Resonance problem of a class of singular quasilinear elliptic equations

Abstract

In this paper, we study the resonance problem of a class of singular quasilinear elliptic equations with respect to its higher near-eigenvalues. Under a generalized Landesman–Lazer condition, it is proved that the resonance problem admits at least one nontrivial solution in weighted Sobolev spaces. The proof is based upon applying the Galerkin-type technique, the Brouwer’s fixed-point theorem and a compact embedding theorem of weighted Sobolev spaces by Shapiro.

Keywords:

• weighted Sobolev space
• resonance problem
• singular quasiliner elliptic equation
• near-eigenvalue

AMS Subject Classifications:

• 35B34
• 35H30
• 35J50

Notes

The work was supported by the National Natural Science Foundation of China [grant number 11171220] and the Shanghai Leading Academic Discipline Project [grant number XTKX2012].