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2010 | 30 | 2 | 169-189
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On the existence of five nontrivial solutions for resonant problems with p-Laplacian

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In this paper we study a nonlinear Dirichlet elliptic differential equation driven by the p-Laplacian and with a nonsmooth potential. The hypotheses on the nonsmooth potential allow resonance with respect to the principal eigenvalue λ₁ > 0 of $(-Δₚ,W₀^{1,p}(Z))$. We prove the existence of five nontrivial smooth solutions, two positive, two negative and the fifth nodal.
  • Jagiellonian University, Institute of Computer Science, Poland
  • National Technical University, Department of Mathematics, Greece
  • [1] A. Ambrosetti, J. García Azorero and I. Peral, Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal. 137 (1996), 219-242. doi: 10.1006/jfan.1996.0045
  • [2] S. Carl and D. Motreanu, Constant sign and sign-changing solutions for nonlinear eigenvalue problems, doi: 10.1016/, 2007.
  • [3] S. Carl and K. Perera, Sign-changing and multiple solutions for the p-Laplacian, Abstr. Appl. Anal. 7 (2002), 613-625.
  • [4] K.-C. Chang, Infinite-Dimensional Morse Theory and Multiple Solution Problems, volume 6 of Progress in Nonlinear Differential Equations and Their Applications (Birkhäuser Verlag, Boston, MA, 1993).
  • [5] F.H. Clarke, Optimization and Nonsmooth Analysis (Wiley, New York, 1983).
  • [6] M. Cuesta, D. de Figueiredo and J.-P. Gossez, The beginning of the Fučik spectrum for the p-Laplacian, J. Differ. Equ. 159 (1999), 212-238.
  • [7] N. Dancer and Y. Du, On sign-changing solutions of certain semilinear elliptic problems, Appl. Anal. 56 (1995), 193-206.
  • [8] N. Dunford and J.T. Schwartz, Linear Operators I, General Theory, volume 7 of Pure and Applied Mathematics (Wiley, New York, 1958).
  • [9] J. García Azorero, J. Manfredi and I. Peral Alonso, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Commun. Contemp. Math. 2 (2000), 385-404.
  • [10] L. Gasiński and N.S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems (Chapman and Hall/CRC Press, Boca Raton, FL, 2005).
  • [11] L. Gasiński and N.S. Papageorgiou, Nonlinear Analysis (Chapman and Hall/ CRC Press, Boca Raton, FL, 2006).
  • [12] Q.-S. Jiu and J.-B. Su, Existence and multiplicity results for Dirichlet problems with p-Laplacian, J. Math. Anal. Appl. 281 (2003), 587-601. doi: 10.1016/S0022-247X(03)00165-3
  • [13] O.A. Ladyzhenskaya and N. Uraltseva, Linear and Quasilinear Elliptic Equations, volume 46 of Mathematics in Science and Engineering (Academic Press, New York, 1968).
  • [14] G.M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), 1203-1219.
  • [15] J.-Q. Liu and S.-B. Liu, The existence of multiple solutions to quasilinear elliptic equations, Bull. London Math. Soc. 37 (2005), 592-600.
  • [16] S.-B. Liu, Multiple solutions for coercive p-Laplacian equations, J. Math. Anal. Appl. 316 (2006), 229-236. doi: 10.1016/j.jmaa.2005.04.034
  • [17] M. Marcus and V.J. Mizel, Every superposition operator mapping one Sobolev space into another is continuous, J. Funct. Anal. 33 (1979), 217-229.
  • [18] E.H. Papageorgiou and N.S. Papageorgiou, A multiplicity theorem for problems with the p-Laplacian, J. Funct. Anal. 244 (2007), 63-77.
  • [19] J.L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), 191-202.
  • [20] Z. Zhang, J.-Q. Chen and S.-J. Li, Construction of pseudo-gradient vector field and sign-changing multiple solutions involving p-Laplacian, J. Differ. Equ. 201 (2004), 287-303.
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