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Uniform stability estimate for the Vlasov-Poisson-Boltzmann system
Asymptotic behavior of minimal solutions of $ -\Delta u = \lambda f(u) $ as $ \lambda\to-\infty $
1. | Università degli Studi Roma Tre, Dipartimento di Matematica e Fisica, Largo S. Leonardo Murialdo 1, 00146 Roma, Italy |
2. | Università degli Studi di Sassari, Dipartimento di Chimica e Farmacia, Via Piandanna 4, 00710 Sassari, Italy |
3. | Sapienza Università di Roma, Dipartimento di Matematica, Piazzale Aldo Moro 5, 00185 Roma, Italy |
$\left\{ \begin{matrix} -\Delta u=\lambda f(u)\ \ \ \ \text{in}\ \Omega \\ u=0\ \ \ \ \ \ \ \ \ \ \ \ \ \text{on}\ \partial \Omega \\\end{matrix} \right.,\ \ \ \ \ \ \ \left( \mathcal{P}_{f}^{\lambda } \right)$ |
$ \lambda<0 $ |
$ f $ |
$ u_\lambda $ |
$ \lambda $ |
$ \lambda\to-\infty $ |
$ u_\lambda $ |
References:
[1] |
C. Bandle and M. Marcus,
On the boundary values of solutions of a problem arising in plasma physics, Nonlinear Anal., 6 (1982), 1287-1294.
doi: 10.1016/0362-546X(82)90104-3. |
[2] |
C. Bandle and M. Marcus,
"Large" solutions of semilinear elliptic equations: Existence, uniqueness and asymptotic behaviour, J. Anal. Math., 58 (1992), 9-24.
doi: 10.1007/BF02790355. |
[3] |
C. Bandle and R. P. Sperb,
Qualitative behavior and bounds in a nonlinear plasma problem, SIAM J. Math. Anal., 14 (1983), 142-151.
doi: 10.1137/0514011. |
[4] |
S. Baraket and F. Pacard,
Construction of singular limits for a semilinear elliptic equation in dimension $2$, Calc. Var. Partial Differential Equations, 6 (1998), 1-38.
doi: 10.1007/s005260050080. |
[5] |
H. Berestycki and H. Brézis,
On a free boundary problem arising in plasma physics, Nonlinear Anal., 4 (1980), 415-436.
doi: 10.1016/0362-546X(80)90083-8. |
[6] |
S. Berhanu, F. Gladiali and G. Porru,
Qualitative properties of solutions to elliptic singular problems, J. Inequal. Appl., 3 (1999), 313-330.
doi: 10.1155/S1025583499000223. |
[7] |
L. A. Caffarelli and A. Friedman, Asymptotic estimates for the plasma problem, Duke Math. J., 47 (1980), 705–742, URLhttp://projecteuclid.org/euclid.dmj/1077314190.
doi: 10.1215/S0012-7094-80-04743-2. |
[8] |
E. Caglioti, P.-L. Lions, C. Marchioro and M. Pulvirenti, A special class of stationary flows for two-dimensional {E}uler equations: a statistical mechanics description, Comm. Math. Phys., 143 (1992), 501–525, URLhttp://projecteuclid.org/euclid.cmp/1104249078.
doi: 10.1007/BF02099262. |
[9] |
E. Caglioti, P.-L. Lions, C. Marchioro and M. Pulvirenti, A special class of stationary flows for two-dimensional {E}uler equations: A statistical mechanics description. Ⅱ, Comm. Math. Phys., 174 (1995), 229–260, URLhttp://projecteuclid.org/euclid.cmp/1104275293.
doi: 10.1007/BF02099602. |
[10] |
M. G. Crandall, P. H. Rabinowitz and L. Tartar,
On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations, 2 (1977), 193-222.
doi: 10.1080/03605307708820029. |
[11] |
M. G. Crandall and P. H. Rabinowitz,
Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180.
doi: 10.1007/BF00282325. |
[12] |
S. Dumont, L. Dupaigne, O. Goubet and V. Rădulescu,
Back to the Keller-Osserman condition for boundary blow-up solutions, Adv. Nonlinear Stud., 7 (2007), 271-298.
doi: 10.1515/ans-2007-0205. |
[13] |
L. Dupaigne, Stable solutions of elliptic partial differential equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 143. Chapman & Hall/CRC, Boca Raton, FL, 2011.
doi: 10.1201/b10802. |
[14] |
P. Esposito, M. Grossi and A. Pistoia,
On the existence of blowing-up solutions for a mean field equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 227-257.
doi: 10.1016/j.anihpc.2004.12.001. |
[15] |
B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209–243, URLhttp://projecteuclid.org/euclid.cmp/1103905359.
doi: 10.1007/BF01221125. |
[16] |
F. Gladiali and M. Grossi,
Some results for the Gelfand's problem, Comm. Partial Differential Equations, 29 (2004), 1335-1364.
doi: 10.1081/PDE-200037754. |
[17] |
D. Gogny and P.-L. Lions,
Sur les états d'équilibre pour les densités électroniques dans les plasmas, RAIRO Modél. Math. Anal. Numér., 23 (1989), 137-153.
doi: 10.1051/m2an/1989230101371. |
[18] |
D. D. Joseph and T. S. Lundgren,
Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1972/73), 241-269.
doi: 10.1007/BF00250508. |
[19] |
J. B. Keller,
On solutions of $\Delta u = f(u)$, Comm. Pure Appl. Math., 10 (1957), 503-510.
doi: 10.1002/cpa.3160100402. |
[20] |
M. Marcus and L. Véron,
Existence and uniqueness results for large solutions of general nonlinear elliptic equations, J. Evol. Equ., 3 (2003), 637-652.
doi: 10.1007/s00028-003-0122-y. |
[21] |
F. Mignot and J.-P. Puel,
Sur une classe de problèmes non linéaires avec non linéairité positive, croissante, convexe, Comm. Partial Differential Equations, 5 (1980), 791-836.
doi: 10.1080/03605308008820155. |
[22] |
P. Mironescu and V. D. Rădulescu,
The study of a bifurcation problem associated to an asymptotically linear function, Nonlinear Anal., 26 (1996), 857-875.
doi: 10.1016/0362-546X(94)00327-E. |
[23] |
R. Osserman, On the inequality $\Delta u\geq f(u)$, Pacific J. Math., 7 (1957), 1641–1647, URLhttp://projecteuclid.org/euclid.pjm/1103043236.
doi: 10.2140/pjm.1957.7.1641. |
[24] |
T. Suzuki,
Global analysis for a two-dimensional elliptic eigenvalue problem with the exponential nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 367-397.
doi: 10.1016/S0294-1449(16)30232-3. |
[25] |
R. Temam,
Remarks on a free boundary value problem arising in plasma physics, Comm. Partial Differential Equations, 2 (1977), 563-585.
doi: 10.1080/03605307708820039. |
show all references
References:
[1] |
C. Bandle and M. Marcus,
On the boundary values of solutions of a problem arising in plasma physics, Nonlinear Anal., 6 (1982), 1287-1294.
doi: 10.1016/0362-546X(82)90104-3. |
[2] |
C. Bandle and M. Marcus,
"Large" solutions of semilinear elliptic equations: Existence, uniqueness and asymptotic behaviour, J. Anal. Math., 58 (1992), 9-24.
doi: 10.1007/BF02790355. |
[3] |
C. Bandle and R. P. Sperb,
Qualitative behavior and bounds in a nonlinear plasma problem, SIAM J. Math. Anal., 14 (1983), 142-151.
doi: 10.1137/0514011. |
[4] |
S. Baraket and F. Pacard,
Construction of singular limits for a semilinear elliptic equation in dimension $2$, Calc. Var. Partial Differential Equations, 6 (1998), 1-38.
doi: 10.1007/s005260050080. |
[5] |
H. Berestycki and H. Brézis,
On a free boundary problem arising in plasma physics, Nonlinear Anal., 4 (1980), 415-436.
doi: 10.1016/0362-546X(80)90083-8. |
[6] |
S. Berhanu, F. Gladiali and G. Porru,
Qualitative properties of solutions to elliptic singular problems, J. Inequal. Appl., 3 (1999), 313-330.
doi: 10.1155/S1025583499000223. |
[7] |
L. A. Caffarelli and A. Friedman, Asymptotic estimates for the plasma problem, Duke Math. J., 47 (1980), 705–742, URLhttp://projecteuclid.org/euclid.dmj/1077314190.
doi: 10.1215/S0012-7094-80-04743-2. |
[8] |
E. Caglioti, P.-L. Lions, C. Marchioro and M. Pulvirenti, A special class of stationary flows for two-dimensional {E}uler equations: a statistical mechanics description, Comm. Math. Phys., 143 (1992), 501–525, URLhttp://projecteuclid.org/euclid.cmp/1104249078.
doi: 10.1007/BF02099262. |
[9] |
E. Caglioti, P.-L. Lions, C. Marchioro and M. Pulvirenti, A special class of stationary flows for two-dimensional {E}uler equations: A statistical mechanics description. Ⅱ, Comm. Math. Phys., 174 (1995), 229–260, URLhttp://projecteuclid.org/euclid.cmp/1104275293.
doi: 10.1007/BF02099602. |
[10] |
M. G. Crandall, P. H. Rabinowitz and L. Tartar,
On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations, 2 (1977), 193-222.
doi: 10.1080/03605307708820029. |
[11] |
M. G. Crandall and P. H. Rabinowitz,
Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180.
doi: 10.1007/BF00282325. |
[12] |
S. Dumont, L. Dupaigne, O. Goubet and V. Rădulescu,
Back to the Keller-Osserman condition for boundary blow-up solutions, Adv. Nonlinear Stud., 7 (2007), 271-298.
doi: 10.1515/ans-2007-0205. |
[13] |
L. Dupaigne, Stable solutions of elliptic partial differential equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 143. Chapman & Hall/CRC, Boca Raton, FL, 2011.
doi: 10.1201/b10802. |
[14] |
P. Esposito, M. Grossi and A. Pistoia,
On the existence of blowing-up solutions for a mean field equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 227-257.
doi: 10.1016/j.anihpc.2004.12.001. |
[15] |
B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209–243, URLhttp://projecteuclid.org/euclid.cmp/1103905359.
doi: 10.1007/BF01221125. |
[16] |
F. Gladiali and M. Grossi,
Some results for the Gelfand's problem, Comm. Partial Differential Equations, 29 (2004), 1335-1364.
doi: 10.1081/PDE-200037754. |
[17] |
D. Gogny and P.-L. Lions,
Sur les états d'équilibre pour les densités électroniques dans les plasmas, RAIRO Modél. Math. Anal. Numér., 23 (1989), 137-153.
doi: 10.1051/m2an/1989230101371. |
[18] |
D. D. Joseph and T. S. Lundgren,
Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1972/73), 241-269.
doi: 10.1007/BF00250508. |
[19] |
J. B. Keller,
On solutions of $\Delta u = f(u)$, Comm. Pure Appl. Math., 10 (1957), 503-510.
doi: 10.1002/cpa.3160100402. |
[20] |
M. Marcus and L. Véron,
Existence and uniqueness results for large solutions of general nonlinear elliptic equations, J. Evol. Equ., 3 (2003), 637-652.
doi: 10.1007/s00028-003-0122-y. |
[21] |
F. Mignot and J.-P. Puel,
Sur une classe de problèmes non linéaires avec non linéairité positive, croissante, convexe, Comm. Partial Differential Equations, 5 (1980), 791-836.
doi: 10.1080/03605308008820155. |
[22] |
P. Mironescu and V. D. Rădulescu,
The study of a bifurcation problem associated to an asymptotically linear function, Nonlinear Anal., 26 (1996), 857-875.
doi: 10.1016/0362-546X(94)00327-E. |
[23] |
R. Osserman, On the inequality $\Delta u\geq f(u)$, Pacific J. Math., 7 (1957), 1641–1647, URLhttp://projecteuclid.org/euclid.pjm/1103043236.
doi: 10.2140/pjm.1957.7.1641. |
[24] |
T. Suzuki,
Global analysis for a two-dimensional elliptic eigenvalue problem with the exponential nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 367-397.
doi: 10.1016/S0294-1449(16)30232-3. |
[25] |
R. Temam,
Remarks on a free boundary value problem arising in plasma physics, Comm. Partial Differential Equations, 2 (1977), 563-585.
doi: 10.1080/03605307708820039. |
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