The singular set for a semilinear unstable problem.

*(English)*Zbl 1407.35232In this interesting paper, the author studies the regularity of solutions and the singular points of the following semilinear problem:
\[
\Delta u=-\lambda_+(x)(u^+)^q +\lambda_-(x)(u^-)^q \;\;\text{ in } \;\;B_1,
\]
where \(B_1\) is the unit ball in \(\mathbb{R}^n\), \(0<q<1\) and \(0\leq \lambda_{\pm}\) are HĂ¶lder continuous, with \(\lambda_+(x)+\lambda_-(x)\geq\lambda_0>0\) for all \(x\). The main concerns are the local regularity of solutions and their nodal set \(\Gamma=\{u=0\}\).

With the additional assumption that \(u\geq 0\), it was shown in [D. Phillips, Commun. Partial Differ. Equations 8, 1409–1454 (1983; Zbl 0555.35128)] that solutions of \(\Delta u=u^q\) have the optimal \(C^{[\kappa],\kappa-[\kappa]}\) regularity, with \(\kappa=2/(1-q)\). Here, the author studies the singular points at which a solution fails to have this regularity. The singular set is divided in two classes: first, the class containing the points where at least one of the derivatives of order less than \(\kappa\) is nonzero; secondly, the class (denoted by \(\mathcal{S}_{\kappa}\)) of the points where all the derivatives of order less than \(\kappa\) exist and vanish.

The author proves that \(\mathcal{S}_{\kappa}=\emptyset\) when \(\kappa\) is not an integer, and exhibits an example where \(S_{\kappa}\neq\emptyset\) when \(\kappa\in\mathbb{N}\) and \(\lambda_+=\lambda_-\). The crucial step in the proof of the existence of a solution with \(S_{\kappa}\neq\emptyset\) is the classification of \(\kappa\)-homogeneous global solutions in \(\mathbb{R}^2\).

Finally, a regularity investigation when \(n=2\) shows that in this case the singular points in \(\mathcal{S}_{\kappa}\) are isolated. Moreover, if \(x_0\in\mathcal{S}_{\kappa}\), \(\Gamma\setminus\{x_0\}\) is, in a neighborhood of \(x_0\) locally \(C^1\). The main ingredients in this proof are an Almgren type monotonicity formula, along with a Weiss type monotonicity formula.

With the additional assumption that \(u\geq 0\), it was shown in [D. Phillips, Commun. Partial Differ. Equations 8, 1409–1454 (1983; Zbl 0555.35128)] that solutions of \(\Delta u=u^q\) have the optimal \(C^{[\kappa],\kappa-[\kappa]}\) regularity, with \(\kappa=2/(1-q)\). Here, the author studies the singular points at which a solution fails to have this regularity. The singular set is divided in two classes: first, the class containing the points where at least one of the derivatives of order less than \(\kappa\) is nonzero; secondly, the class (denoted by \(\mathcal{S}_{\kappa}\)) of the points where all the derivatives of order less than \(\kappa\) exist and vanish.

The author proves that \(\mathcal{S}_{\kappa}=\emptyset\) when \(\kappa\) is not an integer, and exhibits an example where \(S_{\kappa}\neq\emptyset\) when \(\kappa\in\mathbb{N}\) and \(\lambda_+=\lambda_-\). The crucial step in the proof of the existence of a solution with \(S_{\kappa}\neq\emptyset\) is the classification of \(\kappa\)-homogeneous global solutions in \(\mathbb{R}^2\).

Finally, a regularity investigation when \(n=2\) shows that in this case the singular points in \(\mathcal{S}_{\kappa}\) are isolated. Moreover, if \(x_0\in\mathcal{S}_{\kappa}\), \(\Gamma\setminus\{x_0\}\) is, in a neighborhood of \(x_0\) locally \(C^1\). The main ingredients in this proof are an Almgren type monotonicity formula, along with a Weiss type monotonicity formula.

Reviewer: Mariana Vega Smit (Bellingham)

##### MSC:

35R35 | Free boundary problems for PDEs |

35J61 | Semilinear elliptic equations |

35B65 | Smoothness and regularity of solutions to PDEs |

Full Text:
DOI

##### References:

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