Principal eigenvalues of elliptic problems with singular potential and bounded weight function

Year 2023, Volume: 6 Issue: 2, 107 - 127, 15.06.2023

Tomas Godoy

https://doi.org/10.33205/cma.1272110

Abstract

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$ with $C^{0,1}$ boundary, and let $d_{\Omega}:\Omega\rightarrow\mathbb{R}$ be the distance function $d_{\Omega}\left( x\right) :=dist\left( x,\partial\Omega\right) .$ Our aim in this paper is to study the existence and properties of principal eigenvalues of self-adjoint elliptic operators with weight function and singular potential, whose model problem is $-\Delta u+bu=\lambda mu$ in $\Omega,$ $u=0$ on $\partial\Omega,$ $u>0$ in $\Omega,$ where $b:\Omega \rightarrow\mathbb{R}$ is a nonnegative function such that $d_{\Omega}^{2}b\in L^{\infty}\left( \Omega\right) ,$ $m:\Omega\rightarrow\mathbb{R}$ is a nonidentically zero function in $L^{\infty}\left( \Omega\right) $ that may change sign, and the solutions are understood in weak sense.

Keywords

Weighted principal eigenvalue problems, second order elliptic operators, singular potentials

References

• A. Beltramo, P. Hess: On the principal eigenvalue of a periodic-parabolic operator, Comm. Partial Differential Equations 9 (9) (1984), 919–941.
• H. Berestycki, S. R. S. Varadhan and L. Nirenberg: The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math. 47 (1) (1994), 47–92.
• H. Brezis: Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011.
• K. Brown, S. Lin: On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function, J. Math. Anal. Appl., 75 (1980), 112–120.
• D. G. De Figueiredo: Positive solutions of semilinear elliptic equations, Lect. Notes Math., Springer, 957 (1982), 34–87.
• J. Fleckinger, J. Hernandez and F. de Thelin: Existence of multiple eigenvalues for some indefinite linear eigenvalue problems, Bolletino U.M.I., 7 (2004), 159-188.
• D. Gilbarg, N. S. Trudinger: Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin Heidelberg New York, 2001.
• M. Ghergu, V. D. R˘adulescu: Singular Elliptic Problems: Bifurcation and Asymptotic Analysis, Oxford Lecture Series in Mathematics and Its Applications, Oxford University Press, No 37, 2008.
• T. Godoy, A. Guerin: Regularity of the lower positive branch for singular elliptic bifurcqation problems, Electron. J. Differential Equations, 2019 (49) (2019), 1–32.
• J. Hernández, F. J. Mancebo and J. M. Vega: On the linearization of some singular nonlinear elliptic problems and applications, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 19 (6) (2002), 777–813.
• P. Hess: On positive solutions of semilinear periodic-parabolic problems, Infinite-dimensional systems (Retzhof, 1983), 101–114, Lecture Notes in Math. 1076, Springer, Berlin, 1984.
• P. Hess: Periodic parabolic problems and positivity, Pitman Research Notes, 1991.
• P. Hess, T. Kato: On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations, 5 (1980), 999–1030.
• J. Lopez-Gomez: The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems, J. Differential Equations 127 (1) (1996), 263–294.
• A. Manes, A.M. Micheletti: Un’estensione della teoria variazionale classica degli autovalori per operatori ellitici del secondo ordine, Bollettino U.M.I., 7 (1973), 285–301.
• N. S. Papageorgiou, V. D. R˘adulescu and D. D. Repovš: Nonlinear Analysis – Theory and Methods, Springer Monographs in Mathematics, Springer Nature Switzerland, 2019.
• J. Sabina de Lis: Hopf maximum principle revisited, Electron. J. Differential Equations, 2015 (115) (2015), 1–9.
• S. Senn, P. Hess: On positive solutions of a linear elliptic eigenvalue problem with Neumann boundary conditions, Math. Ann., 258 (1982), 459–470.