Asymptotic behavior of minimal solutions of <inline-formula><tex-math id="M1">\begin{document}$ -\Delta u = \lambda f(u) $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M2">\begin{document}$ \lambda\to-\infty $\end{document}</tex-math></inline-formula>

Luca Battaglia; Francesca Gladiali; Massimo Grossi

doi:10.3934/dcds.2020293

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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 $

Luca Battaglia ^1, , Francesca Gladiali ^2, and Massimo Grossi ^3,,

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 February 2021 Early access 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.

Keywords: Asymptotic analysis, minimal solutions, Dirichlet problem, large solutions, entire solutions.

Mathematics Subject Classification: Primary:35B40;Secondary:35B44, 35A01, 35A02.

Citation: Luca Battaglia, Francesca Gladiali, Massimo Grossi. Asymptotic behavior of minimal solutions of $ -\Delta u = \lambda f(u) $ as $ \lambda\to-\infty $. Discrete and Continuous Dynamical Systems, 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.
[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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American Institute of Mathematical Sciences

Asymptotic behavior of minimal solutions of $ -\Delta u = \lambda f(u) $ as $ \lambda\to-\infty $

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Asymptotic behavior of minimal solutions of $ -\Delta u = \lambda f(u) $ as $ \lambda\to-\infty $

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