Araştırma Makalesi
BibTex RIS Kaynak Göster
Yıl 2023, , 107 - 127, 15.06.2023
https://doi.org/10.33205/cma.1272110

Öz

Kaynakça

  • 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.

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

Yıl 2023, , 107 - 127, 15.06.2023
https://doi.org/10.33205/cma.1272110

Öz

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.

Kaynakça

  • 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.
Toplam 18 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Makaleler
Yazarlar

Tomas Godoy 0000-0002-8804-9137

Erken Görünüm Tarihi 5 Haziran 2023
Yayımlanma Tarihi 15 Haziran 2023
Yayımlandığı Sayı Yıl 2023

Kaynak Göster

APA Godoy, T. (2023). Principal eigenvalues of elliptic problems with singular potential and bounded weight function. Constructive Mathematical Analysis, 6(2), 107-127. https://doi.org/10.33205/cma.1272110
AMA Godoy T. Principal eigenvalues of elliptic problems with singular potential and bounded weight function. CMA. Haziran 2023;6(2):107-127. doi:10.33205/cma.1272110
Chicago Godoy, Tomas. “Principal Eigenvalues of Elliptic Problems With Singular Potential and Bounded Weight Function”. Constructive Mathematical Analysis 6, sy. 2 (Haziran 2023): 107-27. https://doi.org/10.33205/cma.1272110.
EndNote Godoy T (01 Haziran 2023) Principal eigenvalues of elliptic problems with singular potential and bounded weight function. Constructive Mathematical Analysis 6 2 107–127.
IEEE T. Godoy, “Principal eigenvalues of elliptic problems with singular potential and bounded weight function”, CMA, c. 6, sy. 2, ss. 107–127, 2023, doi: 10.33205/cma.1272110.
ISNAD Godoy, Tomas. “Principal Eigenvalues of Elliptic Problems With Singular Potential and Bounded Weight Function”. Constructive Mathematical Analysis 6/2 (Haziran 2023), 107-127. https://doi.org/10.33205/cma.1272110.
JAMA Godoy T. Principal eigenvalues of elliptic problems with singular potential and bounded weight function. CMA. 2023;6:107–127.
MLA Godoy, Tomas. “Principal Eigenvalues of Elliptic Problems With Singular Potential and Bounded Weight Function”. Constructive Mathematical Analysis, c. 6, sy. 2, 2023, ss. 107-2, doi:10.33205/cma.1272110.
Vancouver Godoy T. Principal eigenvalues of elliptic problems with singular potential and bounded weight function. CMA. 2023;6(2):107-2.