We study existence and multiplicity of positive ground states for the scalar curvature equation $Delta u+ K(|x|) u^{{n+2}{n-2}}=0$, x in R^n, $n geq 3$ when the function $K:R^+ to R^+$ is bounded above and below by two positive constants, i.e. $\underline{K} leq K(r) leq overline{K}$ for every positive r, it is decreasing in (0,R) and increasing in $(R,+infty)$ for a certain positive constant R. We recall that in this case ground states have to be radial, so the problem is reduced to an ODE and, then, to a dynamical system via Fowler transformation. We provide a smallness non perturbative (i.e. computable) condition on the ratio $overline{K} / underline{K}$ which guarantees the existence of a large number of ground states with fast decay, i.e. such that $u(|x|) sim |x|^{2-n}$ as $|x| to +infty$, which are of bubble-tower type. We emphasize that if K(r) has a unique critical point and it is a maximum the radial ground state with fast decay, if it exists, is unique
Francesca Dalbono, Matteo Franca, Andrea Sfecci (2022). Multiplicity of Radial Ground States for the Scalar Curvature Equation Without Reciprocal Symmetry. JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS, 34(1), 701-720 [10.1007/s10884-020-09895-8].
Multiplicity of Radial Ground States for the Scalar Curvature Equation Without Reciprocal Symmetry
Francesca Dalbono;
2022-03-01
Abstract
We study existence and multiplicity of positive ground states for the scalar curvature equation $Delta u+ K(|x|) u^{{n+2}{n-2}}=0$, x in R^n, $n geq 3$ when the function $K:R^+ to R^+$ is bounded above and below by two positive constants, i.e. $\underline{K} leq K(r) leq overline{K}$ for every positive r, it is decreasing in (0,R) and increasing in $(R,+infty)$ for a certain positive constant R. We recall that in this case ground states have to be radial, so the problem is reduced to an ODE and, then, to a dynamical system via Fowler transformation. We provide a smallness non perturbative (i.e. computable) condition on the ratio $overline{K} / underline{K}$ which guarantees the existence of a large number of ground states with fast decay, i.e. such that $u(|x|) sim |x|^{2-n}$ as $|x| to +infty$, which are of bubble-tower type. We emphasize that if K(r) has a unique critical point and it is a maximum the radial ground state with fast decay, if it exists, is uniqueFile | Dimensione | Formato | |
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