Steklov problems for the fourth order Hénon-Hardy equation
DOI:
https://doi.org/10.12775/TMNA.2025.052Keywords
Steklov problems, fourth order Hénon-Hardy equation, nontrivial nonnegative solutions, regularity of solutions, Caffarelli-Kohn-Nirenberg inequalitiesAbstract
Let $\Omega$ be a bounded smooth domain in $\R^N$ $ (N \geq 5)$ with $0 \in \Omega$. Non-trivial nonnegative weak solutions of Steklov problems of the fourth order Hénon-Hardy equation are obtained via variational methods and new embeddings related to new Caffarelli-Kohn-Nirenberg inequalities. Meanwhile, the further regularities of the weak solutions are studied, especially, the optimal regularity at the singular point $x=0$ of the solutions is obtained.References
E. Berchio, F. Gazzola and E. Mitidieri, Positivity preserving property for a class of biharmonic elliptic problem, J. Differential Equations 229 (2006), 1–23.
E. Berchio, F. Gazzola and D. Pierotti, Gelfand type elliptic problems under Steklov boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), 315–335.
E. Berchio, F. Gazzola and T. Weth, Critical growth biharminic elliptic problems under Steklov-type boundary conditions, Adv. Differential Equations 12 (2007), 381–406.
D. Buoso and J.B. Kennedy, The Bilaplacian with Robin boundary conditions, SIAM J. Math. Anal. 54 (2022), no. 1, 36–78.
L. Caffa relli, R. Kohn, L. Nirenberg, First order interpolation inequalities with weights, Compos. Math. 53 (1984), 259–275.
F. Gazzola and D. Pierotti, Positive solutions to critical growth biharmonic elliptic problems under Steklov boundary conditions, Nonlinear Anal. 71 (2009), 232–238.
F. Gazzola and G. Sweers, On positivity for the biharmonic operator under Steklov boundary conditions, Arch. Rational Mech. Anal. 188 (2008), 399-427.
F. Gazzola, H. -ch. Grunau and G. Sweers, Polyharminic Boundary Value Problems: Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains, Lecture Notes Mathematics, Vol. 1991, Springer, Berlin, 2010, pp. 167–185.
D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1977, pp. 155–158.
Z.M. Guo and F.S. Wan, Weighted fourth order elliptic problems in the unit ball, Electron. Res. Arch. 29 (2021), 3775–3803. (Dedicated to Professor Norman Dancer for his 75th birthday)
Z.M. Guo and F.S. Wan, Optimal regularity of positive solutions of the Hénon–Hardy equation and related equations, Sci. China Math. 67 (2024), 2283–2302.
Z.M. Guo, F.S. Wan and L.P. Wang, Embeddings of weighted Sobolev spaces and a weighted fourth order elliptic equations, Commun. Contemp. Math. 22 (2020), 1950057, 40 pp.
J.R. Kuttler and V.G. Sigillito, Inequalities for membrane and Stekloff eigenvalues, J. Math. Anal. Appl. 23 (1968), 148–160.
P. Hartman, Ordinary Differential Equations, 2nd ed., Birkhäuser, Boston, 1982, pp. 45–92.
G. Migliaccio and H.A. Matevossian, Exterior biharminic problem with the mixed Steklov and Steklov-type boundary conditions, Lobachevskii J. Math. 42 (2021), 1886–1899.
G. Migliaccio and H.A. Matevossian, Solution of the biharmonic problem with the Steklov-type and Farwig boundary conditions, Lobachevskii J. Math. 45 (2024), 2363–2377.
L.E. Payne, Some isoperimetric inequalities for harmonic functions, SIAM J. Math. Anal. 1 (1970), 354–359.
W. Stekloff, Sur les problèmes fondamentaux de la physique mathématique, Ann. Sci. Éc. Norm. Supér. (3) 19 (1902), 191–259.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0
