In this paper the equation -\Delta u+a(x)u=|u|^{p-1}u in \R^N is considered, when N \ge 2, p > 1 and p < N+2/N-2 if N \ge 3. Assuming that the potential a(x) is a positive function belonging to L^{N/2}_loc (R^N) such that a(x)\to a_\infty > 0, as |x|\to\infty, 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 |a(x)-a_\infty|_{L^{N/2}_{loc}}\to 0.
Infinitely many positive solutions to some scalar field equations with nonsymmetric coefficients
PASSASEO, Donato;
2013-01-01
Abstract
In this paper the equation -\Delta u+a(x)u=|u|^{p-1}u in \R^N is considered, when N \ge 2, p > 1 and p < N+2/N-2 if N \ge 3. Assuming that the potential a(x) is a positive function belonging to L^{N/2}_loc (R^N) such that a(x)\to a_\infty > 0, as |x|\to\infty, 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 |a(x)-a_\infty|_{L^{N/2}_{loc}}\to 0.File in questo prodotto:
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