Existence of solutions to a semilinear elliptic boundary value problem with augmented Morse index bigger than two

Alfonso Castro, Ivan Ventura

DOI: http://dx.doi.org/10.12775/TMNA.2016.075


Building on the construction of least energy sign-changing solutions to variational semilinear elliptic boundary value problems introduced in \cite{ccn}, we prove the existence of a solution with {\it augmented Morse index} at least three when a sublevel of the corresponding action functional has nontrivial topology. We provide examples where the set of least energy sign changing solutions is disconnected, hence has nontrivial topology.


Subcritical semilinear elliptic equation; critical point; Morse index; homotopy groups; Nehari manifold; mountain pass lemma; deformation lemma

Full Text:



A. Aftalion and F. Pacella, Qualitative properties of nodal solutions of semilinear elliptic equations in radially symmetric domains, C.R. Acad. Sci. Paris Sér. I (2004).

A. Ambroseti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381.

T. Bartsch and Z.Q. Wang, On the existence of sign changing solutions for semilinear Dirichlet problems, Topol. Methods Nonlinear Anal. 7 (1996), 115–131.

T. Bartsch and T. Weth, A note on additional properties of sign changing solutions to superlinear elliptic equations, Topol. Methods Nonlinear Anal. 22 (2003), 1–14.

A. Castro, J. Cossio and J.M. Neuberger, Sign changing solutions for a superlinear Dirichlet problem, Rocky Mountain J. Math. 27 (1997), 1041–1053.

A. Castro, J. Cossio and J.M. Neuberger, On multiple solutions of a nonlinear Dirichlet problem, Nonlinear Anal. 30 (1997), no. 6, 3657–3662.

A. Castro, J. Cossio and J.M. Neuberger, A minmax principle, index of the critical point, and existence of sign-changing solutions to elliptic boundary value problems, Electron. J. Differential Equations 1998 (1998), no. 2, 1–18.

A. Castro and A.C. Lazer, Critical point theory and the number of solutions of a nonlinear Dirichlet problem, Ann. Mat. Pura Appl. 70 (1979), no. 4, 113–137.

K.C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser, Boston, 1993.

K.C. Chang, S. Li and J. Liu, Remarks on multiple solutions for asymptotically linear elliptic boundary value problems, Topol. Methods Nonlinear Anal. 3 (1994), 179–187.

D. Gilberg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer Verlag, 1997.

H. Hofer, The topological degree at a critical point of mountain pass type, Proc. Sympos. Pure Math. 45 (1986), 501–509.

A.C. Lazer and S. Solimini, Nontrivial solutions of operator equations and Morse indices of critical points of min-max type, Nonlinear Anal. 12 (1988), 761–775.

P.H. Rabinowitz, Some minimax theorems and applications to nonlinear partial differential equations, Nonlinear Anal. 65 (1978), 161–177.

P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, Regional Conference Series in Mathematics, vol. 65, Providence, RI, AMS, 1986.

Z.Q. Wang, On a superlinear elliptic equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991), 43–57.


  • There are currently no refbacks.

Partnerzy platformy czasopism