Abstract This article is directed to the study of the following Schrödinger-Poisson-Slater type equation: − ε 2 Δ u + V ( x ) u + ε − α ( I α ∗ ∣ u ∣ 2 ) u = λ ∣ u ∣ p − 1 u in R N , -{varepsilon }^{2}Delta u+Vleft(x)u+{varepsilon }^{-alpha }left({I}_{alpha }ast | u{| }^{2})u=lambda | u{| }^{p-1}uhspace{1em}hspace{0.1em}text{in}hspace{0.1em}hspace{0.33em}{{mathbb{R}}}^{N}, where ε , λ > 0 varepsilon ,lambda gt 0 are parameters, N ⩾ 2 Ngeqslant 2 , ( α + 6 ) / ( α + 2 ) < p < 2 ∗ − 1 left(alpha +6)hspace{0.1em}text{/}hspace{0.1em}left(alpha +2)lt plt {2}^{ast }-1 , I α {I}_{alpha } is the Riesz potential with 0 < α < N 0lt alpha lt N , and V ∈ C ( R N , R ) Vin {mathcal{C}}left({{mathbb{R}}}^{N},{mathbb{R}}) . By using variational methods, we prove that there is a positive ground state solution for the aforementioned equation concentrating at a global minimum of V V in the semi-classical limit, and then we found that this solution satisfies the property of exponential decay. Finally, the multiplicity and concentration behavior of positive solutions for the aforementioned problem is investigated by the Ljusternik-Schnirelmann theory. Our article improves and extends some existing results in several directions.
{"title":"Existence and concentration behavior of positive solutions to Schrödinger-Poisson-Slater equations","authors":"Yiqing Li, Binlin Zhang, Xiumei Han","doi":"10.1515/anona-2022-0293","DOIUrl":"https://doi.org/10.1515/anona-2022-0293","url":null,"abstract":"Abstract This article is directed to the study of the following Schrödinger-Poisson-Slater type equation: − ε 2 Δ u + V ( x ) u + ε − α ( I α ∗ ∣ u ∣ 2 ) u = λ ∣ u ∣ p − 1 u in R N , -{varepsilon }^{2}Delta u+Vleft(x)u+{varepsilon }^{-alpha }left({I}_{alpha }ast | u{| }^{2})u=lambda | u{| }^{p-1}uhspace{1em}hspace{0.1em}text{in}hspace{0.1em}hspace{0.33em}{{mathbb{R}}}^{N}, where ε , λ > 0 varepsilon ,lambda gt 0 are parameters, N ⩾ 2 Ngeqslant 2 , ( α + 6 ) / ( α + 2 ) < p < 2 ∗ − 1 left(alpha +6)hspace{0.1em}text{/}hspace{0.1em}left(alpha +2)lt plt {2}^{ast }-1 , I α {I}_{alpha } is the Riesz potential with 0 < α < N 0lt alpha lt N , and V ∈ C ( R N , R ) Vin {mathcal{C}}left({{mathbb{R}}}^{N},{mathbb{R}}) . By using variational methods, we prove that there is a positive ground state solution for the aforementioned equation concentrating at a global minimum of V V in the semi-classical limit, and then we found that this solution satisfies the property of exponential decay. Finally, the multiplicity and concentration behavior of positive solutions for the aforementioned problem is investigated by the Ljusternik-Schnirelmann theory. Our article improves and extends some existing results in several directions.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42968190","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 consider anisotropic parabolic systems of p p -Laplace type. The model case is the parabolic p i {p}_{i} -Laplace system u t − ∑ i = 1 n ∂ ∂ x i ( ∣ D i u ∣ p i − 2 D i u ) = 0 {u}_{t}-mathop{sum }limits_{i=1}^{n}frac{partial }{partial {x}_{i}}({| {D}_{i}u| }^{{p}_{i}-2}{D}_{i}u)=0 with exponents p i ≥ 2 {p}_{i}ge 2 . Under the assumption that the exponents are not too far apart, i.e., the difference p max − p min {p}_{max }-{p}_{min } is sufficiently small, we establish a higher integrability result for weak solutions. This extends a result, which was only known for the elliptic setting, to the parabolic setting.
{"title":"Higher integrability for anisotropic parabolic systems of p-Laplace type","authors":"Leon Mons","doi":"10.1515/anona-2022-0308","DOIUrl":"https://doi.org/10.1515/anona-2022-0308","url":null,"abstract":"Abstract In this article, we consider anisotropic parabolic systems of p p -Laplace type. The model case is the parabolic p i {p}_{i} -Laplace system u t − ∑ i = 1 n ∂ ∂ x i ( ∣ D i u ∣ p i − 2 D i u ) = 0 {u}_{t}-mathop{sum }limits_{i=1}^{n}frac{partial }{partial {x}_{i}}({| {D}_{i}u| }^{{p}_{i}-2}{D}_{i}u)=0 with exponents p i ≥ 2 {p}_{i}ge 2 . Under the assumption that the exponents are not too far apart, i.e., the difference p max − p min {p}_{max }-{p}_{min } is sufficiently small, we establish a higher integrability result for weak solutions. This extends a result, which was only known for the elliptic setting, to the parabolic setting.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45131011","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 Our aim of this article is to study qualitative properties of Dirichlet problems involving the Hardy-Leray operator ℒ V ≔ − Δ + V {{mathcal{ {mathcal L} }}}_{V}:= -Delta +V , where V ( x ) = ∑ i = 1 m μ i ∣ x − A i ∣ 2 Vleft(x)={sum }_{i=1}^{m}frac{{mu }_{i}}{{| x-{A}_{i}| }^{2}} , with μ i ≥ − ( N − 2 ) 2 4 {mu }_{i}ge -frac{{left(N-2)}^{2}}{4} being the Hardy-Leray potential containing the polars’ set A m = { A i : i = 1 , … , m } {{mathcal{A}}}_{m}=left{{A}_{i}:i=1,ldots ,mright} in R N {{mathbb{R}}}^{N} ( N ≥ 2 Nge 2 ). Since the inverse-square potentials are critical with respect to the Laplacian operator, the coefficients { μ i } i = 1 m {left{{mu }_{i}right}}_{i=1}^{m} and the locations of polars { A i } left{{A}_{i}right} play an important role in the properties of solutions to the related Poisson problems subject to zero Dirichlet boundary conditions. Let Ω Omega be a bounded domain containing A m {{mathcal{A}}}_{m} . First, we obtain increasing Dirichlet eigenvalues: ℒ V u = λ u in Ω , u = 0 on ∂ Ω , {{mathcal{ {mathcal L} }}}_{V}u=lambda uhspace{1.0em}{rm{in}}hspace{0.33em}Omega ,hspace{1.0em}u=0hspace{1.0em}{rm{on}}hspace{0.33em}partial Omega , and the positivity of the principle eigenvalue depends on the strength μ i {mu }_{i} and polars’ setting. When the spectral does not contain the origin, we then consider the weak solutions of the Poisson problem ( E ) ℒ V u = ν in Ω , u = 0 on ∂ Ω , left(E)hspace{1.0em}hspace{1.0em}{{mathcal{ {mathcal L} }}}_{V}u=nu hspace{1em}{rm{in}}hspace{0.33em}Omega ,hspace{1.0em}u=0hspace{1em}{rm{on}}hspace{0.33em}partial Omega , when ν nu belongs to L p ( Ω ) {L}^{p}left(Omega ) , with p > 2 N N + 2 pgt frac{2N}{N+2} in the variational framework, and we obtain a global weighted L ∞ {L}^{infty } estimate when p > N 2 pgt frac{N}{2} . When the principle eigenvalue is positive and ν nu is a Radon measure, we build a weighted distributional framework to show the existence of weak solutions of problem ( E ) left(E) . Moreover, via this weighted distributional framework, we can obtain a sharp assumption of ν ∈ C γ ( Ω ¯ A m ) nu in {{mathcal{C}}}^{gamma }left(bar{Omega }setminus {{mathcal{A}}}_{m}) for the existence of isolated singular solutions for problem ( E ) left(E) .
摘要本文的目的是研究涉及Hardy-Leray算子的Dirichlet问题的定性性质ℒ V−Δ+V-{A}_{i} |｝^｛2｝｝，其中μi≥−（N−2）2 4｛mu｝_{{A}_{i} ：i=1，ldots，mright｝在R N｛mathbb｛R｝｝^｛N｝中（N≥2Nge2）。由于平方反比势对于拉普拉斯算子是关键的，因此系数｛μi｝i＝1m｛lang1033｛mu｝_｛i｝right｝｝_{{A}_{i} 在零Dirichlet边界条件下相关Poisson问题解的性质中起着重要作用。设ΩOmega是一个包含a m｛mathcal｛a｝｝_｛m｝的有界域。首先，我们获得了增加的狄利克雷特征值：ℒ V u=λu，单位为Ω，u=0，在¦ΒΩ上，｛mathcal｛math L｝｝_{V}u=lambda uhspace｛1.0em｝｛rm｛in｝｝space｛0.33em｝Omega，space{1.0em}u=0hspace｛1.0em｝｛rm｛on｝｝space｛0.33em｝partial Omega，并且主特征值的正性取决于强度μi｛mu｝_｛i｝和polar的设置。当谱不包含原点时，我们考虑泊松问题（E）的弱解ℒ V u=¦Α，u=¦ΒΩ上的0，left（E）hspace｛1.0em｝space｛1.0em｝｛mathcal｛L｝｝_{V}u=nuhspace｛1em｝｛rm｛in｝｝space｛0.33em｝Omega，space{1.0em}u=0hspace｛1em｝｛rm｛on｝｝space｛0.33em｝partialOmega，当Γnu属于Lp（Ω）｛L｝^｛p｝left（Omega）时，在变分框架中p＞2N＋2p｝｛｛N＋2｝，并且当p＞N2｝｛。当主特征值为正，且Γnu为Radon测度时，我们建立了一个加权分布框架来证明问题（E）left（E）弱解的存在性。此外，通过这个加权分布框架，我们可以得到一个关于问题（E）left（E）存在孤立奇异解的尖锐假设，即｛mathcal｛C｝｝｝^｛gamma｝left（bar｛Omega｝setminus｛math cal｛a｝｝｝}_{m｝）中的Γ∈Cγ（Ωam）nu。
{"title":"Dirichlet problems involving the Hardy-Leray operators with multiple polars","authors":"Huyuan Chen, Xiaowei Chen","doi":"10.1515/anona-2022-0320","DOIUrl":"https://doi.org/10.1515/anona-2022-0320","url":null,"abstract":"Abstract Our aim of this article is to study qualitative properties of Dirichlet problems involving the Hardy-Leray operator ℒ V ≔ − Δ + V {{mathcal{ {mathcal L} }}}_{V}:= -Delta +V , where V ( x ) = ∑ i = 1 m μ i ∣ x − A i ∣ 2 Vleft(x)={sum }_{i=1}^{m}frac{{mu }_{i}}{{| x-{A}_{i}| }^{2}} , with μ i ≥ − ( N − 2 ) 2 4 {mu }_{i}ge -frac{{left(N-2)}^{2}}{4} being the Hardy-Leray potential containing the polars’ set A m = { A i : i = 1 , … , m } {{mathcal{A}}}_{m}=left{{A}_{i}:i=1,ldots ,mright} in R N {{mathbb{R}}}^{N} ( N ≥ 2 Nge 2 ). Since the inverse-square potentials are critical with respect to the Laplacian operator, the coefficients { μ i } i = 1 m {left{{mu }_{i}right}}_{i=1}^{m} and the locations of polars { A i } left{{A}_{i}right} play an important role in the properties of solutions to the related Poisson problems subject to zero Dirichlet boundary conditions. Let Ω Omega be a bounded domain containing A m {{mathcal{A}}}_{m} . First, we obtain increasing Dirichlet eigenvalues: ℒ V u = λ u in Ω , u = 0 on ∂ Ω , {{mathcal{ {mathcal L} }}}_{V}u=lambda uhspace{1.0em}{rm{in}}hspace{0.33em}Omega ,hspace{1.0em}u=0hspace{1.0em}{rm{on}}hspace{0.33em}partial Omega , and the positivity of the principle eigenvalue depends on the strength μ i {mu }_{i} and polars’ setting. When the spectral does not contain the origin, we then consider the weak solutions of the Poisson problem ( E ) ℒ V u = ν in Ω , u = 0 on ∂ Ω , left(E)hspace{1.0em}hspace{1.0em}{{mathcal{ {mathcal L} }}}_{V}u=nu hspace{1em}{rm{in}}hspace{0.33em}Omega ,hspace{1.0em}u=0hspace{1em}{rm{on}}hspace{0.33em}partial Omega , when ν nu belongs to L p ( Ω ) {L}^{p}left(Omega ) , with p > 2 N N + 2 pgt frac{2N}{N+2} in the variational framework, and we obtain a global weighted L ∞ {L}^{infty } estimate when p > N 2 pgt frac{N}{2} . When the principle eigenvalue is positive and ν nu is a Radon measure, we build a weighted distributional framework to show the existence of weak solutions of problem ( E ) left(E) . Moreover, via this weighted distributional framework, we can obtain a sharp assumption of ν ∈ C γ ( Ω ¯ A m ) nu in {{mathcal{C}}}^{gamma }left(bar{Omega }setminus {{mathcal{A}}}_{m}) for the existence of isolated singular solutions for problem ( E ) left(E) .","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43333591","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 consider a four-dimensional singular differential system that can describe the dynamics of configurations bearing a small number of vortices in atomic Bose-Einstein condensates. On the basis of the topological degree theory and some analysis methods, we prove that such a system has two distinct families of periodic solutions and two distinct families of quasi-periodic solutions. Some results in the literature are generalized and improved.
{"title":"Periodic and quasi-periodic solutions of a four-dimensional singular differential system describing the motion of vortices","authors":"Zaitao Liang, Shengjun Li, Xin Li","doi":"10.1515/anona-2022-0287","DOIUrl":"https://doi.org/10.1515/anona-2022-0287","url":null,"abstract":"Abstract In this article, we consider a four-dimensional singular differential system that can describe the dynamics of configurations bearing a small number of vortices in atomic Bose-Einstein condensates. On the basis of the topological degree theory and some analysis methods, we prove that such a system has two distinct families of periodic solutions and two distinct families of quasi-periodic solutions. Some results in the literature are generalized and improved.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47145885","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}
F. Farroni, L. Greco, G. Moscariello, Gabriella Zecca
Abstract We prove an existence result for obstacle problems related to convection-diffusion parabolic equations with singular coefficients in the convective term. Our operator is not coercive, the obstacle function is time-dependent irregular, and the coefficients in the lower-order term belong to a borderline mixed Lebesgue-Marcinkiewicz space.
{"title":"Noncoercive parabolic obstacle problems","authors":"F. Farroni, L. Greco, G. Moscariello, Gabriella Zecca","doi":"10.1515/anona-2022-0322","DOIUrl":"https://doi.org/10.1515/anona-2022-0322","url":null,"abstract":"Abstract We prove an existence result for obstacle problems related to convection-diffusion parabolic equations with singular coefficients in the convective term. Our operator is not coercive, the obstacle function is time-dependent irregular, and the coefficients in the lower-order term belong to a borderline mixed Lebesgue-Marcinkiewicz space.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48861225","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 existence of periodic solutions to a class of distributed delay differential equations. We transform the search for periodic solutions with the special symmetry of a delay differential equation to the problem of finding periodic solutions of an associated Hamiltonian system. Using the critical point theory and the pseudo-index theory, we obtain some sufficient conditions for the multiplicity of periodic solutions. This is the first time that critical point theory has been used to study the existence of periodic solutions to distributed delay differential equations.
{"title":"Periodic solutions to a class of distributed delay differential equations via variational methods","authors":"Huafeng Xiao, Zhiming Guo","doi":"10.1515/anona-2022-0305","DOIUrl":"https://doi.org/10.1515/anona-2022-0305","url":null,"abstract":"Abstract In this article, we study the existence of periodic solutions to a class of distributed delay differential equations. We transform the search for periodic solutions with the special symmetry of a delay differential equation to the problem of finding periodic solutions of an associated Hamiltonian system. Using the critical point theory and the pseudo-index theory, we obtain some sufficient conditions for the multiplicity of periodic solutions. This is the first time that critical point theory has been used to study the existence of periodic solutions to distributed delay differential equations.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42369970","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 are concerned with the following Kirchhoff-type equation with exponential critical nonlinearities − a + b ∫ R 2 ∣ ∇ u ∣ 2 d x Δ u + ( h ( x ) + μ V ( x ) ) u = K ( x ) f ( u ) in R 2 , -left(a+bmathop{int }limits_{{{mathbb{R}}}^{2}}| nabla u{| }^{2}{rm{d}}xright)Delta u+left(hleft(x)+mu Vleft(x))u=Kleft(x)fleft(u)hspace{1em}{rm{in}}hspace{0.33em}{{mathbb{R}}}^{2}, where a , b , μ > 0 a,b,mu gt 0 , the potential V V has a bounded set of zero points and decays at infinity as ∣ x ∣ − γ | x{| }^{-gamma } with γ ∈ ( 0 , 2 ) gamma in left(0,2) , the weight K K has finite singular points and may have exponential growth at infinity. By using the truncation technique and working in some weighted Sobolev space, we obtain the existence of a mountain pass solution for μ > 0 mu gt 0 large and the concentration behavior of solutions as μ → + ∞ mu to +infty .
摘要:我们关注以下具有指数临界非线性的kirchhoff型方程- a+b∫r2∣∇u∣2d x Δ u+ (h (x)+ μ V (x))u=K (x)f (u)在r2中，- left (a+b mathop{int }limits ＿ {{{mathbb{R}}} ^{2}}| nabla u{|} ^{2x}{rm{d}}right) Delta u+ left (h left (x)+ mu V left (x))u=K left (x)f left (u)^hspace{1em}{rm{in}}hspace{0.33em}{{mathbb{R}}}2{，}其中a,b， μ > 0 a,b，mugt 0，势V V有一个有界的零点集合，在无穷远处衰减为∣x∣−γ | x| ^{-}{gamma，其中γ}∈(0,2)gammainleft(0,2)，权K K有有限奇点，在无穷远处可以呈指数增长。利用截断技术，在一些权重Sobolev空间中，我们得到了μ > 0 mugt 0大的山口解的存在性和解的集中行为为μ→+∞muto + infty。
{"title":"Existence and concentration of solutions to Kirchhoff-type equations in ℝ2 with steep potential well vanishing at infinity and exponential critical nonlinearities","authors":"Jian Zhang, Xue Bao, Jianjun Zhang","doi":"10.1515/anona-2022-0317","DOIUrl":"https://doi.org/10.1515/anona-2022-0317","url":null,"abstract":"Abstract We are concerned with the following Kirchhoff-type equation with exponential critical nonlinearities − a + b ∫ R 2 ∣ ∇ u ∣ 2 d x Δ u + ( h ( x ) + μ V ( x ) ) u = K ( x ) f ( u ) in R 2 , -left(a+bmathop{int }limits_{{{mathbb{R}}}^{2}}| nabla u{| }^{2}{rm{d}}xright)Delta u+left(hleft(x)+mu Vleft(x))u=Kleft(x)fleft(u)hspace{1em}{rm{in}}hspace{0.33em}{{mathbb{R}}}^{2}, where a , b , μ > 0 a,b,mu gt 0 , the potential V V has a bounded set of zero points and decays at infinity as ∣ x ∣ − γ | x{| }^{-gamma } with γ ∈ ( 0 , 2 ) gamma in left(0,2) , the weight K K has finite singular points and may have exponential growth at infinity. By using the truncation technique and working in some weighted Sobolev space, we obtain the existence of a mountain pass solution for μ > 0 mu gt 0 large and the concentration behavior of solutions as μ → + ∞ mu to +infty .","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"12 1","pages":""},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67260729","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 This article presents new local and global gradient estimates of Li-Yau type for positive solutions to a class of nonlinear elliptic equations on smooth metric measure spaces involving the Witten Laplacian. The estimates are derived under natural lower bounds on the associated Bakry-Émery Ricci curvature tensor and find utility in proving fairly general Harnack inequalities and Liouville-type theorems to name a few. The results here unify, extend and improve various existing results in the literature for special nonlinearities already of huge interest and applications. Some consequences are presented and discussed.
{"title":"Gradient estimates for nonlinear elliptic equations involving the Witten Laplacian on smooth metric measure spaces and implications","authors":"A. Taheri, V. Vahidifar","doi":"10.1515/anona-2022-0288","DOIUrl":"https://doi.org/10.1515/anona-2022-0288","url":null,"abstract":"Abstract This article presents new local and global gradient estimates of Li-Yau type for positive solutions to a class of nonlinear elliptic equations on smooth metric measure spaces involving the Witten Laplacian. The estimates are derived under natural lower bounds on the associated Bakry-Émery Ricci curvature tensor and find utility in proving fairly general Harnack inequalities and Liouville-type theorems to name a few. The results here unify, extend and improve various existing results in the literature for special nonlinearities already of huge interest and applications. Some consequences are presented and discussed.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41408547","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 investigate the existence and multiplicity of weak solutions for a nonlinear Kirchhoff type quasilinear elliptic system on the whole space R N {{mathbb{R}}}^{N} . We assume that the nonlinear term satisfies the locally super- ( m 1 , m 2 ) left({m}_{1},{m}_{2}) condition, that is, lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 = + ∞ {mathrm{lim}}_{| left(u,v)| to +infty }frac{Fleft(x,u,v)}{| u{| }^{{m}_{1}}+| v{| }^{{m}_{2}}}=+infty for a.e. x ∈ G xin G , where G G is a domain in R N {{mathbb{R}}}^{N} , which is weaker than the well-known Ambrosseti-Rabinowitz condition and the naturally global restriction, lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 = + ∞ {mathrm{lim}}_{| left(u,v)| to +infty }frac{Fleft(x,u,v)}{| u{| }^{{m}_{1}}+| v{| }^{{m}_{2}}}=+infty for a.e. x ∈ R N xin {{mathbb{R}}}^{N} . We obtain that the system has at least one weak solution by using the classical mountain pass theorem. To a certain extent, our theorems extend the results of Tang et al. [Nontrivial solutions for Schrodinger equation with local super-quadratic conditions, J. Dynam. Differ. Equ. 31 (2019), no. 1, 369–383]. Moreover, under the aforementioned naturally global restriction, we obtain that the system has infinitely many weak solutions of high energy by using the symmetric mountain pass theorem, which is different from those results of Wang et al. [Existence and multiplicity of solutions for a class of quasilinear elliptic systems in Orlicz-Sobolev spaces, J. Nonlinear Sci. Appl. 10 (2017), no. 7, 3792–3814] even if we consider the system on the bounded domain with Dirichlet boundary condition.
{"title":"Existence and multiplicity of solutions for a quasilinear system with locally superlinear condition","authors":"Cuiling Liu, Xingyong Zhang","doi":"10.1515/anona-2022-0289","DOIUrl":"https://doi.org/10.1515/anona-2022-0289","url":null,"abstract":"Abstract We investigate the existence and multiplicity of weak solutions for a nonlinear Kirchhoff type quasilinear elliptic system on the whole space R N {{mathbb{R}}}^{N} . We assume that the nonlinear term satisfies the locally super- ( m 1 , m 2 ) left({m}_{1},{m}_{2}) condition, that is, lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 = + ∞ {mathrm{lim}}_{| left(u,v)| to +infty }frac{Fleft(x,u,v)}{| u{| }^{{m}_{1}}+| v{| }^{{m}_{2}}}=+infty for a.e. x ∈ G xin G , where G G is a domain in R N {{mathbb{R}}}^{N} , which is weaker than the well-known Ambrosseti-Rabinowitz condition and the naturally global restriction, lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 = + ∞ {mathrm{lim}}_{| left(u,v)| to +infty }frac{Fleft(x,u,v)}{| u{| }^{{m}_{1}}+| v{| }^{{m}_{2}}}=+infty for a.e. x ∈ R N xin {{mathbb{R}}}^{N} . We obtain that the system has at least one weak solution by using the classical mountain pass theorem. To a certain extent, our theorems extend the results of Tang et al. [Nontrivial solutions for Schrodinger equation with local super-quadratic conditions, J. Dynam. Differ. Equ. 31 (2019), no. 1, 369–383]. Moreover, under the aforementioned naturally global restriction, we obtain that the system has infinitely many weak solutions of high energy by using the symmetric mountain pass theorem, which is different from those results of Wang et al. [Existence and multiplicity of solutions for a class of quasilinear elliptic systems in Orlicz-Sobolev spaces, J. Nonlinear Sci. Appl. 10 (2017), no. 7, 3792–3814] even if we consider the system on the bounded domain with Dirichlet boundary condition.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41664225","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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