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February  2021, 41(2): 681-700. doi: 10.3934/dcds.2020293

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

* Corresponding author: Massimo Grossi

Received  March 2020 Published  August 2020

Fund Project: The third author is supported by "Fondi di ateneo Sapienza"

We consider the following Dirichlet problem
$\left\{ \begin{matrix} -\Delta u=\lambda f(u)\ \ \ \ \text{in}\ \Omega \\ u=0\ \ \ \ \ \ \ \ \ \ \ \ \ \text{on}\ \partial \Omega \\\end{matrix} \right.,\ \ \ \ \ \ \ \left( \mathcal{P}_{f}^{\lambda } \right)$
with
$ \lambda<0 $
and
$ f $
non-negative and non-decreasing.
We show existence and uniqueness of solutions
$ u_\lambda $
for any
$ \lambda $
and discuss their asymptotic behavior as
$ \lambda\to-\infty $
. In the expansion of
$ u_\lambda $
large solutions naturally appear.
Citation: Luca Battaglia, Francesca Gladiali, Massimo Grossi. Asymptotic behavior of minimal solutions of $ -\Delta u = \lambda f(u) $ as $ \lambda\to-\infty $. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 681-700. doi: 10.3934/dcds.2020293
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

S. BerhanuF. Gladiali and G. Porru, Qualitative properties of solutions to elliptic singular problems, J. Inequal. Appl., 3 (1999), 313-330.  doi: 10.1155/S1025583499000223.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

M. G. CrandallP. 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.  Google Scholar

[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.  Google Scholar

[12]

S. DumontL. DupaigneO. 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.  Google Scholar

[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.  Google Scholar

[14]

P. EspositoM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

J. B. Keller, On solutions of $\Delta u = f(u)$, Comm. Pure Appl. Math., 10 (1957), 503-510.  doi: 10.1002/cpa.3160100402.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

S. BerhanuF. Gladiali and G. Porru, Qualitative properties of solutions to elliptic singular problems, J. Inequal. Appl., 3 (1999), 313-330.  doi: 10.1155/S1025583499000223.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

M. G. CrandallP. 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.  Google Scholar

[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.  Google Scholar

[12]

S. DumontL. DupaigneO. 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.  Google Scholar

[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.  Google Scholar

[14]

P. EspositoM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

J. B. Keller, On solutions of $\Delta u = f(u)$, Comm. Pure Appl. Math., 10 (1957), 503-510.  doi: 10.1002/cpa.3160100402.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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