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Sign-changing solutions for critical equations with Hardy potential
Analysis & PDE ( IF 1.8 ) Pub Date : 2021-03-20 , DOI: 10.2140/apde.2021.14.533
Pierpaolo Esposito , Nassif Ghoussoub , Angela Pistoia , Giusi Vaira

We consider the following perturbed critical Dirichlet problem involving the Hardy–Schrödinger operator:

Δu γ(u|x|2) 𝜖u = |u| 4 N2 u in Ω, u = 0  on Ω,

when 𝜖 > 0 is small, γ < (N2)2 4 , and where Ω N , N 3, is a smooth bounded domain with 0 Ω. We show that there exists a sequence (γj)j=1 in (,0] with limjγj = such that, if γγj for any j and γ (N2)2 4 1, then the above equation has for 𝜖 small, a positive — in general nonminimizing — solution that develops a bubble at the origin. If moreover γ < (N2)2 4 4, then for any integer k 2, the equation has for small enough 𝜖 a sign-changing solution that develops into a superposition of k bubbles with alternating sign centered at the origin. The above result is optimal in the radial case, where the condition γγj is not necessary. Indeed, it is known that, if γ > (N2)2 4 1 and Ω is a ball B, then there is no radial positive solution for 𝜖 > 0 small. We complete the picture here by showing that, if γ (N2)2 4 4, then the above problem has no radial sign-changing solutions for 𝜖 > 0 small. These results recover and improve what is already known in the nonsingular case, i.e., when γ = 0.



中文翻译:

具有Hardy势的临界方程的符号转换解决方案

我们考虑以下涉及Hardy–Schrödinger算子的扰动的关键Dirichlet问题:

-Δü - γü|X|2个 - 𝜖ü = |ü| 4 ñ-2个 ü 在 Ω ü = 0  上 Ω

什么时候 𝜖 > 0 是小, γ < ñ-2个2个 4 ,以及在哪里 Ω ñ ñ 3,是具有 0 Ω。我们证明存在一个序列γĴĴ=1个-0]ĴγĴ = - 这样,如果 γγĴ 对于任何 Ĵγ ñ-2个2个 4 - 1个,则上述等式具有 𝜖小型的,积极的(通常是非最小化)解决方案,其起因是会产生气泡。如果还有γ < ñ-2个2个 4 - 4,那么对于任何整数 ķ 2个,等式足够小 𝜖 转变为符号叠加的解决方案 ķ以原点为中心带有交替符号的气泡。在径向条件下,上述结果是最佳的,在这种情况下γγĴ没有必要。确实,众所周知,如果γ > ñ-2个2个 4 - 1个Ω 是一个球 ,则没有径向正解 𝜖 > 0小的。我们通过显示以下内容来完成图片:γ ñ-2个2个 4 - 4,则上述问题没有用于 𝜖 > 0小的。这些结果恢复并改进了非奇异情况下(即何时)的已知结果。γ = 0

更新日期:2021-03-21
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