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

In this paper the equation $fontopen=msbm10 at 10ptdefR{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 $fontopen=msbm10 at 10ptdefR{hbox{open R}}L_{{rm loc}}^{N/2} ({R}^N )$ 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 $fontopen=msbm10 at 10ptdefR{hbox{open R}}|a(x) - a_infty |_{L_{{rm loc}}^{N/2} ({R}^N )} to 0$ . © 2012 Wiley Periodicals, Inc.