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Bifurcation results for semilinear elliptic problems in N . (English) Zbl 1067.35027

This paper is devoted to the study of the semilinear bifurcation problem

-Δψ-λψ=a(x)|ψ| p-1 ψ+b(x)|ψ| q-1 ψ

in N , with lim |x| ψ(x)=0, where N1, λ0, 1<p<q(N+2)/(N-2) if N3 (and q< if N=1,2), p<1+4/N, and a,b: N are sign-changing potentials.

The main results of the paper establish, under various assumptions on a and b, the existence of families of solutions bifurcating from the bottom of the spectrum of (-Δ). The proofs rely essentially on a nonlinear reduction method that enables the authors to search for solutions as critical points of suitable functionals defined on finite-dimensional manifolds.

35J60Nonlinear elliptic equations
35B32Bifurcation (PDE)
47J15Abstract bifurcation theory
47J30Variational methods (nonlinear operator equations)
58E07Abstract bifurcation theory