Asymmetric Robin Problems with Indefinite Potential and Concave Terms

Abstract

We consider a parametric semilinear Robin problem driven by the Laplacian plus an indefinite and unbounded potential. In the reaction, we have the competing effects of a concave term appearing with a negative sign and of an asymmetric asymptotically linear term which is resonant in the negative direction. Using variational methods together with truncation and perturbation techniques and Morse theory (critical groups), we prove two multiplicity theorems producing four and five, respectively, nontrivial smooth solutions when the parameter λ>0{\lambda>0} is small.

Keywords

• indefinite and unbounded potential
• concave term
• asymmetric reaction
• critical groups, multiple solutions
• harnack inequality
• 35j20
• 35j60