On noncoercive elliptic problems

Abstract

We consider a nonlinear noncoercive elliptic equation driven by the p-Laplacian. We show that if the L∞-perturbation has small norm, then the problem admits a positive solution. Moreover, if the L∞-perturbation is nonzero and nonnegative, then we find two positive solutions. Also, we consider a class of semilinear equations with an indefinite and unbounded potential. Using critical groups, we show that there is a nontrivial solution and under a global sign condition, we show that this solutions is nodal. Our results extend and improve a recent work of Rădulescu (Discr. Cont. Dyn. Syst. Ser. S , 5:845–856, [14]).