We consider the geometric structure of quasi-normed Calderón–Lozanovskiĭ spaces. First, we study relations between the quasi-norm and the quasi-modular “near zero” and “near one,” which are fundamental for the theory. With their help, we provide a precise description of the basic monotonicity properties. In comparison with the well-known normed case, we develop a number of new techniques and methods, among which the conditions and play a crucial role. From our general results, we conclude the criteria for monotonicity properties in quasi-normed Orlicz spaces, which are new even in this particular context. We consider both the function and the sequence case as well as we admit degenerated Orlicz functions, which provides us with a maximal class of spaces under consideration. We also discuss the applications of suitable properties to the best dominated approximation problems in quasi-Banach lattices.
{"title":"Monotonicities of quasi-normed Calderón–Lozanovskiĭ spaces with applications to approximation problems","authors":"Paweł Foralewski, Paweł Kolwicz","doi":"10.1002/mana.202400013","DOIUrl":"10.1002/mana.202400013","url":null,"abstract":"<p>We consider the geometric structure of quasi-normed Calderón–Lozanovskiĭ spaces. First, we study relations between the quasi-norm and the quasi-modular “near zero” and “near one,” which are fundamental for the theory. With their help, we provide a precise description of the basic monotonicity properties. In comparison with the well-known normed case, we develop a number of new techniques and methods, among which the conditions <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>Δ</mi>\u0000 <mi>ε</mi>\u0000 </msub>\u0000 <annotation>$Delta _{varepsilon }$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>Δ</mi>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mo>−</mo>\u0000 <mi>s</mi>\u0000 <mi>t</mi>\u0000 <mi>r</mi>\u0000 </mrow>\u0000 </msub>\u0000 <annotation>$Delta _{2-str}$</annotation>\u0000 </semantics></math> play a crucial role. From our general results, we conclude the criteria for monotonicity properties in quasi-normed Orlicz spaces, which are new even in this particular context. We consider both the function and the sequence case as well as we admit degenerated Orlicz functions, which provides us with a maximal class of spaces under consideration. We also discuss the applications of suitable properties to the best dominated approximation problems in quasi-Banach lattices.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 11","pages":"4081-4114"},"PeriodicalIF":0.8,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142176864","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
<p>If <span></span><math> <semantics> <mi>C</mi> <annotation>$C$</annotation> </semantics></math> is a smooth projective curve over an algebraically closed field <span></span><math> <semantics> <mi>F</mi> <annotation>$mathbb {F}$</annotation> </semantics></math> and <span></span><math> <semantics> <mi>G</mi> <annotation>$G$</annotation> </semantics></math> is a group of automorphisms of <span></span><math> <semantics> <mi>C</mi> <annotation>$C$</annotation> </semantics></math>, the <i>canonical representation of</i> <span></span><math> <semantics> <mi>C</mi> <annotation>$C$</annotation> </semantics></math> is given by the induced <span></span><math> <semantics> <mi>F</mi> <annotation>$mathbb {F}$</annotation> </semantics></math>-linear action of <span></span><math> <semantics> <mi>G</mi> <annotation>$G$</annotation> </semantics></math> on the vector space <span></span><math> <semantics> <mrow> <msup> <mi>H</mi> <mn>0</mn> </msup> <mfenced> <mi>C</mi> <mo>,</mo> <msub> <mi>Ω</mi> <mi>C</mi> </msub> </mfenced> </mrow> <annotation>$H^0left(C,Omega _Cright)$</annotation> </semantics></math> of holomorphic differentials on <span></span><math> <semantics> <mi>C</mi> <annotation>$C$</annotation> </semantics></math>. Computing it is still an open problem in general when the cover <span></span><math> <semantics> <mrow> <mi>C</mi> <mo>→</mo> <mi>C</mi> <mo>/</mo> <mi>G</mi> </mrow> <annotation>$C rightarrow C/G$</annotation> </semantics></math> is wildly ramified. In this paper, we fix a prime power <span></span><math> <semantics> <mi>q</mi> <annotation>$q$</annotation> </semantics></math>, we consider the Drinfeld curve, that is, the curve <span></span><math> <semantics> <mi>C</mi> <annotation>$C$</annotation> </semantics></math> given by the equation <span></span
如果 是一条代数闭域上的光滑投影曲线，并且 是 的自动形群 ，那么 的典型表示是由 的在全形微分向量空间上的诱导线性作用给出的。在一般情况下，当覆盖有大量斜边时，计算它仍是一个未决问题。在本文中，我们固定一个质幂 ，考虑德林费尔德曲线，即由方程 over 及其标准作用给出的曲线，并将其分解为Ⅳ的不可分解表示的直接和，从而解决了这种情况下的上述问题。
{"title":"The canonical representation of the Drinfeld curve","authors":"Lucas Laurent, Bernhard Köck","doi":"10.1002/mana.202200402","DOIUrl":"10.1002/mana.202200402","url":null,"abstract":"<p>If <span></span><math>\u0000 <semantics>\u0000 <mi>C</mi>\u0000 <annotation>$C$</annotation>\u0000 </semantics></math> is a smooth projective curve over an algebraically closed field <span></span><math>\u0000 <semantics>\u0000 <mi>F</mi>\u0000 <annotation>$mathbb {F}$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> is a group of automorphisms of <span></span><math>\u0000 <semantics>\u0000 <mi>C</mi>\u0000 <annotation>$C$</annotation>\u0000 </semantics></math>, the <i>canonical representation of</i> <span></span><math>\u0000 <semantics>\u0000 <mi>C</mi>\u0000 <annotation>$C$</annotation>\u0000 </semantics></math> is given by the induced <span></span><math>\u0000 <semantics>\u0000 <mi>F</mi>\u0000 <annotation>$mathbb {F}$</annotation>\u0000 </semantics></math>-linear action of <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> on the vector space <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>H</mi>\u0000 <mn>0</mn>\u0000 </msup>\u0000 <mfenced>\u0000 <mi>C</mi>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>Ω</mi>\u0000 <mi>C</mi>\u0000 </msub>\u0000 </mfenced>\u0000 </mrow>\u0000 <annotation>$H^0left(C,Omega _Cright)$</annotation>\u0000 </semantics></math> of holomorphic differentials on <span></span><math>\u0000 <semantics>\u0000 <mi>C</mi>\u0000 <annotation>$C$</annotation>\u0000 </semantics></math>. Computing it is still an open problem in general when the cover <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 <mo>→</mo>\u0000 <mi>C</mi>\u0000 <mo>/</mo>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation>$C rightarrow C/G$</annotation>\u0000 </semantics></math> is wildly ramified. In this paper, we fix a prime power <span></span><math>\u0000 <semantics>\u0000 <mi>q</mi>\u0000 <annotation>$q$</annotation>\u0000 </semantics></math>, we consider the Drinfeld curve, that is, the curve <span></span><math>\u0000 <semantics>\u0000 <mi>C</mi>\u0000 <annotation>$C$</annotation>\u0000 </semantics></math> given by the equation <span></span","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 11","pages":"4115-4120"},"PeriodicalIF":0.8,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.202200402","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142223556","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce the notion of localized operators. We extend the boundedness of the localized operators from the weighted Lebesgue spaces to the weighted Herz spaces. The localized operators include the Hardy operator, the Riemann–Liouville fractional integrals, the general Hardy-type operators, the geometric mean operator, and the one-sided maximal function. Therefore, this paper extends the mapping properties of the the Hardy operator, the Riemann–Liouville fractional integrals, the general Hardy-type operators, the geometric mean operator, and the one-sided maximal function to the weighted Herz spaces.
{"title":"Localized operators on weighted Herz spaces","authors":"Kwok-Pun Ho","doi":"10.1002/mana.202400086","DOIUrl":"10.1002/mana.202400086","url":null,"abstract":"<p>We introduce the notion of localized operators. We extend the boundedness of the localized operators from the weighted Lebesgue spaces to the weighted Herz spaces. The localized operators include the Hardy operator, the Riemann–Liouville fractional integrals, the general Hardy-type operators, the geometric mean operator, and the one-sided maximal function. Therefore, this paper extends the mapping properties of the the Hardy operator, the Riemann–Liouville fractional integrals, the general Hardy-type operators, the geometric mean operator, and the one-sided maximal function to the weighted Herz spaces.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 11","pages":"4067-4080"},"PeriodicalIF":0.8,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142176863","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study a superlinear elliptic boundary value problem involving the -Laplacian operator, with changing sign weights. The problem has positive solutions bifurcating from the trivial solution set at the two principal eigenvalues of the corresponding linear weighted boundary value problem.
Drabek's bifurcation result applies when the nonlinearity is of power growth. We extend Drabek's bifurcation result to slightly subcritical nonlinearities. Compactness in this setting is a delicate issue obtained via Orlicz spaces.
{"title":"Bifurcation for indefinite-weighted \u0000 \u0000 p\u0000 $p$\u0000 -Laplacian problems with slightly subcritical nonlinearity","authors":"Mabel Cuesta, Rosa Pardo","doi":"10.1002/mana.202400184","DOIUrl":"10.1002/mana.202400184","url":null,"abstract":"<p>We study a superlinear elliptic boundary value problem involving the <span></span><math>\u0000 <semantics>\u0000 <mi>p</mi>\u0000 <annotation>$p$</annotation>\u0000 </semantics></math>-Laplacian operator, with changing sign weights. The problem has positive solutions bifurcating from the trivial solution set at the two principal eigenvalues of the corresponding linear weighted boundary value problem.</p><p>Drabek's bifurcation result applies when the nonlinearity is of power growth. We extend Drabek's bifurcation result to <i>slightly subcritical</i> nonlinearities. Compactness in this setting is a delicate issue obtained via Orlicz spaces.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 11","pages":"3982-4002"},"PeriodicalIF":0.8,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.202400184","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142176978","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we extend the well-known concentration–compactness principle of P.L. Lions to Orlicz spaces. As an application, we show an existence result to some critical elliptic problem with nonstandard growth.
{"title":"The concentration–compactness principle for Orlicz spaces and applications","authors":"Julián Fernández Bonder, Analía Silva","doi":"10.1002/mana.202300469","DOIUrl":"10.1002/mana.202300469","url":null,"abstract":"<p>In this paper, we extend the well-known concentration–compactness principle of P.L. Lions to Orlicz spaces. As an application, we show an existence result to some critical elliptic problem with nonstandard growth.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 11","pages":"4044-4066"},"PeriodicalIF":0.8,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142176876","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose a notion of stability for constant -mean curvature hypersurfaces in a general Riemannian manifold and we give some applications. When the ambient manifold is a Space Form, our notion coincides with the known one, given by means of the variational problem.
{"title":"On the stability of constant higher order mean curvature hypersurfaces in a Riemannian manifold","authors":"Maria Fernanda Elbert, Barbara Nelli","doi":"10.1002/mana.202400159","DOIUrl":"10.1002/mana.202400159","url":null,"abstract":"<p>We propose a notion of stability for constant <span></span><math>\u0000 <semantics>\u0000 <mi>k</mi>\u0000 <annotation>$k$</annotation>\u0000 </semantics></math>-mean curvature hypersurfaces in a general Riemannian manifold and we give some applications. When the ambient manifold is a Space Form, our notion coincides with the known one, given by means of the variational problem.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 11","pages":"4031-4043"},"PeriodicalIF":0.8,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142176975","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the fully nonlocal diffusion equations with nonnegative -data. Based on the approximation and energy methods, we prove the existence and uniqueness of nonnegative entropy solutions for such problems. In particular, our results are valid for the time-space fractional Laplacian equations.
{"title":"Entropy solutions to the fully nonlocal diffusion equations","authors":"Ying Li, Chao Zhang","doi":"10.1002/mana.202400130","DOIUrl":"10.1002/mana.202400130","url":null,"abstract":"<p>We consider the fully nonlocal diffusion equations with nonnegative <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mn>1</mn>\u0000 </msup>\u0000 <annotation>$L^1$</annotation>\u0000 </semantics></math>-data. Based on the approximation and energy methods, we prove the existence and uniqueness of nonnegative entropy solutions for such problems. In particular, our results are valid for the time-space fractional Laplacian equations.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 11","pages":"4003-4030"},"PeriodicalIF":0.8,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142176977","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
<p>Let <span></span><math> <semantics> <mrow> <mi>d</mi> <mo>≥</mo> <mn>3</mn> </mrow> <annotation>$dge 3$</annotation> </semantics></math> be an integer. For a holomorphic <span></span><math> <semantics> <mi>d</mi> <annotation>$d$</annotation> </semantics></math>-web <span></span><math> <semantics> <mi>W</mi> <annotation>$mathcal {W}$</annotation> </semantics></math> on a complex surface <span></span><math> <semantics> <mi>M</mi> <annotation>$M$</annotation> </semantics></math>, smooth along an irreducible component <span></span><math> <semantics> <mi>D</mi> <annotation>$D$</annotation> </semantics></math> of its discriminant <span></span><math> <semantics> <mrow> <mi>Δ</mi> <mo>(</mo> <mi>W</mi> <mo>)</mo> </mrow> <annotation>$Delta (mathcal {W})$</annotation> </semantics></math>, we establish an effective criterion for the holomorphy of the curvature of <span></span><math> <semantics> <mi>W</mi> <annotation>$mathcal {W}$</annotation> </semantics></math> along <span></span><math> <semantics> <mi>D</mi> <annotation>$D$</annotation> </semantics></math>, generalizing results on decomposable webs due to Marín, Pereira, and Pirio. As an application, we deduce a complete characterization for the holomorphy of the curvature of the Legendre transform (dual web) <span></span><math> <semantics> <mrow> <mi>Leg</mi> <mi>H</mi> </mrow> <annotation>$mathrm{Leg}mathcal {H}$</annotation> </semantics></math> of a homogeneous foliation <span></span><math> <semantics> <mi>H</mi> <annotation>$mathcal {H}$</annotation> </semantics></math> of degree <span></span><math> <semantics> <mi>d</mi> <annotation>$d$</annotation> </semantics></math> on <span></span><math> <semantics> <msubsup> <mi>P</mi> <mi>C</mi> <mn>2</mn> </msubsup> <annotation>$mathbb {P}^{2}_{mathbb {C}}$</annotation> </semantics></math>, generalizing some of our previous results. This then allows us to study the flatness of the <span></span><math> <semantics> <mi>d</mi> <annotation>$d$</annotation> </semantics></math>-web <span></span><math>
{"title":"A criterion for the holomorphy of the curvature of smooth planar webs and applications to dual webs of homogeneous foliations on \u0000 \u0000 \u0000 P\u0000 C\u0000 2\u0000 \u0000 $mathbb {P}^{2}_{mathbb {C}}$","authors":"Samir Bedrouni, David Marín","doi":"10.1002/mana.202400150","DOIUrl":"10.1002/mana.202400150","url":null,"abstract":"<p>Let <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 <mo>≥</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$dge 3$</annotation>\u0000 </semantics></math> be an integer. For a holomorphic <span></span><math>\u0000 <semantics>\u0000 <mi>d</mi>\u0000 <annotation>$d$</annotation>\u0000 </semantics></math>-web <span></span><math>\u0000 <semantics>\u0000 <mi>W</mi>\u0000 <annotation>$mathcal {W}$</annotation>\u0000 </semantics></math> on a complex surface <span></span><math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$M$</annotation>\u0000 </semantics></math>, smooth along an irreducible component <span></span><math>\u0000 <semantics>\u0000 <mi>D</mi>\u0000 <annotation>$D$</annotation>\u0000 </semantics></math> of its discriminant <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Δ</mi>\u0000 <mo>(</mo>\u0000 <mi>W</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$Delta (mathcal {W})$</annotation>\u0000 </semantics></math>, we establish an effective criterion for the holomorphy of the curvature of <span></span><math>\u0000 <semantics>\u0000 <mi>W</mi>\u0000 <annotation>$mathcal {W}$</annotation>\u0000 </semantics></math> along <span></span><math>\u0000 <semantics>\u0000 <mi>D</mi>\u0000 <annotation>$D$</annotation>\u0000 </semantics></math>, generalizing results on decomposable webs due to Marín, Pereira, and Pirio. As an application, we deduce a complete characterization for the holomorphy of the curvature of the Legendre transform (dual web) <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Leg</mi>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 <annotation>$mathrm{Leg}mathcal {H}$</annotation>\u0000 </semantics></math> of a homogeneous foliation <span></span><math>\u0000 <semantics>\u0000 <mi>H</mi>\u0000 <annotation>$mathcal {H}$</annotation>\u0000 </semantics></math> of degree <span></span><math>\u0000 <semantics>\u0000 <mi>d</mi>\u0000 <annotation>$d$</annotation>\u0000 </semantics></math> on <span></span><math>\u0000 <semantics>\u0000 <msubsup>\u0000 <mi>P</mi>\u0000 <mi>C</mi>\u0000 <mn>2</mn>\u0000 </msubsup>\u0000 <annotation>$mathbb {P}^{2}_{mathbb {C}}$</annotation>\u0000 </semantics></math>, generalizing some of our previous results. This then allows us to study the flatness of the <span></span><math>\u0000 <semantics>\u0000 <mi>d</mi>\u0000 <annotation>$d$</annotation>\u0000 </semantics></math>-web <span></span><math>\u0000 ","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 11","pages":"3964-3981"},"PeriodicalIF":0.8,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142176974","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove new multiplicity results for some critical growth -biharmonic problems in bounded domains. More specifically, we show that each of the problems considered here has arbitrarily many solutions for all sufficiently large values of a certain parameter . In particular, the number of solutions goes to infinity as . We also give an explicit lower bound on in order to have a given number of solutions. This lower bound will be in terms of an unbounded sequence of eigenvalues of a related eigenvalue problem. Our multiplicity results are new even in the semilinear case . The proofs are based on an abstract critical point theorem.
{"title":"Multiplicity results for critical \u0000 \u0000 p\u0000 $p$\u0000 -biharmonic problems","authors":"Said El Manouni, Kanishka Perera","doi":"10.1002/mana.202300535","DOIUrl":"10.1002/mana.202300535","url":null,"abstract":"<p>We prove new multiplicity results for some critical growth <span></span><math>\u0000 <semantics>\u0000 <mi>p</mi>\u0000 <annotation>$p$</annotation>\u0000 </semantics></math>-biharmonic problems in bounded domains. More specifically, we show that each of the problems considered here has arbitrarily many solutions for all sufficiently large values of a certain parameter <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 <mo>></mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$lambda &gt; 0$</annotation>\u0000 </semantics></math>. In particular, the number of solutions goes to infinity as <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 <mo>→</mo>\u0000 <mi>∞</mi>\u0000 </mrow>\u0000 <annotation>$lambda rightarrow infty$</annotation>\u0000 </semantics></math>. We also give an explicit lower bound on <span></span><math>\u0000 <semantics>\u0000 <mi>λ</mi>\u0000 <annotation>$lambda$</annotation>\u0000 </semantics></math> in order to have a given number of solutions. This lower bound will be in terms of an unbounded sequence of eigenvalues of a related eigenvalue problem. Our multiplicity results are new even in the semilinear case <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mo>=</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$p = 2$</annotation>\u0000 </semantics></math>. The proofs are based on an abstract critical point theorem.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 10","pages":"3943-3953"},"PeriodicalIF":0.8,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142176979","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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