Journal of Dynamics and Differential Equations ( IF 1.3 ) Pub Date : 2020-09-23 , DOI: 10.1007/s10884-020-09895-8 Francesca Dalbono , Matteo Franca , Andrea Sfecci
We study existence and multiplicity of positive ground states for the scalar curvature equation
$$\begin{aligned} \varDelta u+ K(|x|)\, u^{\frac{n+2}{n-2}}=0, \quad x\in {{\mathbb {R}}}^n\,, \quad n>2, \end{aligned}$$when the function \(K:{{\mathbb {R}}}^+\rightarrow {{\mathbb {R}}}^+\) is bounded above and below by two positive constants, i.e. \(0<\underline{K} \le K(r) \le \overline{K}\) for every \(r > 0\), it is decreasing in \((0,{{{\mathcal {R}}}})\) and increasing in \(({{{\mathcal {R}}}},+\infty )\) for a certain \({{{\mathcal {R}}}}>0\). 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 \(|x| \rightarrow +\infty \), 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.
中文翻译:
标量曲率方程没有对等对称的径向基态的多重性
我们研究标量曲率方程的正基态的存在性和多重性
$$ \ begin {aligned} \ varDelta u + K(| x |)\,u ^ {\ frac {n + 2} {n-2}} = 0,\ quad x \ in {{\ mathbb {R}} } ^ n \,\ quad n> 2,\ end {aligned} $$当函数\(K:{{\ mathbb {R}}} ^ + \ rightarrow {{\ mathbb {R}}} ^ + \)由两个正常数(即\(0 <\下划线) {K} \ le K(r)\ le \ overline {K} \)每\(r> 0 \),它以\((0,{{{\\ mathcal {R}}}})}递减\ )并以\(({{{mathcal {R}}}},+ \ infty)\)增大一定的\({{{mathcal {R}}}}}> 0 \)。我们记得在这种情况下,基态必须是径向的,因此问题可以简化为ODE,然后通过Fowler变换简化为动力学系统。我们在比率\(\ overline {K} / \ underline {K} \)上提供了一个小非扰动(即可计算)条件保证存在大量具有快速衰减的基态,即\(u(| x |)\ sim | x | ^ {2-n} \)如\(| x | \ rightarrow + \ infty \),是泡泡塔类型的。我们强调,如果K(r)具有唯一的临界点,并且最大,则具有快速衰减的径向基态(如果存在)是唯一的。