Infinitely many positive solutions to some nonsymmetric scalar field equations: the planar case

We show the existence of infinitely many positive solutions u is an element of H-1(R-2) to the equation -Delta u + a(x) u = u(p), with p > 1, without asking, on the positive potential a(x), any symmetry assumption as in Wei and Yan (Calc Var Partial Differ Equ 37, 423-439, 2010) or Devillanova and Solimini (Adv Nonlinear Studies 12, 173-186, 2012) or small oscillation assumption as in Cerami et al. (Commun Pure Appl Math, doi: 10.1002/cpa.21410, 2012) and in Weiwei and Wei (Infinitely many positive solutions for Nonlinear equations with non-symmetric Potential, 2012).