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

when $𝜖>0$ is small, $\gamma <\frac{{\left(N-2\right)}^2}{4}$, and where $\Omega \subset {ℝ}^N$, $N\ge 3$, is a smooth bounded domain with $0\in \Omega$. We show that there exists a sequence ${\left({\gamma }_j\right)}_{j=1}^{\infty }$ in $\left(-\infty ,0\right]$ with $\underset{j\to \infty }{lim}{\gamma }_j=-\infty$ such that, if $\gamma \ne {\gamma }_j$ for any $j$ and $\gamma \le \frac{{\left(N-2\right)}^2}{4}-1$, then the above equation has for $𝜖$ small, a positive — in general nonminimizing — solution that develops a bubble at the origin. If moreover $\gamma <\frac{{\left(N-2\right)}^2}{4}-4$, then for any integer $k\ge 2$, 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 $\gamma \ne {\gamma }_j$ is not necessary. Indeed, it is known that, if $\gamma >\frac{{\left(N-2\right)}^2}{4}-1$ and $\Omega$ is a ball $B$, then there is no radial positive solution for $𝜖>0$ small. We complete the picture here by showing that, if $\gamma \ge \frac{{\left(N-2\right)}^2}{4}-4$, 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 $\gamma =0$.

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

什么时候 $𝜖>0$ 是小， $\gamma <\frac{{（ñ-\mathrm{2个}）}^{\mathrm{2个}}}{4}$，以及在哪里 $\Omega \subset {ℝ}^{ñ}$， $ñ\ge 3$，是具有 $0\in \Omega$。我们证明存在一个序列${（{\gamma }_{Ĵ}）}_{Ĵ=\mathrm{1个}}^{\infty }$ 在 $（-\infty ，0]$ 和 $\underset{Ĵ\to \infty }{林}{\gamma }_{Ĵ}=-\infty$ 这样，如果 $\gamma \ne {\gamma }_{Ĵ}$ 对于任何 $Ĵ$ 和 $\gamma \le \frac{{（ñ-\mathrm{2个}）}^{\mathrm{2个}}}{4}-\mathrm{1个}$，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$\gamma <\frac{{（ñ-\mathrm{2个}）}^{\mathrm{2个}}}{4}-4$，那么对于任何整数 $ķ\ge \mathrm{2个}$，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$\gamma \ne {\gamma }_{Ĵ}$没有必要。确实，众所周知，如果$\gamma >\frac{{（ñ-\mathrm{2个}）}^{\mathrm{2个}}}{4}-\mathrm{1个}$ 和 $\Omega$ 是一个球 $乙$，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：$\gamma \ge \frac{{（ñ-\mathrm{2个}）}^{\mathrm{2个}}}{4}-4$，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。$\gamma =0$。