Bounded Solutions for Some Dirichlet Problems with L 1 ( Ω ) Data

Tommaso Leonori

Bollettino dell'Unione Matematica Italiana (2007)

  • Volume: 10-B, Issue: 3, page 785-795
  • ISSN: 0392-4033

Abstract

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In this paper we prove the existence of a solution for a problem whose model is: { - Δ u + u σ - | u | = γ | u | 2 + f ( x ) in  Ω u = 0 on  Ω with f ( x ) in L 1 ( Ω ) and σ , γ > 0 .

How to cite

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Leonori, Tommaso. "Bounded Solutions for Some Dirichlet Problems with $L^1(\Omega)$ Data." Bollettino dell'Unione Matematica Italiana 10-B.3 (2007): 785-795. <http://eudml.org/doc/290364>.

@article{Leonori2007,
abstract = {In this paper we prove the existence of a solution for a problem whose model is: \begin\{equation*\} \begin\{cases\} -\Delta u + \frac\{u\}\{\sigma - |u|\} = \gamma |\nabla u|^\{2\} + f(x) & \text\{in \} \Omega \\ u = 0 & \text\{on \} \partial \Omega \end\{cases\} \end\{equation*\} with $f(x)$ in $L^\{1\}(\Omega)$ and $\sigma$, $\gamma > 0$.},
author = {Leonori, Tommaso},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {785-795},
publisher = {Unione Matematica Italiana},
title = {Bounded Solutions for Some Dirichlet Problems with $L^1(\Omega)$ Data},
url = {http://eudml.org/doc/290364},
volume = {10-B},
year = {2007},
}

TY - JOUR
AU - Leonori, Tommaso
TI - Bounded Solutions for Some Dirichlet Problems with $L^1(\Omega)$ Data
JO - Bollettino dell'Unione Matematica Italiana
DA - 2007/10//
PB - Unione Matematica Italiana
VL - 10-B
IS - 3
SP - 785
EP - 795
AB - In this paper we prove the existence of a solution for a problem whose model is: \begin{equation*} \begin{cases} -\Delta u + \frac{u}{\sigma - |u|} = \gamma |\nabla u|^{2} + f(x) & \text{in } \Omega \\ u = 0 & \text{on } \partial \Omega \end{cases} \end{equation*} with $f(x)$ in $L^{1}(\Omega)$ and $\sigma$, $\gamma > 0$.
LA - eng
UR - http://eudml.org/doc/290364
ER -

References

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  1. BÉNILAN, P. - BOCCARDO, L.. GALLOUËTT, T. - GARIEPY, R. - PIERRE, M. - VAZQUEZ, J. L., An L 1 -theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 22 (1995), 241-273. MR1354907
  2. BÉNILAN, P. - BREZIS, H. - CRANDALL, M. C., A semilinear equation in L 1 ( N ) , Ann. Scuola Norm. Sup. Pisa Cl. Sci., 2, no. 4 (1975), 523-555. MR390473
  3. BOCCARDO, L., On the regularizing effect of strongly increasing lower order terms, J. Evol. Equ., 3, no. 2 (2003), 225-236 Zbl1225.35085MR1980975DOI10.1201/9780203910108.ch12
  4. BOCCARDO, L. - GALLOUËT, T., Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87 (1989), 149-169. MR1025884DOI10.1016/0022-1236(89)90005-0
  5. BOCCARDO, L. - MURAT, F. - PUEL, J. P., Existence of bounded solutions for nonlinear elliptic unilateral problems, Ann. Mat. Pura Appl., 152 (1988), 183-196. Zbl0687.35042MR980979DOI10.1007/BF01766148
  6. 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, no. 2 (1992), 326-333. Zbl0785.35033MR1147866DOI10.1137/0523016
  7. BREZIS, H. - STRAUSS, W. A., Semi-linear second-order elliptic equations in L 1 , J. Math. Soc. Japan, 25 (1973), 565-590. Zbl0278.35041MR336050DOI10.2969/jmsj/02540565
  8. DUPAIGNE, L. - PONCE, A. C. - PORRETTA, A., Elliptic equations with vertical asymptotes in the nonlinear term, J. Anal. Math., 98 (2006), 349-396. Zbl1132.35366MR2254490DOI10.1007/BF02790280
  9. LERAY, J. - LIONS, J. L., Quelque resultat de Višik sur les problémes elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97-107. Zbl0132.10502MR194733

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