Abstract The article is concerned with systems of fractional discrete equations Δ α x ( n + 1 ) = F n ( n , x ( n ) , x ( n − 1 ) , … , x ( n 0 ) ) , n = n 0 , n 0 + 1 , … , {Delta }^{alpha }xleft(n+1)={F}_{n}left(n,xleft(n),xleft(n-1),ldots ,xleft({n}_{0})),hspace{1em}n={n}_{0},{n}_{0}+1,ldots , where n 0 ∈ Z {n}_{0}in {mathbb{Z}} , n n is an independent variable, Δ α {Delta }^{alpha } is an α alpha -order fractional difference, α ∈ R alpha in {mathbb{R}} , F n : { n } × R n − n 0 + 1 → R s {F}_{n}:left{nright}times {{mathbb{R}}}^{n-{n}_{0}+1}to {{mathbb{R}}}^{s} , s ⩾ 1 sgeqslant 1 is a fixed integer, and x : { n 0 , n 0 + 1 , … } → R s x:left{{n}_{0},{n}_{0}+1,ldots right}to {{mathbb{R}}}^{s} is a dependent (unknown) variable. A retract principle is used to prove the existence of solutions with graphs remaining in a given domain for every n ⩾ n 0 ngeqslant {n}_{0} , which then serves as a basis for further proving the existence of bounded solutions to a linear nonhomogeneous system of discrete equations Δ α x ( n + 1 ) = A ( n ) x ( n ) + δ ( n ) , n = n 0 , n 0 + 1 , … , {Delta }^{alpha }xleft(n+1)=Aleft(n)xleft(n)+delta left(n),hspace{1em}n={n}_{0},{n}_{0}+1,ldots , where A ( n ) Aleft(n) is a square matrix and δ ( n ) delta left(n) is a vector function. Illustrative examples accompany the statements derived, possible generalizations are discussed, and open problems for future research are formulated as well.
摘要本文讨论分数阶离散方程组Δαx（n+1）=Fn（n，x（n），x（n-1），…，x（n0）），n=n0，n0+1，…，｛Delta｝={F}_{n} left（n，xleft（n），xlift（n-1），ldots，xlef({n}_｛0｝））， hspace{1em}n={n}_｛0｝，{n}_｛0｝+1，ldots，其中n 0∈Z{n}_｛0｝在｛mathbb｛Z｝｝中，n n是自变量，Δα｛Delta｝^｛alpha｝是αalpha阶分数差，α∈Ralpha在｛ mathbb｛R｝中｝，Fn:｛n｝×Rn−n0+1→ Rs{F}_{n} ：left-{n}_｛0｝+1｝to｛mathbb｛R｝｝^｛s｝，s⩾1sgeqslant 1是一个固定整数，x:｛n 0，n 0+1，…｝→ R s x:left{{n}_｛0｝，{n}_｛0｝+1，ldotsright｝to是一个因变量（未知）。对于每个n⩾n0ngeqslant，使用收回原理来证明图保留在给定域中的解的存在性{n}_｛0｝，然后作为进一步证明线性非齐次离散方程组Δαx（n+1）=a（n）x（n）+δ{1em}n={n}_｛0｝，{n}_｛0｝+1，ldots，其中A（n）Aleft（n）是一个平方矩阵，δ（n）deltaleft是一个向量函数。举例说明了所导出的陈述，讨论了可能的概括，并提出了未来研究的悬而未决的问题。
{"title":"Bounded solutions to systems of fractional discrete equations","authors":"J. Diblík","doi":"10.1515/anona-2022-0260","DOIUrl":"https://doi.org/10.1515/anona-2022-0260","url":null,"abstract":"Abstract The article is concerned with systems of fractional discrete equations Δ α x ( n + 1 ) = F n ( n , x ( n ) , x ( n − 1 ) , … , x ( n 0 ) ) , n = n 0 , n 0 + 1 , … , {Delta }^{alpha }xleft(n+1)={F}_{n}left(n,xleft(n),xleft(n-1),ldots ,xleft({n}_{0})),hspace{1em}n={n}_{0},{n}_{0}+1,ldots , where n 0 ∈ Z {n}_{0}in {mathbb{Z}} , n n is an independent variable, Δ α {Delta }^{alpha } is an α alpha -order fractional difference, α ∈ R alpha in {mathbb{R}} , F n : { n } × R n − n 0 + 1 → R s {F}_{n}:left{nright}times {{mathbb{R}}}^{n-{n}_{0}+1}to {{mathbb{R}}}^{s} , s ⩾ 1 sgeqslant 1 is a fixed integer, and x : { n 0 , n 0 + 1 , … } → R s x:left{{n}_{0},{n}_{0}+1,ldots right}to {{mathbb{R}}}^{s} is a dependent (unknown) variable. A retract principle is used to prove the existence of solutions with graphs remaining in a given domain for every n ⩾ n 0 ngeqslant {n}_{0} , which then serves as a basis for further proving the existence of bounded solutions to a linear nonhomogeneous system of discrete equations Δ α x ( n + 1 ) = A ( n ) x ( n ) + δ ( n ) , n = n 0 , n 0 + 1 , … , {Delta }^{alpha }xleft(n+1)=Aleft(n)xleft(n)+delta left(n),hspace{1em}n={n}_{0},{n}_{0}+1,ldots , where A ( n ) Aleft(n) is a square matrix and δ ( n ) delta left(n) is a vector function. Illustrative examples accompany the statements derived, possible generalizations are discussed, and open problems for future research are formulated as well.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"11 1","pages":"1614 - 1630"},"PeriodicalIF":4.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45448471","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 the non-linear Choquard equation − Δ u + V ( ∣ x ∣ ) u = ∫ R 3 ∣ u ( y ) ∣ 2 ∣ x − y ∣ d y u in R 3 , -Delta u+Vleft(| x| )u=left(mathop{int }limits_{{{mathbb{R}}}^{3}}frac{| u(y){| }^{2}}{| x-y| }{rm{d}}yright)uhspace{1.0em}hspace{0.1em}text{in}hspace{0.1em}hspace{0.33em}{{mathbb{R}}}^{3}, where V ( r ) Vleft(r) is a positive bounded function. Under some proper assumptions on V ( r ) Vleft(r) , we are able to establish the existence of infinitely many non-radial solutions.
摘要本文考虑非线性Choquard方程- Δ u+V(∣x∣)u=∫R 3∣u (y)∣2∣x - y∣d y u在R 3中，- Delta u+V left (| x|)u= left (mathop{int }limits ＿ {{{mathbb{R}}} ^{3}}frac{| u(y){| }^{2}}{| x-y| }{rm{d}} y right)u hspace{1.0em}hspace{0.1em}text{in}hspace{0.1em}hspace{0.33em}{{mathbb{R}}} ^{3}，其中V (R) V left (R)是一个正有界函数。在V (r) V left (r)的适当假设下，我们能够建立无穷多个非径向解的存在性。
{"title":"Infinitely many non-radial solutions for a Choquard equation","authors":"Fashun Gao, Minbo Yang","doi":"10.1515/anona-2022-0224","DOIUrl":"https://doi.org/10.1515/anona-2022-0224","url":null,"abstract":"Abstract In this article, we consider the non-linear Choquard equation − Δ u + V ( ∣ x ∣ ) u = ∫ R 3 ∣ u ( y ) ∣ 2 ∣ x − y ∣ d y u in R 3 , -Delta u+Vleft(| x| )u=left(mathop{int }limits_{{{mathbb{R}}}^{3}}frac{| u(y){| }^{2}}{| x-y| }{rm{d}}yright)uhspace{1.0em}hspace{0.1em}text{in}hspace{0.1em}hspace{0.33em}{{mathbb{R}}}^{3}, where V ( r ) Vleft(r) is a positive bounded function. Under some proper assumptions on V ( r ) Vleft(r) , we are able to establish the existence of infinitely many non-radial solutions.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"11 1","pages":"1085 - 1096"},"PeriodicalIF":4.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43707346","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 interested in the following Kirchhoff-type problem (0.1) − a + b ∫ R N ∣ ∇ u ∣ 2 d x Δ u + V ( ∣ x ∣ ) u = ∣ u ∣ 2 u in R N , u ∈ H 1 ( R N ) , left{begin{array}{l}-left(a+bmathop{displaystyle int }limits_{{{mathbb{R}}}^{N}}| nabla uhspace{-0.25em}{| }^{2}{rm{d}}xright)Delta u+Vleft(| x| )u=| uhspace{-0.25em}{| }^{2}uhspace{1.0em}{rm{in}}hspace{0.33em}{{mathbb{R}}}^{N}, uin {H}^{1}left({{mathbb{R}}}^{N}),end{array}right. where a , b > 0 , N = 2 a,bgt 0,N=2 or 3, the potential function V V is radial and bounded from below by a positive number. Because the nonlocal b ∣ ∇ u ∣ L 2 ( R N ) 2 Δ u b| nabla uhspace{-0.25em}{| }_{{L}^{2}left({{mathbb{R}}}^{N})}^{2}Delta u is 3-homogeneous which is in complicated competition with the nonlinear term ∣ u ∣ 2 u | uhspace{-0.25em}{| }^{2}u . This causes that not all function in H 1 ( R N ) {H}^{1}left({{mathbb{R}}}^{N}) can be projected on the Nehari manifold and thereby the classical Nehari manifold method does not work. By introducing the Gersgorin Disk theorem and the Miranda theorem, via a limit approach and subtle analysis, we prove that for each positive integer k k , equation (0.1) admits a radial nodal solution U k , 4 b {U}_{k,4}^{b} having exactly k k nodes. Moreover, we show that the energy of U k , 4 b {U}_{k,4}^{b} is strictly increasing in k k and for any sequence { b n } left{{b}_{n}right} with b n → 0 + , {b}_{n}to {0}_{+}, up to a subsequence, U k , 4 b n {U}_{k,4}^{{b}_{n}} converges to U k , 4 0 {U}_{k,4}^{0} in H 1 ( R N ) {H}^{1}left({{mathbb{R}}}^{N}) , which is a radial nodal solution with exactly k k nodes of the classical Schrödinger equation − a Δ u + V ( ∣ x ∣ ) u = ∣ u ∣ 2 u in R N , u ∈ H 1 ( R N ) . left{begin{array}{l}-aDelta u+Vleft(| x| )u=| uhspace{-0.25em}{| }^{2}uhspace{1.0em}{rm{in}}hspace{0.33em}{{mathbb{R}}}^{N}, uin {H}^{1}left({{mathbb{R}}}^{N}).end{array}right. Our results extend the existence result from the super-cubic case to the cubic case.
在本文中，我们对以下kirchhoff型问题(0.1)−a + b∫R N∣∇u∣2d x Δ u + V(∣x∣)u =∣u∣2u In R N, u∈h1 (R N)， left ｛begin{array}{l}-left(a+bmathop{displaystyle int }limits_{{{mathbb{R}}}^{N}}| nabla uhspace{-0.25em}{| }^{2}{rm{d}}xright)Delta u+Vleft(| x| )u=| uhspace{-0.25em}{| }^{2}uhspace{1.0em}{rm{in}}hspace{0.33em}{{mathbb{R}}}^{N}, uin {H}^{1}left({{mathbb{R}}}^{N}),end{array}right感兴趣。其中，a,b > 0,N=2, a,b gt 0,N=2或3，势函数V V是径向的，从下面开始有一个正数。因为非局部的b∣∇u∣l2 (R N) 2 Δ u b| nabla u＿Lhspace{-0.25em}{| }^{{2 }{}left ({{mathbb{R}}} ^{N})}^{2 }Delta u是3齐次的这与非线性项∣u∣2u | uhspace{-0.25em}{| }^{2u是复杂的竞争关系。这导致不是所有在h1 (rn) H}^{1}{}left ({{mathbb{R}}} ^{N})中的函数都可以投影到Nehari流形上，因此经典的Nehari流形方法不起作用。通过引入Gersgorin圆盘定理和Miranda定理，通过极限逼近和精细分析，证明了对于每一个正整数k k，方程(0.1)承认一个径向节点解U k, 4b U {k,4}^{b}恰好有k k个节点。此外，我们证明了uk, 4b {U＿k},4{^}b{的能量在k k中是严格递增的，对于任意序列}b n{}{}left ｛{b＿n}{}right｝，当b n→0 +，{b＿n}{}to 0＿{+}时，直到一个子序列，uk, 4b n {U＿k},4{^}b＿n{收敛于uk, 40 }U＿k{{,4}^b＿n在H 1 (R n) {H}}^{1}{}{}{}{}left ({{mathbb{R}}} ^ {n})，这是经典Schrödinger方程- a Δ U + V(∣x∣)U =∣U∣2u在R n, U∈H 1 (R n)中具有精确k个节点的径向节点解。left ｛begin{array}{l}-aDelta u+Vleft(| x| )u=| uhspace{-0.25em}{| }^{2}uhspace{1.0em}{rm{in}}hspace{0.33em}{{mathbb{R}}}^{N}, uin {H}^{1}left({{mathbb{R}}}^{N}).end{array}right。我们的结果将超三次情况下的存在性结果推广到三次情况。
{"title":"Multiple nodal solutions of the Kirchhoff-type problem with a cubic term","authors":"Tao Wang, Yanling Yang, Hui Guo","doi":"10.1515/anona-2022-0225","DOIUrl":"https://doi.org/10.1515/anona-2022-0225","url":null,"abstract":"Abstract In this article, we are interested in the following Kirchhoff-type problem (0.1) − a + b ∫ R N ∣ ∇ u ∣ 2 d x Δ u + V ( ∣ x ∣ ) u = ∣ u ∣ 2 u in R N , u ∈ H 1 ( R N ) , left{begin{array}{l}-left(a+bmathop{displaystyle int }limits_{{{mathbb{R}}}^{N}}| nabla uhspace{-0.25em}{| }^{2}{rm{d}}xright)Delta u+Vleft(| x| )u=| uhspace{-0.25em}{| }^{2}uhspace{1.0em}{rm{in}}hspace{0.33em}{{mathbb{R}}}^{N}, uin {H}^{1}left({{mathbb{R}}}^{N}),end{array}right. where a , b > 0 , N = 2 a,bgt 0,N=2 or 3, the potential function V V is radial and bounded from below by a positive number. Because the nonlocal b ∣ ∇ u ∣ L 2 ( R N ) 2 Δ u b| nabla uhspace{-0.25em}{| }_{{L}^{2}left({{mathbb{R}}}^{N})}^{2}Delta u is 3-homogeneous which is in complicated competition with the nonlinear term ∣ u ∣ 2 u | uhspace{-0.25em}{| }^{2}u . This causes that not all function in H 1 ( R N ) {H}^{1}left({{mathbb{R}}}^{N}) can be projected on the Nehari manifold and thereby the classical Nehari manifold method does not work. By introducing the Gersgorin Disk theorem and the Miranda theorem, via a limit approach and subtle analysis, we prove that for each positive integer k k , equation (0.1) admits a radial nodal solution U k , 4 b {U}_{k,4}^{b} having exactly k k nodes. Moreover, we show that the energy of U k , 4 b {U}_{k,4}^{b} is strictly increasing in k k and for any sequence { b n } left{{b}_{n}right} with b n → 0 + , {b}_{n}to {0}_{+}, up to a subsequence, U k , 4 b n {U}_{k,4}^{{b}_{n}} converges to U k , 4 0 {U}_{k,4}^{0} in H 1 ( R N ) {H}^{1}left({{mathbb{R}}}^{N}) , which is a radial nodal solution with exactly k k nodes of the classical Schrödinger equation − a Δ u + V ( ∣ x ∣ ) u = ∣ u ∣ 2 u in R N , u ∈ H 1 ( R N ) . left{begin{array}{l}-aDelta u+Vleft(| x| )u=| uhspace{-0.25em}{| }^{2}uhspace{1.0em}{rm{in}}hspace{0.33em}{{mathbb{R}}}^{N}, uin {H}^{1}left({{mathbb{R}}}^{N}).end{array}right. Our results extend the existence result from the super-cubic case to the cubic case.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"11 1","pages":"1030 - 1047"},"PeriodicalIF":4.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48120538","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 paper we study the existence and the nonexistence of solutions to an inhomogeneous non-linear elliptic problem (P) −Δu+u=F(u)+κμ in RN, u>0 in RN, u(x)→0 as |x|→∞, - Delta u + u = F(u) + kappa mu quad {kern 1pt} {rm in}{kern 1pt} quad {{bf R}^N},quad u > 0quad {kern 1pt} {rm in}{kern 1pt} quad {{bf R}^N},quad u(x) to 0quad {kern 1pt} {rm as}{kern 1pt} quad |x| to infty , where F = F(t) grows up (at least) exponentially as t → ∞. Here N ≥ 2, κ > 0, and μ∈Lc1(RN){0} mu in L_{rm{c}}^1({{bf R}^N})backslash { 0} is nonnegative. Then, under a suitable integrability condition on μ, there exists a threshold parameter κ* > 0 such that problem (P) possesses a solution if 0 < κ < κ* and it does not possess no solutions if κ > κ*. Furthermore, in the case of 2 ≤ N ≤ 9, problem (P) possesses a unique solution if κ = κ*.
{"title":"Thresholds for the existence of solutions to inhomogeneous elliptic equations with general exponential nonlinearity","authors":"Kazuhiro Ishige, S. Okabe, Tokushi Sato","doi":"10.1515/anona-2021-0220","DOIUrl":"https://doi.org/10.1515/anona-2021-0220","url":null,"abstract":"Abstract In this paper we study the existence and the nonexistence of solutions to an inhomogeneous non-linear elliptic problem (P) −Δu+u=F(u)+κμ in RN, u>0 in RN, u(x)→0 as |x|→∞, - Delta u + u = F(u) + kappa mu quad {kern 1pt} {rm in}{kern 1pt} quad {{bf R}^N},quad u > 0quad {kern 1pt} {rm in}{kern 1pt} quad {{bf R}^N},quad u(x) to 0quad {kern 1pt} {rm as}{kern 1pt} quad |x| to infty , where F = F(t) grows up (at least) exponentially as t → ∞. Here N ≥ 2, κ > 0, and μ∈Lc1(RN){0} mu in L_{rm{c}}^1({{bf R}^N})backslash { 0} is nonnegative. Then, under a suitable integrability condition on μ, there exists a threshold parameter κ* > 0 such that problem (P) possesses a solution if 0 < κ < κ* and it does not possess no solutions if κ > κ*. Furthermore, in the case of 2 ≤ N ≤ 9, problem (P) possesses a unique solution if κ = κ*.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"11 1","pages":"968 - 992"},"PeriodicalIF":4.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45919650","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 discrete Kirchhoff-type problems when the nonlinearity is resonant at both zero and infinity. We establish a series of results on the existence of nontrivial solutions by combining variational method with Morse theory. Several examples are provided to illustrate applications of our results.
{"title":"Nontrivial solutions of discrete Kirchhoff-type problems via Morse theory","authors":"Y. Long","doi":"10.1515/anona-2022-0251","DOIUrl":"https://doi.org/10.1515/anona-2022-0251","url":null,"abstract":"Abstract In this article, we study discrete Kirchhoff-type problems when the nonlinearity is resonant at both zero and infinity. We establish a series of results on the existence of nontrivial solutions by combining variational method with Morse theory. Several examples are provided to illustrate applications of our results.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"11 1","pages":"1352 - 1364"},"PeriodicalIF":4.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45001118","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 paper is devoted to investigate the existence and multiplicity of the normalized solutions for the following fractional Schrödinger equation: (P) ( − Δ ) s u + λ u = μ ∣ u ∣ p − 2 u + ∣ u ∣ 2 s ∗ − 2 u , x ∈ R N , u > 0 , ∫ R N ∣ u ∣ 2 d x = a 2 , left{begin{array}{l}{left(-Delta )}^{s}u+lambda u=mu | u{| }^{p-2}u+| u{| }^{{2}_{s}^{ast }-2}u,hspace{1em}xin {{mathbb{R}}}^{N},hspace{1.0em} ugt 0,hspace{1em}mathop{displaystyle int }limits_{{{mathbb{R}}}^{N}}| u{| }^{2}{rm{d}}x={a}^{2},hspace{1.0em}end{array}right. where 0 < s < 1 0lt slt 1 , a a , μ > 0 mu gt 0 , N ≥ 2 Nge 2 , and 2 < p < 2 s ∗ 2lt plt {2}_{s}^{ast } . We consider the L 2 {L}^{2} -subcritical and L 2 {L}^{2} -supercritical cases. More precisely, in L 2 {L}^{2} -subcritical case, we obtain the multiplicity of the normalized solutions for problem ( P ) left(P) by using the truncation technique, concentration-compactness principle, and genus theory. In L 2 {L}^{2} -supercritical case, we obtain a couple of normalized solution for ( P ) left(P) by using a fiber map and concentration-compactness principle. To some extent, these results can be viewed as an extension of the existing results from Sobolev subcritical growth to Sobolev critical growth.
{"title":"The existence and multiplicity of the normalized solutions for fractional Schrödinger equations involving Sobolev critical exponent in the L2-subcritical and L2-supercritical cases","authors":"Quanqing Li, W. Zou","doi":"10.1515/anona-2022-0252","DOIUrl":"https://doi.org/10.1515/anona-2022-0252","url":null,"abstract":"Abstract This paper is devoted to investigate the existence and multiplicity of the normalized solutions for the following fractional Schrödinger equation: (P) ( − Δ ) s u + λ u = μ ∣ u ∣ p − 2 u + ∣ u ∣ 2 s ∗ − 2 u , x ∈ R N , u > 0 , ∫ R N ∣ u ∣ 2 d x = a 2 , left{begin{array}{l}{left(-Delta )}^{s}u+lambda u=mu | u{| }^{p-2}u+| u{| }^{{2}_{s}^{ast }-2}u,hspace{1em}xin {{mathbb{R}}}^{N},hspace{1.0em} ugt 0,hspace{1em}mathop{displaystyle int }limits_{{{mathbb{R}}}^{N}}| u{| }^{2}{rm{d}}x={a}^{2},hspace{1.0em}end{array}right. where 0 < s < 1 0lt slt 1 , a a , μ > 0 mu gt 0 , N ≥ 2 Nge 2 , and 2 < p < 2 s ∗ 2lt plt {2}_{s}^{ast } . We consider the L 2 {L}^{2} -subcritical and L 2 {L}^{2} -supercritical cases. More precisely, in L 2 {L}^{2} -subcritical case, we obtain the multiplicity of the normalized solutions for problem ( P ) left(P) by using the truncation technique, concentration-compactness principle, and genus theory. In L 2 {L}^{2} -supercritical case, we obtain a couple of normalized solution for ( P ) left(P) by using a fiber map and concentration-compactness principle. To some extent, these results can be viewed as an extension of the existing results from Sobolev subcritical growth to Sobolev critical growth.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"11 1","pages":"1531 - 1551"},"PeriodicalIF":4.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46463248","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 paper, a class of Schrödinger equations is investigated, which can be stated as −Δu+V(x)u=f(u), x∈ℝN. - Delta u + V(x)u = f(u),;;;;x in {{rm{mathbb R}}^N}. If the external potential V is radial and coercive, then we give the local Ambrosetti-Rabinowitz super-linear condition on the nonlinearity term f ∈ C(ℝ, ℝ) which assures the problem has not only infinitely many radial sign-changing solutions, but also infinitely many non-radial sign-changing solutions.
{"title":"Infinitely many radial and non-radial sign-changing solutions for Schrödinger equations","authors":"Gui-Dong Li, Yong-Yong Li, Chunlei Tang","doi":"10.1515/anona-2021-0221","DOIUrl":"https://doi.org/10.1515/anona-2021-0221","url":null,"abstract":"Abstract In the present paper, a class of Schrödinger equations is investigated, which can be stated as −Δu+V(x)u=f(u), x∈ℝN. - Delta u + V(x)u = f(u),;;;;x in {{rm{mathbb R}}^N}. If the external potential V is radial and coercive, then we give the local Ambrosetti-Rabinowitz super-linear condition on the nonlinearity term f ∈ C(ℝ, ℝ) which assures the problem has not only infinitely many radial sign-changing solutions, but also infinitely many non-radial sign-changing solutions.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"11 1","pages":"907 - 920"},"PeriodicalIF":4.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47358939","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}
Vincenzo Amato, Alba Lia Masiello, C. Nitsch, C. Trombetti
Abstract We study the behaviour, when p → + ∞ pto +infty , of the first p-Laplacian eigenvalues with Robin boundary conditions and the limit of the associated eigenfunctions. We prove that the limit of the eigenfunctions is a viscosity solution to an eigenvalue problem for the so-called ∞ infty -Laplacian. Moreover, in the second part of the article, we focus our attention on the p-Poisson equation when the datum f f belongs to L ∞ ( Ω ) {L}^{infty }left(Omega ) and we study the behaviour of solutions when p → ∞ pto infty .
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Abstract In this article, we aimed to study a class of nonhomogeneous fractional (p, q)-Laplacian systems with critical nonlinearities as well as critical Hardy nonlinearities in R N {{mathbb{R}}}^{N} . By appealing to a fixed point result and fractional Hardy-Sobolev inequality, the existence of nontrivial nonnegative solutions is obtained. In particular, we also consider Choquard-type nonlinearities in the second part of this article. More precisely, with the help of Hardy-Littlewood-Sobolev inequality, we obtain the existence of nontrivial solutions for the related systems based on the same approach. Finally, we obtain the corresponding existence results for the fractional (p, q)-Laplacian systems in the case of N = s p = l q N=sp=lq . It is worth pointing out that using fixed point argument to seek solutions for a class of nonhomogeneous fractional (p, q)-Laplacian systems is the main novelty of this article.
摘要本文研究了一类具有临界非线性和临界Hardy非线性的非齐次分数(p, q)-拉普拉斯系统在R N {{mathbb{R}}}^{N}上的问题。利用不动点结果和分数阶Hardy-Sobolev不等式，得到了非平凡非负解的存在性。特别地，我们还在本文的第二部分中考虑了choquard型非线性。更准确地说，我们利用Hardy-Littlewood-Sobolev不等式，基于相同的方法，得到了相关系统非平凡解的存在性。最后，我们得到了N=sp=lq N=sp=lq情况下分数阶(p, q)-拉普拉斯系统的存在性结果。值得指出的是，利用不动点参数求一类非齐次分数(p, q)-拉普拉斯系统的解是本文的主要新颖之处。
{"title":"Solutions for nonhomogeneous fractional (p, q)-Laplacian systems with critical nonlinearities","authors":"Mengfei Tao, Binlin Zhang","doi":"10.1515/anona-2022-0248","DOIUrl":"https://doi.org/10.1515/anona-2022-0248","url":null,"abstract":"Abstract In this article, we aimed to study a class of nonhomogeneous fractional (p, q)-Laplacian systems with critical nonlinearities as well as critical Hardy nonlinearities in R N {{mathbb{R}}}^{N} . By appealing to a fixed point result and fractional Hardy-Sobolev inequality, the existence of nontrivial nonnegative solutions is obtained. In particular, we also consider Choquard-type nonlinearities in the second part of this article. More precisely, with the help of Hardy-Littlewood-Sobolev inequality, we obtain the existence of nontrivial solutions for the related systems based on the same approach. Finally, we obtain the corresponding existence results for the fractional (p, q)-Laplacian systems in the case of N = s p = l q N=sp=lq . It is worth pointing out that using fixed point argument to seek solutions for a class of nonhomogeneous fractional (p, q)-Laplacian systems is the main novelty of this article.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"11 1","pages":"1332 - 1351"},"PeriodicalIF":4.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46139127","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 Let n ≥ 2 nge 2 and Ω ⊂ R n Omega subset {{mathbb{R}}}^{n} be a bounded nontangentially accessible domain. In this article, the authors investigate (weighted) global gradient estimates for Dirichlet boundary value problems of second-order elliptic equations of divergence form with an elliptic symmetric part and a BMO antisymmetric part in Ω Omega . More precisely, for any given p ∈ ( 2 , ∞ ) pin left(2,infty ) , the authors prove that a weak reverse Hölder inequality with exponent p p implies the global W 1 , p {W}^{1,p} estimate and the global weighted W 1 , q {W}^{1,q} estimate, with q ∈ [ 2 , p ] qin left[2,p] and some Muckenhoupt weights, of solutions to Dirichlet boundary value problems. As applications, the authors establish some global gradient estimates for solutions to Dirichlet boundary value problems of second-order elliptic equations of divergence form with small BMO {rm{BMO}} symmetric part and small BMO {rm{BMO}} antisymmetric part, respectively, on bounded Lipschitz domains, quasi-convex domains, Reifenberg flat domains, C 1 {C}^{1} domains, or (semi-)convex domains, in weighted Lebesgue spaces. Furthermore, as further applications, the authors obtain the global gradient estimate, respectively, in (weighted) Lorentz spaces, (Lorentz–)Morrey spaces, (Musielak–)Orlicz spaces, and variable Lebesgue spaces. Even on global gradient estimates in Lebesgue spaces, the results obtained in this article improve the known results via weakening the assumption on the coefficient matrix.
{"title":"Global gradient estimates for Dirichlet problems of elliptic operators with a BMO antisymmetric part","authors":"Sibei Yang, Dachun Yang, Wen Yuan","doi":"10.1515/anona-2022-0247","DOIUrl":"https://doi.org/10.1515/anona-2022-0247","url":null,"abstract":"Abstract Let n ≥ 2 nge 2 and Ω ⊂ R n Omega subset {{mathbb{R}}}^{n} be a bounded nontangentially accessible domain. In this article, the authors investigate (weighted) global gradient estimates for Dirichlet boundary value problems of second-order elliptic equations of divergence form with an elliptic symmetric part and a BMO antisymmetric part in Ω Omega . More precisely, for any given p ∈ ( 2 , ∞ ) pin left(2,infty ) , the authors prove that a weak reverse Hölder inequality with exponent p p implies the global W 1 , p {W}^{1,p} estimate and the global weighted W 1 , q {W}^{1,q} estimate, with q ∈ [ 2 , p ] qin left[2,p] and some Muckenhoupt weights, of solutions to Dirichlet boundary value problems. As applications, the authors establish some global gradient estimates for solutions to Dirichlet boundary value problems of second-order elliptic equations of divergence form with small BMO {rm{BMO}} symmetric part and small BMO {rm{BMO}} antisymmetric part, respectively, on bounded Lipschitz domains, quasi-convex domains, Reifenberg flat domains, C 1 {C}^{1} domains, or (semi-)convex domains, in weighted Lebesgue spaces. Furthermore, as further applications, the authors obtain the global gradient estimate, respectively, in (weighted) Lorentz spaces, (Lorentz–)Morrey spaces, (Musielak–)Orlicz spaces, and variable Lebesgue spaces. Even on global gradient estimates in Lebesgue spaces, the results obtained in this article improve the known results via weakening the assumption on the coefficient matrix.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":"11 1","pages":"1496 - 1530"},"PeriodicalIF":4.2,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43249751","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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