Annals of Functional Analysis

Triple solutions for quasilinear one-dimensional p-Laplacian elliptic equations in the whole space

Gabriele Bonanno, Donal O’Regan, and Francesca Vetro

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Abstract

In this paper, we establish the existence of three possibly nontrivial solutions for a Dirichlet problem on the real line without assuming on the nonlinearity asymptotic conditions at infinity. As a particular case, when the nonlinearity is superlinear at zero and sublinear at infinity, the existence of two nontrivial solutions is obtained. This approach is based on variational methods and, more precisely, a critical points theorem, which assumes a more general condition than the classical Palais–Smale condition, is exploited.

Article information

Source
Ann. Funct. Anal., Volume 8, Number 2 (2017), 248-258.

Dates
Received: 2 August 2016
Accepted: 16 September 2016
First available in Project Euclid: 1 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.afa/1488358820

Digital Object Identifier
doi:10.1215/20088752-0000010X

Mathematical Reviews number (MathSciNet)
MR3619320

Zentralblatt MATH identifier
06694413

Subjects
Primary: 34B40: Boundary value problems on infinite intervals
Secondary: 47H14: Perturbations of nonlinear operators [See also 47A55, 58J37, 70H09, 70K60, 81Q15] 49J40: Variational methods including variational inequalities [See also 47J20]

Keywords
nonlinear differential problems in unbounded domains operators without compactness critical points three solutions

Citation

Bonanno, Gabriele; O’Regan, Donal; Vetro, Francesca. Triple solutions for quasilinear one-dimensional $p$ -Laplacian elliptic equations in the whole space. Ann. Funct. Anal. 8 (2017), no. 2, 248--258. doi:10.1215/20088752-0000010X. https://projecteuclid.org/euclid.afa/1488358820


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References

  • [1] R. P. Agarwal and D. O’Regan, Infinite Interval Problems for Differential, Difference and Integral Equations, Kluwer Academic, Dordrecht, 2001.
  • [2] R. P. Agarwal and D. O’Regan, Infinite interval problems arising in non-linear mechanics and non-Newtonian fluid flows, Internat. J. Non-Linear Mech. 38 (2003), no. 9, 1369–1376.
  • [3] A. Ambrosetti and Z.-Q. Wang, Positive solutions to a class of quasilinear elliptic equations on $\mathbb{R}$, Discrete Contin. Dyn. Syst. 9 (2003), no. 1, 55–68.
  • [4] G. Bonanno, A critical point theorem via the Ekeland variational principle, Nonlinear Anal. 75 (2012), no. 5, 2992–3007.
  • [5] G. Bonanno, G. Barletta, and D. O’Regan, A variational approach to multiplicity results for boundary value problems on the real line, Proc. Roy. Soc. Edinburgh Sect. A 145 (2015), no. 1, 13–29.
  • [6] A. Constantin, On an infinite interval value problem, Ann. Mat. Pura Appl. (4) 176 (1999), 379–394.
  • [7] R. W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations, Z. Physik B 37 (1980), no. 1, 83–87.
  • [8] S. Kurihura, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan 50 (1981), 3262–3267.
  • [9] R. Ma, Existence of positive solutions for second-order boundary value problems on infinity intervals, Appl. Math. Lett. 16 (2003), no. 1, 33–39.
  • [10] L. Ma and X. Xu, Positive solutions of a logistical equation on unbounded intervals, Proc. Amer. Math. Soc. 130 (2002), no. 10, 2947–2958.
  • [11] V. G. Makhankov and V. K. Fedyanin, Non-linear effects in quasi-one-dimensional models of condensed matter theory, Phys. Rep. 104 (1984), no. 1, 1–86.
  • [12] M. Poppenberg, On the local well posedness of quasilinear Schrödinger equations in arbitrary space dimension, J. Differential Equations 172 (2001), no. 1, 83–115.
  • [13] G. R. W. Quispel and H. W. Capel, Equation of motion for the Heisenberg spin chain, Phys. A 110 (1982), no. 1–2, 41–80.
  • [14] B. Ricceri, On a three critical points theorem, Arch. Math. (Basel) 75 (2000), no. 3, 220–226.
  • [15] M. Willem, Minimax Theorems, Progr. Nonlinear Differential Equations Appl. 24, Birkhäuser, Berlin, 1996.
  • [16] L. Zima, On positive solutions of boundary value problems on the half line, J. Math. Anal. Appl. 259 (2001), no. 1, 127–136.