Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-28T05:59:05.440Z Has data issue: false hasContentIssue false
  • English
  • Français

A biharmonic equation with singular nonlinearity

Part of: Nonlinear integral equations Nonlinear operators and their properties Elliptic equations and systems

Published online by Cambridge University Press:  14 June 2011

Marius Ghergu
Marius Ghergu
Affiliation:
School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland ([email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the biharmonic equation Δ2u = u−α, 0 < α < 1, in a smooth and bounded domain Ω ⊂ ℝn, n ≥ 2, subject to Dirichlet boundary conditions. Under some suitable assumptions on Ω related to the positivity of the Green function for the biharmonic operator, we prove the existence and uniqueness of a solution.

Keywords

biharmonic operatorsingular nonlinearityGreen functionintegral equation

MSC classification

Secondary: 35J40: Boundary value problems for higher-order elliptic equations 35J08: Green's functions 45G05: Singular nonlinear integral equations 47H10: Fixed-point theorems
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 55 , Issue 1 , February 2012 , pp. 155 - 166
Copyright
Copyright © Edinburgh Mathematical Society 2011

References

1.Boggio, T., Sull'equilibrio delle piastre elastiche incastrate, Rend. Acc. Lincei 10 (1901), 197–205.Google Scholar
2.Boggio, T., Sulle funzioni di Green d'ordine m, Rend. Circ. Mat. Palermo 20 (1905), 97–135.CrossRefGoogle Scholar
3.Coffman, C. V. and Duffin, R. J., On the structure of biharmonic functions satisfying the clamped condition on a right angle, Adv. Appl. Math. 1 (1950), 373–389.CrossRefGoogle Scholar
4.Coffman, C. V. and Grover, C. L., Obtuse cones in Hilbert spaces and applications to partial differential equations, J. Funct. Analysis 35 (1980), 369–396.CrossRefGoogle Scholar
5.Crandall, M. G., Rabinowitz, P. H. and Tartar, L., On a Dirichlet problem with a singular nonlinearity, Commun. PDEs 2 (1977), 193–222.CrossRefGoogle Scholar
6.Dall'Acqua, A. and Sweers, G., Estimates for Green function and Poisson kernels of higher order Dirichlet boundary value problems, J. Diff. Eqns 205 (2004), 466–487.CrossRefGoogle Scholar
7.Garabedian, P. R., A partial differential equation arising in conformal mapping, Pac. J. Math. 1 (1951), 485–524.CrossRefGoogle Scholar
8.Ghergu, M. and Rădulescu, V., Singular elliptic equations: bifurcation and asymptotic analysis, Oxford Lecture Series in Mathematics and Its Applications, Volume 37 (Oxford University Press, 2008).Google Scholar
9.Gilbarg, D. and Trudinger, N. S., Elliptic partial differential equations of second order, 3rd edn (Springer, 2001).CrossRefGoogle Scholar
10.Grunau, H.-Ch. and Robert, F., Positivity and almost positivity of biharmonic Green's functions under Dirichlet boundary conditions, Arch. Ration. Mech. Analysis 195 (2010), 865–898.CrossRefGoogle Scholar
11.Grunau, H.-Ch., Robert, F. and Sweers, G., Optimal estimates from below for biharmonic Green functions, Proc. Am. Math. Soc. 139 (2011), 2151–2161.CrossRefGoogle Scholar
12.Grunau, H.-Ch. and Sweers, G., Positivity for perturbations of polyharmonic operators with Dirichlet boundary conditions in two dimensions, Math. Nachr. 179 (1996), 89–102.CrossRefGoogle Scholar
13.Grunau, H.-Ch. and Sweers, G., Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions, Math. Annalen 307 (1997), 589–626.CrossRefGoogle Scholar
14.Grunau, H.-Ch. and Sweers, G., Sign change for the Green function and for the first eigenfunction of equations of clamped plate type, Arch. Ration. Mech. Analysis 150 (1999), 179–190.CrossRefGoogle Scholar
15.Grunau, H.-Ch. and Sweers, G., Regions of positivity for polyharmonic Green functions in arbitrary domains, Proc. Am. Math. Soc. 135 (2007), 3537–3546.CrossRefGoogle Scholar
16.Gui, C. and Lin, F., Regularity of an elliptic problem with a singular nonlinearity, Proc. R. Soc. Edinb. A123 (1993), 1021–1029.CrossRefGoogle Scholar
17.Hadamard, J., Mémoire sur le problème d'analyse relatif à l'équilibre des plaques élastiques encastrées, in Oeuvres de Jacques Hadamard, II, pp. 515–641 (Centre National de la Recherche Scientifique, Paris, 1968).Google Scholar
18.Kawohl, B. and Sweers, G., On anti-eigenvalues for elliptic systems and a question of McKenna and Walter, Indiana Univ. Math. J. 51 (2002), 1023–1040.CrossRefGoogle Scholar
19.Krasovskiĭ, J. P., Isolation of singularities of the Green function, Math. USSR Izv. 1 (1967), 935–966.CrossRefGoogle Scholar
20.Protter, M. H. and Weinberger, H. F., Maximum principles in differential equations (Prentice Hall, Englewood Cliffs, NJ, 1967).Google Scholar
21.Shapiro, H. S. and Tegmark, M., An elementary proof that the biharmonic Green function of an eccentric ellipse changes sign, SIAM Rev. 36 (1994), 99–101.CrossRefGoogle Scholar