Positive solutions for some quasilinear elliptic equations with natural growths

Lucio Boccardo

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2000)

  • Volume: 11, Issue: 1, page 31-39
  • ISSN: 1120-6330

Abstract

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We shall prove an existence result for a class of quasilinear elliptic equations with natural growth. The model problem is - div ⁢ 1 + u r ⁢ ∇ ⁡ u + u m - 2 ⁢ u ⁢ ∇ ⁡ u 2 = f in ⁢ Ω u = 0 su ⁢ ∂ ⁡ Ω .

How to cite

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Boccardo, Lucio. "Positive solutions for some quasilinear elliptic equations with natural growths." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 11.1 (2000): 31-39. <http://eudml.org/doc/252345>.

@article{Boccardo2000,
abstract = {We shall prove an existence result for a class of quasilinear elliptic equations with natural growth. The model problem is $$ \begin\{cases\} - \text\{div\} ((1+ |u|^\{r\}) \nabla u) + |u|^\{m-2\} u |\nabla u|^\{2\} = f \quad &\text\{in\} \, \Omega \\ u = 0 &\text\{su\} \, \partial\Omega. \end\{cases\} $$},
author = {Boccardo, Lucio},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Quasilinear elliptic equations; Natural growth coefficients; Euler-Lagrange equations; natural growth coefficients},
language = {eng},
month = {3},
number = {1},
pages = {31-39},
publisher = {Accademia Nazionale dei Lincei},
title = {Positive solutions for some quasilinear elliptic equations with natural growths},
url = {http://eudml.org/doc/252345},
volume = {11},
year = {2000},
}

TY - JOUR
AU - Boccardo, Lucio
TI - Positive solutions for some quasilinear elliptic equations with natural growths
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2000/3//
PB - Accademia Nazionale dei Lincei
VL - 11
IS - 1
SP - 31
EP - 39
AB - We shall prove an existence result for a class of quasilinear elliptic equations with natural growth. The model problem is $$ \begin{cases} - \text{div} ((1+ |u|^{r}) \nabla u) + |u|^{m-2} u |\nabla u|^{2} = f \quad &\text{in} \, \Omega \\ u = 0 &\text{su} \, \partial\Omega. \end{cases} $$
LA - eng
KW - Quasilinear elliptic equations; Natural growth coefficients; Euler-Lagrange equations; natural growth coefficients
UR - http://eudml.org/doc/252345
ER -

References

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  2. Bensoussan, A. - Boccardo, L. - Murat, F., On a nonlinear partial differential equation having natural growth terms and unbounded solution. Ann. Inst. H. Poincaré Anal. Non Linéaire, 5, 1988, 347-364. Zbl0696.35042MR963104
  3. Boccardo, L., Calcolo delle Variazioni. Roma 1 University PhD course, 1996. 
  4. Boccardo, L., Some nonlinear Dirichlet problems in L 1 involving lower order terms in divergence form. In: A. Alvino et al. (eds), Progress in elliptic and parabolic partial differential equations (Capri, 1994). Pitman Res. Notes Math. Ser., 350, Longman, Harlow1996, 43-57. Zbl0889.35034MR1430139
  5. Boccardo, L. - Gallouët, T., Strongly nonlinear elliptic equations having natural growth terms and L 1 data. Nonlinear Anal., 19, 1992, 573-579. Zbl0795.35031MR1183664DOI10.1016/0362-546X(92)90022-7
  6. Boccardo, L. - Gallouët, T. - Orsina, L., Existence and nonexistence of solutions for some nonlinear elliptic equations. J. Anal. Math., 73, 1997, 203-223. Zbl0898.35035MR1616410DOI10.1007/BF02788144
  7. Boccardo, L. - Murat, F. - Puel, J.-P., Existence de solutions non bornées pour certaines équations quasi-lineaires. Portugal. Math., 41, 1982, 507-534. Zbl0524.35041MR766873
  8. Boccardo, L. - Murat, F. - Puel, J.-P., L ∞ estimate for some nonlinear elliptic partial differential equations and application to an existence result. SIAM J. Math. Anal., 23, 1992, 326-333. Zbl0785.35033MR1147866DOI10.1137/0523016
  9. Brezis, H. - Browder, F. E., Some properties of higher order Sobolev spaces. J. Math. Pures Appl., 61, 1982, 245-259. Zbl0512.46034MR690395
  10. Brezis, H. - Nirenberg, L., Removable singularities for nonlinear elliptic equations. Topol. Methods Nonlinear Anal., 9, 1997, 201-219. Zbl0905.35027MR1491843
  11. Dacorogna, B., Direct methods in the calculus of variations. Applied Mathematical Sciences, 78. Springer-Verlag, Berlin-New York1989. Zbl0703.49001MR990890
  12. Del Vecchio, T., Strongly nonlinear problems with Hamiltonian having natural growth. Houston J. Math., 16, 1990, 7-24. Zbl0714.35035MR1071263
  13. Lions, J.-L., Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris1969. Zbl0189.40603
  14. Porretta, A., Some remarks on the regularity of solutions for a class of elliptic equations with measure data. Preprint, Dip. Mat. Roma 1. Zbl0974.35032MR1814734
  15. Porretta, A., Existence for elliptic equations in L 1 having lower order terms with natural growth. Preprint, Dip. Mat. Roma 1. Zbl0963.35068MR1759814

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