Rocky Mountain Journal of Mathematics

A class of nonlinear elliptic systems with Steklov-Neumann nonlinear boundary conditions

Juliano D.B. de Godoi, Olimpio H. Miyagaki, and Rodrigo S. Rodrigues

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Abstract

We will study a class of nonlinear elliptic systems involving Steklov-Neumann boundary conditions. We obtain results ensuring the existence of solutions when resonance and nonresonance conditions occur. The results were obtained by using variational arguments.

Article information

Source
Rocky Mountain J. Math., Volume 46, Number 5 (2016), 1519-1545.

Dates
First available in Project Euclid: 7 December 2016

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1481101223

Digital Object Identifier
doi:10.1216/RMJ-2016-46-5-1519

Mathematical Reviews number (MathSciNet)
MR3580798

Zentralblatt MATH identifier
1360.35064

Subjects
Primary: 35J50: Variational methods for elliptic systems
Secondary: 35J15: Second-order elliptic equations 35J57: Boundary value problems for second-order elliptic systems 45C05: Eigenvalue problems [See also 34Lxx, 35Pxx, 45P05, 47A75]

Keywords
Steklov-Neumann eigenvalue variational methods ellip­tic system

Citation

Godoi, Juliano D.B. de; Miyagaki, Olimpio H.; Rodrigues, Rodrigo S. A class of nonlinear elliptic systems with Steklov-Neumann nonlinear boundary conditions. Rocky Mountain J. Math. 46 (2016), no. 5, 1519--1545. doi:10.1216/RMJ-2016-46-5-1519. https://projecteuclid.org/euclid.rmjm/1481101223


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References

  • G.A. Afrouzi, S. Heidarkhani and D. O'Regan, Three solutions to a class of Neumann doubly eigenvalue elliptic ystems driven by a $(p_{1},p_{2},\ldots ,p_{n})$-Laplacian, Bull. Korean Math. Soc. 6 (2010), 1235–1250.
  • H. Amann, Maximum principles and principal eigenvalues, in Ten mathematical essays on approximation in analysis and topology, J. Ferrera, J. López-Gomez and F.R. Ruiz del Portal, eds., Elsevier, Amsterdam, The Netherlands, 2005.
  • G. Auchmuty, Bases and comparison results for linear elliptic eigenproblems, J. Math. Anal. Appl. 390 (2012), 394–406.
  • ––––, Finite energy solutions of mixed elliptic boundary value problems, Math. Meth. Appl. Sci. 33 (2010), 1446–1462.
  • ––––, Steklov eigenproblems and the representation of solutions of elliptic boundary value problems, Numer. Funct. Anal. Optim. 25 (2004), 321–348.
  • ––––, Spectral characterization of the trace spaces $H^{s}(\partial \Omega)$, SIAM J. Math. Anal. 38 (2006), 894–905.
  • J.F. Bonder, S. Martinez and J.D. Rossi, Existence results for gradient elliptic systems with nonlinear boundary conditions, Nonlin. Diff. Equat. Appl. 14 (2007), 153–179.
  • K.J. Brown and T.F. Wu, A semilinear elliptic system involving nonlinear boundary condition and sign-changing weight function, J. Math. Anal. Appl. 337 (2008), 1326–1336.
  • D.G. Costa and C.A Magalhães, Variational elliptic problems which are nonquadratic at infinity, Nonlin. Anal. 23 (1994), 1401–1412.
  • G. Cerami, Un Criterio de Esistenza per i Punti Critici su Varietá Ilimitate, Rc. Ist. Lomb. Sci. Lett. 112 (1978), 332–336.
  • F.O. De Paiva, M.F. Furtado, Multiplicity of solutions for resonant elliptic systems, J. Math. Anal. Appl.319 (2006), 435–449.
  • J.D.B. Godoi, O.H. Miyagaki and R.S. Rodrigues, Steklov-Neumann eigenproblens: A spectral characterization of the Sobolev trace spaces, submitted.
  • N.C. Kourogenis and N.S. Papageorgiou, Nonsmooth critical point theory and nonlinear elliptic equations at resonance, J. Austral. Math. Soc. 69 (2000), 245–271.
  • P.D. Lamberti, Steklov-type eigenvalues associated with best Sobolev trace constants: domain perturbation and overdetermined systems, Complex Var. Ellip. Equat. 59 (2014), 309–323.
  • C. Li, The existence of infinitely many solutions of a class of nonlinear elliptic equations with Neumann boundary condition for both resonance and oscillation problems, Nonlin. Anal. 54 (2003), 441–443.
  • C. Li and S. Li, Multiple solutions and sign-changing solutions of a class of nonlinear elliptic equations with Neumann boundary condition, J. Math. Anal. Appl. 298 (2004), 14–32.
  • N. Mavinga and M.N. Nkashama, Steklov-Neumann eigenproblems and nonlinear elliptic equations with nonlinear boundary conditions, J. Differ. Equat. 248 (2010), 1212–1229.
  • P. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Reg. Conf. Ser. Math. 65, American Mathematical Society, Providence, RI, 1986.
  • M. Willem, Minimax theory, progress in nonlinear differential equations and their applications, Birkhäuser, Boston, 1996.
  • J. Zhang, S. Li, Y. Wang and X. Xue, Multiple solutions for semilinear elliptic equations with Neumann boundary condition and jumping nonlinearities, J. Math. Anal. Appl. 371 (2010), 682–690.