Infinitely many positive solutions for nonlinear equations with non-symmetric potentials

Abstract

We consider the following nonlinear Schrodinger equa-where V is a potential satisfying some decay condition and f (u) is a superlinear nonlinearity satisfying some nondegeneracy condition. Using localized energy method, we prove that there exists some δ 0 such that for 0 < δ < δ 0 , the above problem has infinitely many positive solutions. This generalizes and gives a new proof of the results by Cerami-Passaseo-Solimini [11]. The new techniques allow us to establish the existence of infinitely many positive bound states for elliptic systems.

Intermediate reduction method and infinitely many positive solutions of nonlinear Schrödinger equations with non-symmetric potentials

“…Inspired by the results in [14,4], the answer is very likely yes. But in this situation, the idea of uniformly distribution of points on curves does not work.…”

“…It seems that our argument here can only deal with the case of polynomial decay. Inspired by [14,4], it is reasonable to believe that there are infinitely many positive solutions when the potential V satisfies the following decay assumption:…”

“…The results in [14,4] make us think that condition (1.11) could be improved. In fact we believe that the optimal condition should be (1.12).…”

“…(In [4], Ao and Wei gave a new proof of this result, using localized energy method. The new techniques also allow them to deal with more general nonlinearity.…”

“…( 1.11) Results in this direction with non-symmetric potentials, as far as we know, there are only perturbative results (cf. [14,4]). For instance, if V (x) tends to V ∞ from above with a suitable rate:…”