) , left{begin{array}{l}-Delta u+lambda Vleft(x)u=mu | u{| }^{p-2}u+mathop{displaystyle sum }limits_{i=1}^{k}left(| x{| }^{-left(N-{alpha }_{i})}ast | u{| }^{{2}_{i}^{ast }})| u{| }^{{2}_{i}^{ast }-2}uhspace{1.0em}hspace{0.1em}text{in}hspace{0.1em}hspace{0.33em}{{mathbb{R}}}^{N},hspace{1.0em} uhspace{0.33em}in {H}^{1}left({{mathbb{R}}}^{N}),hspace{1.0em}end{array}right. where λ , μ ∈ R + , N ≥ 4 lambda ,mu in {{mathbb{R}}}^{+},Nge 4 , and 2 i ∗ = N + α i N − 2 {2}_{i}^{ast }=frac{N+{alpha }_{i}}{N-2} with N − 4 < α i < N N-4lt {alpha }_{i}lt N , i = 1 , 2 , … , k i=1,2,ldots ,k are critical exponents and 2 < p < 2 min ∗ = min { 2 i ∗ : i = 1 , 2 , … , k } 2lt plt {2}_{min }^{ast }={rm{min }}left{{2}_{i}^{ast }:i=1,2,ldots ,kright} . Suppose that Ω = int V − 1 ( 0 ) ⊂ R N Omega ={rm{int}}hspace{0.33em}{V}^{-1}left(0)subset {{mathbb{R}}}^{N} is a bounded domain, we show that for λ lambda large, problem (0.1) possesses at least cat Ω ( Ω ) {{rm{cat}}}_{Omega }left(Omega ) positive solutions.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"22 1","pages":"273 - 288"},"PeriodicalIF":1.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44038297","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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 study multiplicity of semi-classical solutions of a Yukawa-coupled massive Dirac-Klein-Gordon system with the general nonlinear self-coupling, which is either subcritical or critical growth. The number of solutions obtained is described by the ratio of maximum and behavior at infinity of the potentials. We use the variational method that relies upon a delicate cutting off technique. It allows us to overcome the lack of convexity of the nonlinearities.
{"title":"Multiplicity and concentration of semi-classical solutions to nonlinear Dirac-Klein-Gordon systems","authors":"Yanheng Ding, Yuanyang Yu, Xiaojing Dong","doi":"10.1515/ans-2022-0011","DOIUrl":"https://doi.org/10.1515/ans-2022-0011","url":null,"abstract":"Abstract In the present article, we study multiplicity of semi-classical solutions of a Yukawa-coupled massive Dirac-Klein-Gordon system with the general nonlinear self-coupling, which is either subcritical or critical growth. The number of solutions obtained is described by the ratio of maximum and behavior at infinity of the potentials. We use the variational method that relies upon a delicate cutting off technique. It allows us to overcome the lack of convexity of the nonlinearities.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"22 1","pages":"248 - 272"},"PeriodicalIF":1.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47829714","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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 paper, we study the existence of the normalized solutions for the following coupled elliptic system with quadratic nonlinearity − Δ u − λ 1 u = μ 1 ∣ u ∣ u + β u v in R N , − Δ v − λ 2 v = μ 2 ∣ v ∣ v + β 2 u 2 in R N , left{begin{array}{ll}-Delta u-{lambda }_{1}u={mu }_{1}| u| u+beta uvhspace{1.0em}& hspace{0.1em}text{in}hspace{0.1em}hspace{0.33em}{{mathbb{R}}}^{N}, -Delta v-{lambda }_{2}v={mu }_{2}| v| v+frac{beta }{2}{u}^{2}hspace{1.0em}& hspace{0.1em}text{in}hspace{0.1em}hspace{0.33em}{{mathbb{R}}}^{N},end{array}right. where u , v u,v satisfying the additional condition ∫ R N u 2 d x = a 1 , ∫ R N v 2 d x = a 2 . mathop{int }limits_{{{mathbb{R}}}^{N}}{u}^{2}{rm{d}}x={a}_{1},hspace{1em}mathop{int }limits_{{{mathbb{R}}}^{N}}{v}^{2}{rm{d}}x={a}_{2}. On the one hand, we prove the existence of minimizer for the system with L 2 {L}^{2} -subcritical growth ( N ≤ 3 Nle 3 ). On the other hand, we prove the existence results for different ranges of the coupling parameter β > 0 beta gt 0 with L 2 {L}^{2} -supercritical growth ( N = 5 N=5 ). Our argument is based on the rearrangement techniques and the minimax construction.
摘要在本文中，我们研究了以下具有二次非线性的耦合椭圆系统的归一化解的存在性——R N中的−Δu−λ1 u=μ1ÜuÜu+βu v，R N中−Δv−λ2 v=μ2ÜvÜv+β2 u 2， left ｛ begin｛array｝{ll}-增量u｛lambda｝_{1}u=｛mu｝_｛1｝|u|u+beta-uvhspace｛1.0em｝&hspace{0.1em｝text｛in｝space｛0.1em｝hspace｝0.33em｝｛mathbb｛R｝｝^｛N｝，-Delta v-｛lambda｝_{2}v＝｛mu｝_｛2｝|v|v+frac｛beta｝｛2}｛u｝^｛2}space｛1.0em｝&space｛｛0.1em｝text｛in｝space{0.1em｝ hspace｛0.33em｝{mathbb｛R｝}^｛N｝，end｛array｝right。式中u，vu，v满足附加条件：。mathop｛int｝limits_{｛mathbb｛R｝｝}^｛N｝}｛u｝^｛2｝｛rm｛d｝x={a}_{1} ，space｛1em｝mathop｛int｝limits_{｛mathbb｛R｝｝}^｛N｝}｛v｝^｛2｝｛rm｛d｝x={a}_{2} 。一方面，我们证明了具有L2｛L｝^｛2｝-次临界增长（N≤3Nle3）系统的极小值的存在性。另一方面，我们证明了在L2｛L｝^｛2｝-超临界生长（N＝5 N＝5）的不同范围内，耦合参数β＞0β＞0的存在性结果。我们的论点是基于重排技术和极小极大构造。
{"title":"Existence of normalized solutions for the coupled elliptic system with quadratic nonlinearity","authors":"Jun Wang, Xuan Wang, Song Wei","doi":"10.1515/ans-2022-0010","DOIUrl":"https://doi.org/10.1515/ans-2022-0010","url":null,"abstract":"Abstract In the present paper, we study the existence of the normalized solutions for the following coupled elliptic system with quadratic nonlinearity − Δ u − λ 1 u = μ 1 ∣ u ∣ u + β u v in R N , − Δ v − λ 2 v = μ 2 ∣ v ∣ v + β 2 u 2 in R N , left{begin{array}{ll}-Delta u-{lambda }_{1}u={mu }_{1}| u| u+beta uvhspace{1.0em}& hspace{0.1em}text{in}hspace{0.1em}hspace{0.33em}{{mathbb{R}}}^{N}, -Delta v-{lambda }_{2}v={mu }_{2}| v| v+frac{beta }{2}{u}^{2}hspace{1.0em}& hspace{0.1em}text{in}hspace{0.1em}hspace{0.33em}{{mathbb{R}}}^{N},end{array}right. where u , v u,v satisfying the additional condition ∫ R N u 2 d x = a 1 , ∫ R N v 2 d x = a 2 . mathop{int }limits_{{{mathbb{R}}}^{N}}{u}^{2}{rm{d}}x={a}_{1},hspace{1em}mathop{int }limits_{{{mathbb{R}}}^{N}}{v}^{2}{rm{d}}x={a}_{2}. On the one hand, we prove the existence of minimizer for the system with L 2 {L}^{2} -subcritical growth ( N ≤ 3 Nle 3 ). On the other hand, we prove the existence results for different ranges of the coupling parameter β > 0 beta gt 0 with L 2 {L}^{2} -supercritical growth ( N = 5 N=5 ). Our argument is based on the rearrangement techniques and the minimax construction.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"22 1","pages":"203 - 227"},"PeriodicalIF":1.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49225130","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We study the existence of solutions for the quasilinear Schrödinger equation with the critical exponent and steep potential well. By using a change of variables, the quasilinear equations are reduced to a semilinear one, whose associated functionals satisfy the geometric conditions of the Mountain Pass Theorem for suitable assumptions. The existence of a ground state solution is obtained, and its concentration behavior is also considered.
{"title":"Existence of ground state solutions for critical quasilinear Schrödinger equations with steep potential well","authors":"Yan-Fang Xue, Xiao-Jing Zhong, Chunlei Tang","doi":"10.1515/ans-2022-0030","DOIUrl":"https://doi.org/10.1515/ans-2022-0030","url":null,"abstract":"Abstract We study the existence of solutions for the quasilinear Schrödinger equation with the critical exponent and steep potential well. By using a change of variables, the quasilinear equations are reduced to a semilinear one, whose associated functionals satisfy the geometric conditions of the Mountain Pass Theorem for suitable assumptions. The existence of a ground state solution is obtained, and its concentration behavior is also considered.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"22 1","pages":"619 - 634"},"PeriodicalIF":1.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47292872","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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