Resonant Robin problems with indefinite and unbounded potential

Nikolaos S. Papageorgiou, George Smyrlis

DOI: http://dx.doi.org/10.12775/TMNA.2016.062

Abstract


We study a semilinear Robin problem with an indefinite and unbounded potential and a reaction term which asymptotically at $ \pm \infty $ is resonant with respect to any nonprincipal nonnegative eigenvalue. We prove two multiplicity theorems producing three and four nontrivial solutions respectively. Our approach uses variational methods based on the critical point theory, truncation and perturbation techniques, and Morse theory (critical groups).

Keywords


Indefinite and unbounded potential; resonance; Robin boundary condition; maximum principle; critical groups; multiple solutions

Full Text:

PREVIEW FULL TEXT

References


S. Aizicovici, N.S. Papageorgiou and V. Staicu, Degree Theory for Operators of Monotone Type and Nonlinear Elliptic Equations with Inequality Constraints, Mem. Amer. Math. Soc., Vol. 196, No. 915, November 2008.

T. Bartsch and S.Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear Anal. 28 (1997), 419–441.

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.

A. Castro, J. Cossio and C. Velez, Existence of seven solutions for asymptotically linear Dirichlet problems without symmetries, Ann. Mat. Pura Appl.

D. de Figueiredo and J.P. Gossez, Strict monotonicity of eigenvalues and unique continuation, Comm. Partial Differential Equations 17 (1992), 339–346.

L. Gasinski and N.S. Papageorgiou, Nonlinear Analysis, Chapman Hall, CRC, Boca Raton, Fl (2006).

L. Gasinski and N.S. Papageorgiou, Pairs of nontrivial solutions for Neumann problems, J. Math. Anal. Appl. 398 (2013), 649–663.

H. Hofer, Variational and topological methods in partially ordered Hilbert spaces’, Math. Ann. 261 (1982), 493–514.

S. Kyritsi and N.S. Papageorgiou, Multiple solutions for Dirichlet problems with an indefinite potential, Ann.i Mat. Pura Appl. 192 (2013), 297–315.

S. Liu and S. Li, Critical groups at infinity, saddle point reduction and elliptic resonant problems, Commun. Contemp. Math. 5 (2003), 761–773.

C. Li, S. Li and J. Liu, Splitting theorem, Poincaré–Hopf theorem and jumping nonlinear problems, J. Funct.l Anal. 221 (2005), 439–455.

D. Motreanu, V.V. Motreanu and N.S. Papageorgiou, On resonant Neumann problems, Math. Ann. 354 (2012), 1117–1145.

D. Motreanu, V.V. Motreanu and N.S. Papageorgiou, Topological and Variatonal Methods with Applications to Nonlinear Boundary Value Problems, Springer, New York (2014).

R. Palais, Homotopy theory for infinite dimensional manifolds, Topology 5 (1966), 1–16.

N.S. Papageorgiou and F. Papalini, Seven solutions with sign information for sublinear equations with unbounded and indefinite potential and no symmetries, Israel J. Math. 201 (2014), 761–796.

N.S. Papageorgiou and V. Radulescu, Multiple solutions with precise sign for nonlinear parametric Robin problems, J. Differential Equations 256 (2014), 2449–2479.

N.S. Papageorgiou and G. Smyrlis, On a class of parametric Neumann problems with indefinite and unbounded potential, Forum Math. DOI: 10.1515/forum-2012-0042.

J. Su and C. Tang, Multiplicity results for semilinear elliptic equations with resonance at higher eigenvalues, Nonlinear Anal. 44 (2001), 311–321.

C.-L. Tang and X.P. Wu, Existence and multiplicity for solutions of Neumann problems for elliptic equations, J. Math. Anal. Appl. 288 (2003), 660–670.

X. Wang, Neumann problems of semilinear elliptic equations involving critical Sobolev exponents, J. Differential Equations 93 (1991), 283–310.


Refbacks

  • There are currently no refbacks.

Partnerzy platformy czasopism