Infinitely Many Positive Solutions to Some Scalar Field Equations with Nonsymmetric Coefficients

Abstract

In this paper the equation $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}} - \Delta u + a(x)u = |u|^{p - 1} u\;{\rm in }\;{\R}^N $ is considered, when N ≥ 2, p > 1, and $p < {{N + 2} \over {N - 2}}$ if N ≥ 3. Assuming that the potential a(x) is a positive function belonging to $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}L_{{\rm loc}}^{N/2} ({\R}^N )$ such that a(x) → a∞ > 0 as |x|→∞ and satisfies slow decay assumptions but does not need to fulfill any symmetry property, the existence of infinitely many positive solutions, by purely variational methods, is proved. The shape of the solutions is described as is, and furthermore, their asymptotic behavior when $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}|a(x) - a_\infty |_{L_{{\rm loc}}^{N/2} ({\R}^N )} \to 0$. © 2012 Wiley Periodicals, Inc.

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).…”

“…( 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:…”

“…= +∞, for someη ∈ (0, V ∞ ), (1.12) and V satisfies a global condition: 13) where V is a sufficiently small positive constant (with no explicit expression), Cerami, Passaseo and Solimini [14] proved that problem (1.3) has infinitely many positive solutions by purely variational methods. (In [4], Ao and Wei gave a new proof of this result, using localized energy method.…”