The equation - Δ 𝑢 - λ 𝑢 | 𝑥 | 2 = | ∇ 𝑢 | 𝑝 + 𝑐 𝑓 ( 𝑥 ) : The optimal power

Boumediene Abdellaoui; Ireneo Peral

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2007)

  • Volume: 6, Issue: 1, page 159-183
  • ISSN: 0391-173X

Abstract

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We will consider the following problem - Δ u - λ u | x | 2 = | ∇ u | p + c f , u > 0 in Ω , where Ω ⊂ ℝ N is a domain such that 0 ∈ Ω , N ≥ 3 , c > 0 and λ > 0 . The main objective of this note is to study the precise threshold p + = p + ( λ ) for which there is novery weak supersolutionif p ≥ p + ( λ ) . The optimality of p + ( λ ) is also proved by showing the solvability of the Dirichlet problem when 1 ≤ p < p + ( λ ) , for c > 0 small enough and f ≥ 0 under some hypotheses that we will prescribe.

How to cite

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Abdellaoui, Boumediene, and 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 6.1 (2007): 159-183. <http://eudml.org/doc/272289>.

@article{Abdellaoui2007,
abstract = {We will consider the following problem\[-\Delta u-\lambda \frac\{u\}\{|x|^2\}=|\nabla u|^p+c\,f,\quad u&gt;0 \hbox\{ in \} \Omega ,\quad \]where $ \Omega \subset \mathbb \{R\}^N$ is a domain such that $0\in \Omega $, $N\ge 3$, $c&gt;0$ and $\lambda &gt;0$. The main objective of this note is to study the precise threshold $p_+=p_+(\lambda )$ for which there is novery weak supersolutionif $ p\ge p_+(\lambda )$. The optimality of $p_+(\lambda )$ is also proved by showing the solvability of the Dirichlet problem when $1\le p&lt;p_+(\lambda )$, for $c&gt;0$ small enough and $f\ge 0$ under some hypotheses that we will prescribe.},
author = {Abdellaoui, Boumediene, Peral, Ireneo},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {optimal power; supersolution; solvability; Dirichlet problem},
language = {eng},
number = {1},
pages = {159-183},
publisher = {Scuola Normale Superiore, Pisa},
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},
url = {http://eudml.org/doc/272289},
volume = {6},
year = {2007},
}

TY - JOUR
AU - Abdellaoui, Boumediene
AU - Peral, Ireneo
TI - 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
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2007
PB - Scuola Normale Superiore, Pisa
VL - 6
IS - 1
SP - 159
EP - 183
AB - We will consider the following problem\[-\Delta u-\lambda \frac{u}{|x|^2}=|\nabla u|^p+c\,f,\quad u&gt;0 \hbox{ in } \Omega ,\quad \]where $ \Omega \subset \mathbb {R}^N$ is a domain such that $0\in \Omega $, $N\ge 3$, $c&gt;0$ and $\lambda &gt;0$. The main objective of this note is to study the precise threshold $p_+=p_+(\lambda )$ for which there is novery weak supersolutionif $ p\ge p_+(\lambda )$. The optimality of $p_+(\lambda )$ is also proved by showing the solvability of the Dirichlet problem when $1\le p&lt;p_+(\lambda )$, for $c&gt;0$ small enough and $f\ge 0$ under some hypotheses that we will prescribe.
LA - eng
KW - optimal power; supersolution; solvability; Dirichlet problem
UR - http://eudml.org/doc/272289
ER -

References

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  1. [1] B. Abdellaoui and I. Peral, Some results for semilinear elliptic equations with critical potential, Proc. Roy. Soc. Edinburgh 132A (2002), 1–24. Zbl1014.35023MR1884469
  2. [2] B. Abdellaoui and I. Peral, A note on a critical problem with natural growth in the gradient, J. Eur. Math. Soc. Sect. A8 (2006), 157–170. Zbl1245.35032MR2239296
  3. [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. Zbl1148.35324MR2019067
  4. [4] B. Abdellaoui, A. Dall’Aglio and I. Peral, Some remarks on elliptic problems with critical growth on the gradient, J. Differential Equations222 (2006), 21–62. Zbl05013584MR2200746
  5. [5] B. Abdellaoui, E. Colorado and I. Peral, Some improved Caffarelli-Kohn-Nirenberg inequalities, Cal. Var. Partial Differential Equations23 (2005), 327–345. Zbl1207.35114MR2142067
  6. [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. Zbl0809.35021MR1199524
  7. [7] H. Berestycki, S. Kamin and G. Sivashinsky, Metastability in a flame front evolution equation, Interfaces Free Bound.3 (2001) 361–392. Zbl0991.35097MR1869585
  8. [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. Zbl0806.35033MR1248641
  9. [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. Zbl0898.35035MR1616410
  10. [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. Zbl0524.35041MR766873
  11. [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. Zbl05141172MR2257147
  12. [12] H. Brezis and X. Cabré, Some simple nonlinear PDE’s without solution, Boll. Unione. Mat. Ital. Sez. B8 (1998), 223–262. Zbl0907.35048MR1638143
  13. [13] H. Brezis, L. Dupaigne and A. Tesei, On a semilinear equation with inverse-square potential, Selecta Math.11 (2005), 1–7. Zbl1161.35383MR2179651
  14. [14] H. Brezis and A. Ponce, Kato’s inequality when Δ u is a measure, C.R. Math. Acad. Sci. Paris338 (2004), 599–604. Zbl1101.35028MR2056467
  15. [15] L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math.53 (1984), 259–275. Zbl0563.46024MR768824
  16. [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. Zbl1158.35358MR1780731
  17. [17] K. Hansson, V. G. Maz’ya and I. E. Verbitsky, Criteria of solvability for multidimensional Riccati equations, Ark. Mat.37 (1999), 87–120. Zbl1087.35513MR1673427
  18. [18] M. Kardar, G. Parisi and Y. C. Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Lett.56 (1986), 889–892. Zbl1101.82329
  19. [19] T. Kato, Schrödinger operators with singular potentials, Israel J. Math.13 (1972), 135–148. Zbl0246.35025
  20. [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. Zbl0368.35031MR477446
  21. [21] P. L. Lions, “Generalized Solutions of Hamilton-Jacobi Equations”, Pitman Res. Notes Math., Vol. 62, 1982. Zbl0497.35001MR667669
  22. [22] F. Murat, L’injection du cone positif de H - 1 dans W - 1 , q est compacte pour tout q l t ; 2 , J. Math. Pures Appl.60 (1981) 309–322. Zbl0471.46020
  23. [23] V. G. Maz’ja“Sobolev Spaces”, Springer Verlag, Berlin, 1985. MR817985
  24. [24] Z.Q. Wang and M. Willem, Caffarelli-Kohn-Nirenberg inequalities with remainder terms, J. Funct. Anal.203 (2003), 550–568. Zbl1037.26014MR2003359

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