The equation -őĒūĚĎĘ-őĽūĚĎĘ |ūĚĎ•| 2 =|‚ąáūĚĎĘ| ūĚĎĚ +ūĚĎźūĚĎď(ūĚĎ•): The optimal power
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 6 (2007) no. 1, pp. 159-183.

We will consider the following problem

-őĒu-őĽu |x| 2 =|‚ąáu| p +cf,u>0inő©,
where ő©‚äā‚ĄĚ N is a domain such that 0‚ąąő©, N‚Č•3, c>0 and őĽ>0. The main objective of this note is to study the precise threshold p + =p + (őĽ) for which there is no very weak supersolution if p‚Č•p + (őĽ). The optimality of p + (őĽ) is also proved by showing the solvability of the Dirichlet problem when 1‚ȧp<p + (őĽ), for c>0 small enough and f‚Č•0 under some hypotheses that we will prescribe.

Classification: 35D05,  35J10,  35J60,  46E30
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     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},
     pages = {159--183},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 6},
     number = {1},
     year = {2007},
     zbl = {1181.35080},
     mrnumber = {2341519},
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     url = {http://archive.numdam.org/item/ASNSP_2007_5_6_1_159_0/}
}
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%A Peral, Ireneo
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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://archive.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 | Zbl

[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. | EuDML | MR | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

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

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

[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 | Zbl

[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

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

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

[14] H. Brezis and A. Ponce, Kato‚Äôs inequality when őĒu is a measure, C.R. Math. Acad. Sci. Paris 338 (2004), 599-604. | MR | Zbl

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

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl

[21] P. L. Lions, ‚ÄúGeneralized Solutions of Hamilton-Jacobi Equations‚ÄĚ, Pitman Res. Notes Math., Vol. 62, 1982. | MR | Zbl

[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

[23] V. G. Maz'Ja ‚ÄúSobolev Spaces‚ÄĚ, Springer Verlag, Berlin, 1985. | MR

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