Quasilinear elliptic problems interacting with its asymptotic spectrum

Abstract

Under suitable assumptions on the coefficients of the matrix A( x, u) and on the nonlinear term f( x, u), we study the quasilinear problem in bounded domains Ω⊂ R N − div(A(x,u) ∇u) = f(x,u), x∈Ω, u = 0, x∈∂Ω. We extend the semilinear results of Landesman–Lazer (J. Math. Mech. 19 (1970) 609) and of Ambrosetti–Prodi (in: A Primer on Nonlinear Analysis, Cambridge University Press, Cambridge, 1993) for resonant problems. The existence of positive solution is also considered extending to the quasilinear case the classical result by Ambrosetti–Rabinowitz (J. Funct. Anal. 14 (1973) 349). In this case, the result is obtained as a corollary of the previous multiplicity result in the Ambrosetti–Prodi framework.