Free boundary problems arising in the theory of maximal solutions of equations with exponential nonlinearities
Séminaire Laurent Schwartz — EDP et applications (2017-2018), Talk no. 10, 12 p.

We consider equations of the form Δu+λ 2 V(x)e u =ρ in various two dimensional settings. We assume that V>0 is a given function, λ>0 is a small parameter and ρ=𝒪(1) or ρ+ as λ0. In a recent paper [27] we proved the existence of the maximal solutions for a particular choice V1, ρ=0 when the problem is posed in doubly connected domains under Dirichlet boundary conditions. We related the maximal solutions with a novel free boundary problem. The purpose of this note is to derive the corresponding free boundary problems in other settings. Solvability of such problems is, viewed formally, the necessary condition for the existence of the maximal solution.

Published online:
DOI: 10.5802/slsedp.122
Kowalczyk, Michał 1; Pistoia, Angela 2; Rybka, Piotr 3; Vaira, Giusi 4

1 Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS) Universidad de Chile Casilla 170 Correo 3 Santiago Chile
2 Dipartimento SBAI Sapienza Universitá di Roma via Antonio Scarpa 16 00161 Roma Italy
3 Institute of Applied Mathematics and Mechanics The University of Warsaw Banacha 2 02-097 Warsaw Poland
4 Dipartimento di Matematica e Fisica Universitá della Campania “L. Vanvitelli” viale Lincoln 5 81100 Caserta Italy
@article{SLSEDP_2017-2018____A10_0,
     author = {Kowalczyk, Micha{\l} and Pistoia, Angela and Rybka, Piotr and Vaira, Giusi},
     title = {Free boundary problems arising in the theory of maximal solutions of equations with exponential nonlinearities},
     journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
     note = {talk:10},
     publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2017-2018},
     doi = {10.5802/slsedp.122},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/slsedp.122/}
}
TY  - JOUR
AU  - Kowalczyk, Michał
AU  - Pistoia, Angela
AU  - Rybka, Piotr
AU  - Vaira, Giusi
TI  - Free boundary problems arising in the theory of maximal solutions of equations with exponential nonlinearities
JO  - Séminaire Laurent Schwartz — EDP et applications
N1  - talk:10
PY  - 2017-2018
DA  - 2017-2018///
PB  - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
UR  - http://archive.numdam.org/articles/10.5802/slsedp.122/
UR  - https://doi.org/10.5802/slsedp.122
DO  - 10.5802/slsedp.122
LA  - en
ID  - SLSEDP_2017-2018____A10_0
ER  - 
%0 Journal Article
%A Kowalczyk, Michał
%A Pistoia, Angela
%A Rybka, Piotr
%A Vaira, Giusi
%T Free boundary problems arising in the theory of maximal solutions of equations with exponential nonlinearities
%J Séminaire Laurent Schwartz — EDP et applications
%Z talk:10
%D 2017-2018
%I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
%U https://doi.org/10.5802/slsedp.122
%R 10.5802/slsedp.122
%G en
%F SLSEDP_2017-2018____A10_0
Kowalczyk, Michał; Pistoia, Angela; Rybka, Piotr; Vaira, Giusi. Free boundary problems arising in the theory of maximal solutions of equations with exponential nonlinearities. Séminaire Laurent Schwartz — EDP et applications (2017-2018), Talk no. 10, 12 p. doi : 10.5802/slsedp.122. http://archive.numdam.org/articles/10.5802/slsedp.122/

[1] Baraket, S., Pacard, F.: Construction of singular limits for a semilinear elliptic equation in dimension 2, Calculus of Variations and Partial Differential Equations 6 (1997), no. 1, 1–38. | DOI | MR | Zbl

[2] Bonheure, D., Grossi M., Noris, B., Terracini, S., Multi-layer radial solutions for a supercritical Neumann problem, Journal of Differential Equations 261 (2016), n. 1, 455–504. | DOI | MR | Zbl

[3] Bonheure, D., Casteras, J.-B., Noris, B., Multiple positive solutions of the stationary Keller–Segel system, Calculus of Variations and Partial Differential Equations 56 (2017), no. 3, 74–109 | DOI | MR | Zbl

[4] Bonheure, D., Casteras, J.-B., Noris, B., Layered solutions with unbounded mass for the Keller-Segel equation J. Fixed Point Theory Appl. 19 (2017) no. 1, 529–558. | DOI | MR | Zbl

[5] Bonheure, D., Casteras, J.-B., Román, C., Unbounded mass radial solutions for the Keller-Segel equation in the disk, | arXiv

[6] Brezis, H., Merle, F.: Uniform estimates and blow–up behavior for solutions of -Δu=V(x)e u in two dimensions, Communications in Partial Differential Equations 16 (1991), no. 8-9, 1223–1253. | DOI | Zbl

[7] Caglioti, E., Lions, P.-L., Marchioro, C., Pulvirenti, M.: A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description, Communications in Mathematical Physics 143 (1992), no. 3, 501–525. | DOI | MR | Zbl

[8] —, A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description. part II, Communications in Mathematical Physics 174 (1995), no. 2, 229–260. | DOI | Zbl

[9] Chang, S.-A., Yang, P.-C.: Prescribing gaussian curvature on 𝕊 2 , Acta Math. 159 (1987), 215–259. | DOI | Zbl

[10] Chang, S.-A.: Non-linear elliptic equations in conformal geometry, Zurich lectures in advanced mathematics, European Mathematical Society, 2004. | DOI

[11] Chen, C.-C., Lin, C.-S.: Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces, Communications on Pure and Applied Mathematics 55 (2002), no. 6, 728–771. | DOI | MR | Zbl

[12] —, Topological degree for a mean field equation on Riemann surfaces, Communications on Pure and Applied Mathematics 56 (2003), no. 12, 1667–1727. | DOI | MR | Zbl

[13] De Marchis, F.: Generic multiplicity for a scalar field equation on compact surfaces, J. Funct. Anal. 259 (2010), no. 8, 2165–2192 | DOI | MR | Zbl

[14] del Pino, M., Kowalczyk, M., Musso, M.: Singular limits in Liouville-type equations, Calculus of Variations and Partial Differential Equations 24 (2005), no. 1, 47–81. | DOI | MR | Zbl

[15] del Pino, M., Pistoia, A., Vaira, G.: Large mass boundary condensation patterns in the stationary Keller–Segel system, Journal of Differential Equations 261 (2016), no. 6, 3414 – 3462. | DOI | MR | Zbl

[16] del Pino, M., Wei, J.: Collapsing steady states of the Keller–Segel system, Nonlinearity 19 (2006), 601–684. | DOI | MR | Zbl

[17] Ding, W., Jost, J., Li, J., Wang, G.: Existence results for mean field equations, Annales de l’I.H.P. Analyse non linéaire 16 (1999), no. 5, 653 – 666. | DOI | Numdam | MR | Zbl

[18] Djadli, Z.: Existence result for the mean field problem on Riemann surfaces of all genuses, Commun. Contemp. Math. 10 (2008), no. 2, 205–220. | DOI | MR | Zbl

[19] Djadli, Z., Malchiodi, A.: Existence of conformal metrics with constant Q-curvature Ann. of Math. (2) 168 (2008), no. 3, 813–858. | DOI | MR | Zbl

[20] Esposito, P., Figueroa, P.,: Singular mean field equations on compact Riemann surfaces, Nonlinear Anal. 111 (2014), 33–65. | DOI | MR | Zbl

[21] Grossi, M., Pistoia, A., Esposito, P.: On the existence of blowing-up solutions for a mean field equation, Annales de l’I.H.P. Analyse non linéaire 22 (2005), no. 2, 227–257 (eng). | DOI | Numdam | MR | Zbl

[22] Gladiali, F., Grossi, M.: Singular limits of radial solutions in an annulus, Asymptot. Anal. 55 (2007), 73–83. | Zbl

[23] Hong, J., Kim, Y., Pac, P.-Y.: Multivortex solutions of the abelian Chern-Simons-Higgs theory, Phys. Rev. Lett. 64 (1990), 2230–2233. | DOI | MR | Zbl

[24] Jackiw, R., Weinberg, E.-J.: Self-dual Chern-Simons vortices, Phys. Rev. Lett. 64 (1990), 2234–2237. | DOI | MR | Zbl

[25] Kiessling, M.: Statistical mechanics of classical particles with logarithmic interactions, Communications on Pure and Applied Mathematics 46, no. 1, 27–56. | DOI | MR | Zbl

[26] Kazdan, J.-L., Warner, F.-W.: Curvature Functions for Compact 2-Manifolds, Annals of Mathematics 99 (1974), no. 1, 14–47. | DOI | MR | Zbl

[27] Kowalczyk, M., Pistoia, A., Vaira, G.,: Maximal solutions of the Liouville equation in doubly connected domains, submitted. | DOI | MR | Zbl

[28] Li, Y-Y.: Harnack type inequality: the method of moving planes, Communications in Mathematical Physics 200 (1999), no. 2, 421–444. | DOI | MR | Zbl

[29] Li, Y.-Y., Shafrir, I.: Blow-up analysis for solutions of -Δu=Ve u in dimension two, Indiana Univ. Math. J. 43 (1994), no. 4, 1255–1270. | DOI | Zbl

[30] López-Soriano, R., Malchiodi, A., Ruiz, D.: Conformal metrics with prescribed Gaussion and geodesic curvatures, preprint 2018.

[31] Malchiodi, A.: Morse theory and a scalar field equation on compact surfaces, Adv. Differential Equations, 13 (2008) no. 11-12, 1109–1129 | Zbl

[32] Nagasaki, K., Suzuki, T.: Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities, Asymptotic Anal. 3 (1990) no. 2, 173–188. . | DOI | MR | Zbl

[33] —, Radial and nonradial solutions for the nonlinear eigenvalue problem Δu+λe u =0 on annuli in 2 , Journal of Differential Equations 87 (1990), no. 1, 144 – 168. | DOI

[34] Pistoia, A., Vaira, G.: Steady states with unbounded mass of the Keller-Segel system. Proc. Roy. Soc. Edinburgh Sect. A 145 (2015), no. 1, 203–222. | DOI | MR | Zbl

[35] Krantz, S., Geometric Function Theory, Birkhäuser Boston Basel Berlin, 2005.

[36] Tarantello, G., Struwe, M.: On multivortex solutions in Chern-Simons gauge theory, Bollettino dell’Unione Matematica Italiana 1-B (1998), no. 1, 109–121 (eng). | Zbl

Cited by Sources: