Existence and nonexistence of solutions for an approximation of the Paneitz problem on spheres
Abstract This paper is devoted to studying the nonlinear problem with slightly subcritical and supercritical exponents ( S ± ε ) : Δ 2 u − c n Δ u + d n u = K u n + 4 n − 4 ± ε $(S_{\pm \varepsilon}): \Delta ^{2}u-c_{n}\Delta u+d_{n}u = Ku^{ \frac{n+4}{n-4}\pm \varepsilon}$ , u > 0 $u>0$ on S...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2023-10-01
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| Series: | Boundary Value Problems |
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| Online Access: | https://doi.org/10.1186/s13661-023-01789-0 |
| _version_ | 1797452015835545600 |
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| author | Kamal Ould Bouh |
| author_facet | Kamal Ould Bouh |
| author_sort | Kamal Ould Bouh |
| collection | DOAJ |
| description | Abstract This paper is devoted to studying the nonlinear problem with slightly subcritical and supercritical exponents ( S ± ε ) : Δ 2 u − c n Δ u + d n u = K u n + 4 n − 4 ± ε $(S_{\pm \varepsilon}): \Delta ^{2}u-c_{n}\Delta u+d_{n}u = Ku^{ \frac{n+4}{n-4}\pm \varepsilon}$ , u > 0 $u>0$ on S n $S^{n}$ , where n ≥ 5 $n\geq 5$ , ε is a small positive parameter and K is a smooth positive function on S n $S^{n}$ . We construct some solutions of ( S − ε ) $(S_{-\varepsilon})$ that blow up at one critical point of K. However, we prove also a nonexistence result of single-peaked solutions for the supercritical equation ( S + ε ) $(S_{+\varepsilon})$ . |
| first_indexed | 2024-03-09T15:01:43Z |
| format | Article |
| id | doaj.art-1ac4beb63903408ea13b1f9554ae31c8 |
| institution | Directory Open Access Journal |
| issn | 1687-2770 |
| language | English |
| last_indexed | 2024-03-09T15:01:43Z |
| publishDate | 2023-10-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Boundary Value Problems |
| spelling | doaj.art-1ac4beb63903408ea13b1f9554ae31c82023-11-26T13:50:59ZengSpringerOpenBoundary Value Problems1687-27702023-10-012023111410.1186/s13661-023-01789-0Existence and nonexistence of solutions for an approximation of the Paneitz problem on spheresKamal Ould Bouh0Institut Supérieur de Comptabilité et d’Administration d’Entreprises, ISCAEAbstract This paper is devoted to studying the nonlinear problem with slightly subcritical and supercritical exponents ( S ± ε ) : Δ 2 u − c n Δ u + d n u = K u n + 4 n − 4 ± ε $(S_{\pm \varepsilon}): \Delta ^{2}u-c_{n}\Delta u+d_{n}u = Ku^{ \frac{n+4}{n-4}\pm \varepsilon}$ , u > 0 $u>0$ on S n $S^{n}$ , where n ≥ 5 $n\geq 5$ , ε is a small positive parameter and K is a smooth positive function on S n $S^{n}$ . We construct some solutions of ( S − ε ) $(S_{-\varepsilon})$ that blow up at one critical point of K. However, we prove also a nonexistence result of single-peaked solutions for the supercritical equation ( S + ε ) $(S_{+\varepsilon})$ .https://doi.org/10.1186/s13661-023-01789-0Critical pointsCritical exponentVariational problemPaneitz curvature |
| spellingShingle | Kamal Ould Bouh Existence and nonexistence of solutions for an approximation of the Paneitz problem on spheres Boundary Value Problems Critical points Critical exponent Variational problem Paneitz curvature |
| title | Existence and nonexistence of solutions for an approximation of the Paneitz problem on spheres |
| title_full | Existence and nonexistence of solutions for an approximation of the Paneitz problem on spheres |
| title_fullStr | Existence and nonexistence of solutions for an approximation of the Paneitz problem on spheres |
| title_full_unstemmed | Existence and nonexistence of solutions for an approximation of the Paneitz problem on spheres |
| title_short | Existence and nonexistence of solutions for an approximation of the Paneitz problem on spheres |
| title_sort | existence and nonexistence of solutions for an approximation of the paneitz problem on spheres |
| topic | Critical points Critical exponent Variational problem Paneitz curvature |
| url | https://doi.org/10.1186/s13661-023-01789-0 |
| work_keys_str_mv | AT kamalouldbouh existenceandnonexistenceofsolutionsforanapproximationofthepaneitzproblemonspheres |