A Neumann problem of Ambrosetti–Prodi type

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• 作者：Adilson Eduardo Presoto…
• 关键词：Semilinear problem ; Neumann problem ; gradient nonlinearity ; upper and lower solutions method
• 刊名：Journal of Fixed Point Theory and Applications
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：18
• 期：1
• 页码：189-200
• 全文大小：544 KB
• 参考文献：1.Agmon S., Douglis A., Nirenberg L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Comm. Pure Appl. Math. 12, 623–727 (1959)CrossRef MathSciNet MATH
  2.Ambrosetti A., Prodi G.: On the inversion of some differentiable mappings with singularities between Banach spaces. Ann. Mat. Pura Appl. (4) 93, 231–246 (1972)CrossRef MathSciNet MATH
  3.Arcoya D., Carmona J.: On two problems studied by A. Ambrosetti. J. Eur. Math. Soc. (JEMS) 8, 181–188 (2006)CrossRef MathSciNet MATH
  4.Berestycki H., Lions P.-L.: Sharp existence results for a class of semilinear elliptic problems. Bol. Soc. Brasil. Mat. 12, 9–19 (1981)CrossRef MathSciNet MATH
  5.Bony J.-M.: Principe du maximum dans les espaces de Sobolev. C. R. Acad. Sci. Paris Sér. A-B 265, A333–A336 (1967)MathSciNet
  6.Deimling K.: Nonlinear Functional Analysis. Springer-Verlag, Berlin (1985)CrossRef MATH
  7.D. Gilbarg and N. S.Trudinger, Elliptic Partial Differential Equations of Second Order. Classics in Mathematics, Springer-Verlag, Berlin, 2001.
  8.Habets P., Omari P.: Existence and localization of solutions of second order elliptic problems using lower and upper solutions in the reversed order. Topol. Methods Nonlinear Anal. 8, 25–56 (1996)MathSciNet MATH
  9.Hess P.: On a nonlinear elliptic boundary value problem of the Ambrosetti-Prodi type. Boll. Unione Mat. Ital. A (5) 17, 187–192 (1980)MathSciNet MATH
  10.Hofer H.: Existence and multiplicity result for a class of second order elliptic equations. Proc. Roy. Soc. Edinburgh Sect. A 88, 83–92 (1981)CrossRef MathSciNet MATH
  11.Kannan R., Ortega R.: Superlinear elliptic boundary value problems. Czechoslovak Math. J. 37, 386–399 (1987)MathSciNet
  12.J. Mawhin, Ambrosetti-Prodi type results in nonlinear boundary value problems. In: Differential Equations and Mathematical Physics (Birmingham, Ala., 1986), Lecture Notes in Math. 1285, Springer, Berlin, 1987, 290–313.
  13.Mawhin J.: The periodic Ambrosetti-Prodi problem for nonlinear perturbations of the p-Laplacian. J. Eur. Math. Soc. (JEMS) 8, 375–388 (2006)CrossRef MathSciNet MATH
  
• 作者单位：Adilson Eduardo Presoto (1)
  Francisco Odair de Paiva (1)
  
  1. Departamento de Matemática, Universidade Federal de São Carlos, 13565-905, São Carlos, SP, Brazil
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Analysis
  Mathematical Methods in Physics
  
• 出版者：Birkh盲user Basel
• ISSN：1661-7746

文摘

The aim of this paper is to establish an Ambrosetti–Proditype result for the problem $$\left\{ \begin{array}{ll}-\Delta{u} = g(x, u,\nabla{u}) + t\varphi \quad {\rm in}\, \Omega,\\ \frac{\partial{u}}{\partial\eta} = 0 \qquad\qquad\qquad\quad {\rm on}\, \partial\Omega ;\end{array} \right.$$i.e., under appropriate conditions, we will show that there exists a constant t ₀ such that the problem above has no solution if t > t ₀, at least a solution if t =  t ₀ and at least two solutions if t < t ₀. The proof is based on a combination of upper and lower solutions method and the Leray–Schauder degree. Keywords Semilinear problem Neumann problem gradient nonlinearity upper and lower solutions method Mathematics Subject Classification 35J25 35J61 Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (13) References1.Agmon S., Douglis A., Nirenberg L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Comm. Pure Appl. Math. 12, 623–727 (1959)CrossRefMathSciNetMATH2.Ambrosetti A., Prodi G.: On the inversion of some differentiable mappings with singularities between Banach spaces. Ann. Mat. Pura Appl. (4) 93, 231–246 (1972)CrossRefMathSciNetMATH3.Arcoya D., Carmona J.: On two problems studied by A. Ambrosetti. J. Eur. Math. Soc. (JEMS) 8, 181–188 (2006)CrossRefMathSciNetMATH4.Berestycki H., Lions P.-L.: Sharp existence results for a class of semilinear elliptic problems. Bol. Soc. Brasil. Mat. 12, 9–19 (1981)CrossRefMathSciNetMATH5.Bony J.-M.: Principe du maximum dans les espaces de Sobolev. C. R. Acad. Sci. Paris Sér. A-B 265, A333–A336 (1967)MathSciNet6.Deimling K.: Nonlinear Functional Analysis. Springer-Verlag, Berlin (1985)CrossRefMATH7.D. Gilbarg and N. S.Trudinger, Elliptic Partial Differential Equations of Second Order. Classics in Mathematics, Springer-Verlag, Berlin, 2001.8.Habets P., Omari P.: Existence and localization of solutions of second order elliptic problems using lower and upper solutions in the reversed order. Topol. Methods Nonlinear Anal. 8, 25–56 (1996)MathSciNetMATH9.Hess P.: On a nonlinear elliptic boundary value problem of the Ambrosetti-Prodi type. Boll. Unione Mat. Ital. A (5) 17, 187–192 (1980)MathSciNetMATH10.Hofer H.: Existence and multiplicity result for a class of second order elliptic equations. Proc. Roy. Soc. Edinburgh Sect. A 88, 83–92 (1981)CrossRefMathSciNetMATH11.Kannan R., Ortega R.: Superlinear elliptic boundary value problems. Czechoslovak Math. J. 37, 386–399 (1987)MathSciNet12.J. Mawhin, Ambrosetti-Prodi type results in nonlinear boundary value problems. In: Differential Equations and Mathematical Physics (Birmingham, Ala., 1986), Lecture Notes in Math. 1285, Springer, Berlin, 1987, 290–313.13.Mawhin J.: The periodic Ambrosetti-Prodi problem for nonlinear perturbations of the p-Laplacian. J. Eur. Math. Soc. (JEMS) 8, 375–388 (2006)CrossRefMathSciNetMATH About this Article Title A Neumann problem of Ambrosetti–Prodi type Journal Journal of Fixed Point Theory and Applications Volume 18, Issue 1 , pp 189-200 Cover Date2016-03 DOI 10.1007/s11784-015-0277-5 Print ISSN 1661-7738 Online ISSN 1661-7746 Publisher Springer International Publishing Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Mathematics, general Analysis Mathematical Methods in Physics Keywords 35J25 35J61 Semilinear problem Neumann problem gradient nonlinearity upper and lower solutions method Authors Adilson Eduardo Presoto (1) Francisco Odair de Paiva (1) Author Affiliations 1. Departamento de Matemática, Universidade Federal de São Carlos, 13565-905, São Carlos, SP, Brazil Continue reading... To view the rest of this content please follow the download PDF link above.