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Existence and multiplicity results for classes of elliptic resonant problems. (English) Zbl 1121.35059

In this paper, the author establishes existence and multiplicity results for elliptic resonant problems

-Δu=λ k u+f(u)inΩ,u=0onΩ;

where λ k is an eigenvalue of -Δ, and

lim inf vE k ,v ±1 v Ω 0 v(x) f(s)dsdxc(f ± )

for an appropriate constant c, where E k is the eigenspace associated to λ k . Let f 0 (u)=(λ k -λ m )u+f(u) and F 0 (u)= 0 u f 0 (s)ds, and consider the conditions

±F 0 (u)>0for|u|small·(f 0 ± )

For k=1 and 2, under the conditions (f + ) or (f - ) with (f 0 + ) or (f 0 - ), and suitable values of m (with extra assumptions), the author obtain multiplicity results. The proofs rely on the Morse theory and new observations on the critical groups of degenerate critical points of a local linking type.

MSC:
35J60Nonlinear elliptic equations
35J20Second order elliptic equations, variational methods
35J65Nonlinear boundary value problems for linear elliptic equations
47J30Variational methods (nonlinear operator equations)
58E05Abstract critical point theory