Symmetry of solutions to singular fractional elliptic equations and applications

Arora, Rakesh; Giacomoni, Jacques; Goel, Divya; Sreenadh, Konijeti

Comptes Rendus. Mathématique, Tome 358 (2020) no. 2, pp. 237-243.

Dans cet article, nous étudions la symétrie et la monotonie des solutions positives d’une équation elliptique semi-linéaire singulière dont le modèle type est

$$\begin{array}{c}\hfill \left(P\right)\left\{\begin{array}{cc}{(-\Delta )}^{s}u=\frac{1}{u^{\delta }}+f\left(u\right),u>0\hfill & \text{in}\Omega ;\hfill \\ u=0\hfill & \text{in}{ℝ}^n\setminus \Omega ,\hfill \end{array}\right.\end{array}$$

où $0<s<1$, $\Omega =B_r\left(0\right)\subset {ℝ}^n$, $n\ge 2s$, $\delta >0$, $f:{ℝ}^{+}\to {ℝ}^{+}$ est localement Lipschitz. Nous démontrons que les solutions classiques de ce problème type sont à symétrie radiale et radialement décroissantes (Théorème 1). Pour cela, nous mettons en oeuvre la méthode du “moving plane”. Nous utilisons ensuite ce résultat général de symétrie pour étudier le comportement global de solutions d’équations elliptiques singulières non locales  : existence d’estimations a priori uniformes (Théorème 2), convergence de solutions à énergie non bornée vers une solution singulière (Théorème 3).

In this article, we study the symmetry of positive solutions to a class of singular semilinear elliptic equations whose prototype is

$$\begin{array}{c}\hfill \left(P\right)\left\{\begin{array}{cc}{(-\Delta )}^{s}u=\frac{1}{u^{\delta }}+f\left(u\right),u>0\hfill & \text{in}\Omega ;\hfill \\ u=0\hfill & \text{in}{ℝ}^n\setminus \Omega ,\hfill \end{array}\right.\end{array}$$

where $0<s<1$, $n\ge 2s$, $\Omega =B_r\left(0\right)\subset {ℝ}^n,\delta >0$, $f\left(u\right)$ is a locally Lipschitz function. We prove that classical solutions are radial and radially decreasing (see Theorem 1). The proof uses the moving plane method adapted to the non local setting. We then give two applications of this main result: Theorem 2 establishes the uniform apriori bound for classical solutions in case of polynomial growth nonlinearities whereas Theorem 3 ensures in case of exponential growth nonlinearities the convergence of large solutions with unbounded energy to a singular solution.

Classification : 35B40, 35B45, 35J75, 35B06

[1] Adimurthi; Giacomoni, Jacques; Santra, Sanjiban Positive solutions to a fractional equation with singular nonlinearity, J. Differ. Equations, Volume 265 (2018) no. 4, pp. 1191-1226 | DOI | MR

[2] Arora, Rakesh; Giacomoni, Jacques; Goel, Divya; Sreenadh, Konijeti Positive solutions of $1$-D half Laplacian equation with singular and critical exponential nonlinearity (to appear in Asymptotic Anal.) | DOI

[3] Barrios, Begoña; De Bonis, Ida; Medina, María; Peral, Ireneo Semilinear problems for the fractional laplacian with a singular nonlinearity, Open Math., Volume 13 (2015), pp. 390-407 | MR | Zbl

[4] Brézis, Haïm; Lions, Pierre-Louis A note on isolated singularities for linear elliptic equations, Mathematical analysis and applications. Part A. (Advances in Mathematics. Supplementary Studies), Volume 7a, Academic Press Inc., 1981, pp. 263-266 | Zbl

[5] Chen, Huyuan; Quaas, Alexander Classification of isolated singularities of nonnegative solutions to fractional semi-linear elliptic equations and the existence results, J. Lond. Math. Soc., Volume 97 (2018) no. 2, pp. 196-221 | DOI | MR | Zbl

[6] Chen, Wenxiong; Li, Congming; Li, Yan A direct method of moving planes for the fractional Laplacian, Adv. Math., Volume 308 (2017), pp. 404-437 | DOI | MR | Zbl

[7] de Figueiredo, Djairo G.; Lions, Pierre-Louis; Nussbaum, Roger D. Apriori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl., Volume 61 (1982), pp. 263-275 | Zbl

[8] Giacomoni, Jacques; Mishra, Pawan K.; Sreenadh, Konijeti Critical growth problems for $\frac{1}{2}$-Laplacian in $ℝ$, Differ. Equ. Appl., Volume 8 (2016) no. 3, pp. 295-317 | MR | Zbl

[9] Giacomoni, Jacques; Mukherjee, Tuhina; Sreenadh, Konijeti Positive solutions of fractional elliptic equation with critical and singular nonlinearty, Adv. Nonlinear Anal., Volume 6 (2017) no. 3, pp. 327-354 | DOI | Zbl

[10] Jarohs, Sven; Weth, Tobias Symmetry via antisymmetric maximum principles in nonlocal problems of variable order, Ann. Mat. Pura Appl., Volume 195 (2016) no. 1, pp. 273-291 | DOI | MR | Zbl

[11] Mukherjee, Tuhina; Sreenadh, Konijeti Fractional elliptic equations with critical growth and singular nonlinearities, Electron. J. Differ. Equ., Volume 2016 (2016), 54, 23 pages | MR | Zbl

[12] Ros-Oton, Xavier; Serra, Joaquim The extremal solution for the fractional Laplacian, Calc. Var. Partial Differ. Equ., Volume 50 (2014) no. 3-4, pp. 723-750 | DOI | MR | Zbl

[13] Takahashi, Futoshi Critical and subcritical fractional Trudinger-–Moser-type inequalities on $ℝ$, Adv. Nonlinear Anal., Volume 8 (2019), pp. 868-884 | DOI | MR | Zbl