The equation $-\Delta 𝑢-\lambda \frac{𝑢}{{|𝑥|}^{\mathbf{2}}}={|\nabla 𝑢|}^{𝑝}+𝑐𝑓\left(𝑥\right)$: The optimal power
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 6 (2007) no. 1, p. 159-183

We will consider the following problem $-\Delta u-\lambda \frac{u}{{|x|}^{2}}={|\nabla u|}^{p}+c\phantom{\rule{0.166667em}{0ex}}f,\phantom{\rule{1em}{0ex}}u>0\phantom{\rule{4pt}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}\Omega ,\phantom{\rule{1em}{0ex}}$ where $\Omega \subset {ℝ}^{N}$ is a domain such that $0\in \Omega$, $N\ge 3$, $c>0$ and $\lambda >0$. The main objective of this note is to study the precise threshold ${p}_{+}={p}_{+}\left(\lambda \right)$ for which there is no very weak supersolution if $p\ge {p}_{+}\left(\lambda \right)$. The optimality of ${p}_{+}\left(\lambda \right)$ is also proved by showing the solvability of the Dirichlet problem when $1\le p<{p}_{+}\left(\lambda \right)$, for $c>0$ small enough and $f\ge 0$ under some hypotheses that we will prescribe.

Classification:  35D05,  35J10,  35J60,  46E30
@article{ASNSP_2007_5_6_1_159_0,
author = {Abdellaoui, Boumediene and Peral, Ireneo},
title = {The equation $-\Delta \textit {u}-\lambda \dfrac{\textit {u}}{|\textit {x}|^{\bf 2}}=|\nabla \textit {u}|^{\textit {p}}+ \textit {c} \textit {f}(\textit {x})$: The optimal power},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 6},
number = {1},
year = {2007},
pages = {159-183},
zbl = {1181.35080},
mrnumber = {2341519},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2007_5_6_1_159_0}
}

Abdellaoui, Boumediene; Peral, Ireneo. The equation $-\Delta \textit {u}-\lambda \dfrac{\textit {u}}{|\textit {x}|^{\bf 2}}=|\nabla \textit {u}|^{\textit {p}}+ \textit {c} \textit {f}(\textit {x})$: The optimal power. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 6 (2007) no. 1, pp. 159-183. http://www.numdam.org/item/ASNSP_2007_5_6_1_159_0/

[1] B. Abdellaoui and I. Peral, Some results for semilinear elliptic equations with critical potential, Proc. Roy. Soc. Edinburgh 132A (2002), 1-24. | MR 1884469 | Zbl 1014.35023

[2] B. Abdellaoui and I. Peral, A note on a critical problem with natural growth in the gradient, J. Eur. Math. Soc. Sect. A 8 (2006), 157-170. | MR 2239296 | Zbl 1245.35032

[3] B. Abdelahoui and I. Peral, Nonexistence results for quasilinear elliptic equations related to Caffarelli-Kohn-Nirenberg inequalities, Commun. Pure Appl. Anal. 2 (2003), 539-566. | MR 2019067 | Zbl 1148.35324

[4] B. Abdellaoui, A. Dall'Aglio and I. Peral, Some remarks on elliptic problems with critical growth on the gradient, J. Differential Equations 222 (2006), 21-62. | MR 2200746 | Zbl 1357.35089 | Zbl pre05013584

[5] B. Abdellaoui, E. Colorado and I. Peral, Some improved Caffarelli-Kohn-Nirenberg inequalities, Cal. Var. Partial Differential Equations 23 (2005), 327-345. | MR 2142067 | Zbl 1207.35114

[6] N. E. Alaa and M. Pierre, Weak solutions of some quasilinear elliptic equations with data measures, SIAM J. Math. Anal. 24 (1993), 23-35. | MR 1199524 | Zbl 0809.35021

[7] H. Berestycki, S. Kamin and G. Sivashinsky, Metastability in a flame front evolution equation, Interfaces Free Bound. 3 (2001) 361-392. | MR 1869585 | Zbl 0991.35097

[8] L. Boccardo, T. Gallouet and F. Murat, “A Unified Presentation of Two Existence Results for Problems with Natural Growth”, Research Notes in Mathematics, Vol. 296, 1993, 127-137, Longman. | MR 1248641 | Zbl 0806.35033

[9] L. Boccardo, T. Gallouët and L. Orsina, Existence and nonexistence of solutions for some nonlinear elliptic equations, J. Anal. Math. 73 (1997), 203-223. | MR 1616410 | Zbl 0898.35035

[10] L. Boccardo, F. Murat and J.-P. Puel, Existence des solutions non bornées pour certains équations quasi-linéaires, Port. Math., 41 (1982), 507-534. | MR 766873 | Zbl 0524.35041

[11] L. Boccardo, L. Orsina and I. Peral, A remark on existence and optimal summability of solutions of elliptic problems involving Hardy potential, Discrete Cont. Dyn. Syst. 16 (2006), 513-523. | MR 2257147 | Zbl pre05141172

[12] H. Brezis and X. Cabré, Some simple nonlinear PDE's without solution, Boll. Unione. Mat. Ital. Sez. B 8 (1998), 223-262. | MR 1638143 | Zbl 0907.35048

[13] H. Brezis, L. Dupaigne and A. Tesei, On a semilinear equation with inverse-square potential, Selecta Math. 11 (2005), 1-7. | MR 2179651 | Zbl 1161.35383

[14] H. Brezis and A. Ponce, Kato’s inequality when $\Delta u$ is a measure, C.R. Math. Acad. Sci. Paris 338 (2004), 599-604. | MR 2056467 | Zbl 1101.35028

[15] L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math. 53 (1984), 259-275. | Numdam | MR 768824 | Zbl 0563.46024

[16] V. Ferone and F. Murat, Nonlinear problems having natural growth in the gradient: an existence result when the source terms are small, Nonlinear Anal. 42 (2000), 1309-1326. | MR 1780731 | Zbl 1158.35358

[17] K. Hansson, V. G. Maz'Ya and I. E. Verbitsky, Criteria of solvability for multidimensional Riccati equations, Ark. Mat. 37 (1999), 87-120. | MR 1673427 | Zbl 1087.35513

[18] M. Kardar, G. Parisi and Y. C. Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Lett. 56 (1986), 889-892. | Zbl 1101.82329

[19] T. Kato, Schrödinger operators with singular potentials, Israel J. Math. 13 (1972), 135-148. | Zbl 0246.35025

[20] J. L. Kazdan and R. J. Kramer, Invariant criteria for existence of solutions to second-order quasilinear elliptic equations, Comm. Pure Appl. Math. 31 (1978), 619-645. | MR 477446 | Zbl 0368.35031

[21] P. L. Lions, “Generalized Solutions of Hamilton-Jacobi Equations”, Pitman Res. Notes Math., Vol. 62, 1982. | MR 667669 | Zbl 0497.35001

[22] F. Murat, L’injection du cone positif de ${H}^{-1}$ dans ${W}^{-1,q}$ est compacte pour tout $q<2$, J. Math. Pures Appl. 60 (1981) 309-322. | Zbl 0471.46020

[23] V. G. Maz'Ja “Sobolev Spaces”, Springer Verlag, Berlin, 1985. | MR 817985

[24] Z.Q. Wang and M. Willem, Caffarelli-Kohn-Nirenberg inequalities with remainder terms, J. Funct. Anal. 203 (2003), 550-568. | MR 2003359 | Zbl 1037.26014