| dc.contributor.supervisor |
Cowan, Craig (Mathematics) |
en_US |
| dc.contributor.author |
Zaherparandaz, Aidin
|
|
| dc.contributor.author |
Zaherparandaz, Aidin
|
|
| dc.date.accessioned |
2020-09-09T12:23:54Z |
|
| dc.date.available |
2020-09-09T12:23:54Z |
|
| dc.date.copyright |
2020-08-23 |
|
| dc.date.issued |
2020 |
en_US |
| dc.date.submitted |
2020-08-23T08:54:49Z |
en_US |
| dc.identifier.citation |
Aghajani, A., and Cowan, C. Some elliptic problems involving the gradient on general bounded and exterior domains. |
en_US |
| dc.identifier.citation |
Agmon, S., Douglis, A., and Nirenberg, L. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Commun. Pure Appl. Math. 12 (1959), 623–727. |
en_US |
| dc.identifier.citation |
Agmon, S., Douglis, A., and Nirenberg, L. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions ii. Communications on Pure and Applied Mathematics 17, 1 (1964), 35–92. |
en_US |
| dc.identifier.citation |
Ai, S., and Cowan, C. Perturbations of lane–emden and hamilton–jacobi equations ii: Exterior domains. Journal of Differential Equations 260, 11 (2016), 8025–8050. |
en_US |
| dc.identifier.citation |
Ai, S., and Cowan, C. Perturbations of lane–emden and hamilton–jacobi equations: The full space case. Nonlinear Analysis 151 (2017), 227–251. |
en_US |
| dc.identifier.citation |
Boggio, T. Sulle funzioni di Green d’ordine m: memorie. Tip. Matematica, 1905. |
en_US |
| dc.identifier.citation |
Bozhkov, Y. The group classification of lane–emden systems. Journal of Mathematical Analysis and Applications 426, 1 (2015), 89–104. |
en_US |
| dc.identifier.citation |
Brezis, H., and Nirenberg, L. Positive solutions of nonlinear elliptic equations involving critical sobolev exponents |
en_US |
| dc.identifier.citation |
Caffarelli, L. A., Gidas, B., and Spruck, J. Asymptotic symmetry and local behavior of semilinear elliptic equations with critical sobolev growth. |
en_US |
| dc.identifier.citation |
Cajori, F. The early history of partial differential equations and of partial differentiation and integration. The American Mathematical Monthly 35, 9 (1928), 459–467. |
en_US |
| dc.identifier.citation |
Chen, W., and Li, C. Classification of solutions of some nonlinear elliptic equations. Duke Mathematical Journal 63, 3 (1991), 615–622. |
en_US |
| dc.identifier.citation |
Chung, W.-J., and Nelson, L. Bose-einstein condensate & degenerate fermi cored dark matter halos. Journal of Cosmology and Astroparticle Physics 2018, 06 (Jun 2018), 010–010. |
en_US |
| dc.identifier.citation |
Clément, P., de Figueiredo, D. G., and Mitidieri, E. Positive splutions of semilinear elliptic systems. Communications in Partial Differential Equations 17, 5-6 (1992), 923–940. |
en_US |
| dc.identifier.citation |
Cowan, C. Uniqueness of solutions for elliptic systems and fourth order equations involving a parameter. |
en_US |
| dc.identifier.citation |
Cowan, C. Liouville theorems for stable lane–emden systems and biharmonic problems. Nonlinearity 26, 8 (2013), 2357–2371. |
en_US |
| dc.identifier.citation |
Cowan, C. Stability of entire solutions to supercritical elliptic problems involving advection. Nonlinear Analysis 104 (2014), 1–11. |
en_US |
| dc.identifier.citation |
Cowan, C. A short remark regarding pohozaev-type results on general domains assuming finite morse index. Proceedings of the Royal Society of Edinburgh: Section A Mathematics 147, 2 (2017), 293–297. |
en_US |
| dc.identifier.citation |
Cowan, C., and Fazly, M. On stable entire solutions of semi-linear elliptic equations with weights. Proceedings of the American Mathematical Society 140, 6 (2012), 2003–2012. |
en_US |
| dc.identifier.citation |
Dacorogna, B. Direct Methods in the Calculus of Variations, 2nd ed. ed. Applied Mathematical Sciences, 78. Springer New York, New York, NY, 2008. |
en_US |
| dc.identifier.citation |
Dalmasso, R. Problème de dirichlet homogène pour une équation biharmonique semi- linéaire dans une boule. (homogeneous dirichlet problem for a semilinear biharmonic equation in a ball). Bulletin des Sciences Mathématiques. Deuxième Série 114 (01 1990). |
en_US |
| dc.identifier.citation |
Dancer, E. N., and Wei, J. Sign-changing solutions for supercritical elliptic problems in domains with small holes. manuscripta mathematica 123, 4 (2007), 493–511. |
en_US |
| dc.identifier.citation |
del Pino, M., Dolbeault, J., and Musso, M. A phase plane analysis of the “multi-bubbling” phenomenon in some slightly supercritical equations. Monatshefte für Mathematik 142, 1 (2004), 57–79. |
en_US |
| dc.identifier.citation |
del Pino, M., Felmer, P., and Musso, M. Multi-peak solutions for supercritical elliptic problems in domains with small holes. Journal of Differential Equations 182, 2 (2002), 511–540. |
en_US |
| dc.identifier.citation |
del Pino, M., and Wei, J. Supercritical elliptic problems in domains with small holes. Annales de l’Institut Henri Poincare / Analyse non lineaire 24, 4 (2007), 507–520. |
en_US |
| dc.identifier.citation |
Douglas, R. G. Banach Algebra Techniques in Operator Theory, second edition. ed., vol. 179 of Graduate Texts in Mathematics. Springer New York, New York, NY, 1998. |
en_US |
| dc.identifier.citation |
Duong, A. T. A liouville type theorem for non-linear elliptic systems involving advection terms. Complex Variables and Elliptic Equations 63, 12 (2018), 1704– 1720. |
en_US |
| dc.identifier.citation |
Dávila, J., Dupaigne, L., Wang, K., and Wei, J. A monotonicity formula and a liouville-type theorem for a fourth order supercritical problem. Advances in Mathematics 258 (Jun 2014), 240–285. |
en_US |
| dc.identifier.citation |
Emden, R. J. Gaskugeln: Anwendungen der Mechanischen Warmetheorie auf Kosmologische und Meteorologische Probleme. Teubner, Berlin, Germany. |
en_US |
| dc.identifier.citation |
Evans, L., and of the Mathematical Sciences, C. B. Weak Convergence Methods for Nonlinear Partial Differential Equations. CBMS Regional Conference Series. Conference Board of the Mathematical Sciences, 1990. |
en_US |
| dc.identifier.citation |
Evans, L. C. Partial differential equations. Graduate studies in mathematics v. 19. American Mathematical Society, Providence, R.I, 1998. |
en_US |
| dc.identifier.citation |
Farina, A. On the classification of solutions of the lane-emden equation on unbounded domains of rn. Journal des Mathematiques Pures et Appliquees 87, 5 (2007), 537–561. |
en_US |
| dc.identifier.citation |
García-Huidobro, M., Manásevich, R., Mitidieri, E., and Yarur, C. S. Existence and nonexistence of positive singular solutions for a class of semilinear elliptic systems. Archive for Rational Mechanics and Analysis 140, 3 (1997), 253–284. |
en_US |
| dc.identifier.citation |
Gazzola, F., Grunau, H.-C., and Sweers, G. Polyharmonic Boundary Value Problems. Springer Berlin Heidelberg, Berlin/Heidelberg, 2010. |
en_US |
| dc.identifier.citation |
Gidas, B., ming Ni, W., and Nirenberg, L. Symmetry and related properties via the maximum principle, 1979. |
en_US |
| dc.identifier.citation |
Gidas, B., and Spruck, J. Global and local behavior of positive solutions of nonlinear elliptic equations. Communications on Pure and Applied Mathematics 34, 4 (1981), 525–598. |
en_US |
| dc.identifier.citation |
Gohberg, I., Goldberg, S., and Kaashoek, M. A. Classes of Linear Operators Vol. I. Birkhäuser Basel, 1990. |
en_US |
| dc.identifier.citation |
Grafakos, L. Modern Fourier Analysis, 3rd ed. 2014 ed., vol. 250 of Graduate Texts in Mathematics. Springer, New York, NY, 2014. |
en_US |
| dc.identifier.citation |
Guerra, I. A. Asymptotic behaviour of a semilinear elliptic system with a large exponent. Journal of Dynamics and Differential Equations 19, 1 (2007), 243–263. |
en_US |
| dc.identifier.citation |
Higson, N., and Roe, J. Analytic K-Homology. Oxford Mathematical Monographs. OUP Oxford, 2000. |
en_US |
| dc.identifier.citation |
LANE, J. H. Art. ix.–on the theoretical temperature of the sun; under the hypothesis of a gaseous mass maintaining its volume by its internal heat, and depending on the laws of gases as known to terrestrial experiment. American Journal of Science and Arts (1820-1879) 50, 148 (1870), 57–74. |
en_US |
| dc.identifier.citation |
Lui, S. H. S. H. Numerical analysis of partial differential equations. Pure and applied mathematics, a Wiley-Interscience series of texts, monographs, and tracts. Wiley, Hoboken, N.J. |
en_US |
| dc.identifier.citation |
Mazzeo, R., and Pacard, F. A construction of singular solutions for a semilinear elliptic equation using asymptotic analysis. |
en_US |
| dc.identifier.citation |
Melrose, R. Introduction to functional analysis, Spring 2009. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA. |
en_US |
| dc.identifier.citation |
Mitidieri, E. A rellich type identity and applications. Communications in Partial Differential Equations 18 (01 1993), 125–151. |
en_US |
| dc.identifier.citation |
Mitidieri, E. Nonexistence of positive solutions of semilinear elliptic systems in Rn. Differential Integral Equations 9, 3 (1996), 465–479. |
en_US |
| dc.identifier.citation |
Oswald, P. On a priori estimates for positive solutions of a semilinear biharmonic equation in a ball. Commentationes Mathematicae Universitatis Carolinae 026, 3 (1985), 565–577. |
en_US |
| dc.identifier.citation |
Passaseo, D. Nonexistence results for elliptic problems with supercritical nonlinearity in nontrivial domains. Journal of Functional Analysis 114, 1 (1993), 97–105. |
en_US |
| dc.identifier.citation |
Pokhozhaev, S. I. Eigenfunctions of the equation u + λf(u) = 0. Sov. Math., Dokl. 6 (1965), 1408–1411. |
en_US |
| dc.identifier.citation |
Salsa, S. Partial Differential Equations in Action From Modelling to Theory. Universitext. Springer Milan, Milan, 2008. |
en_US |
| dc.identifier.citation |
Soranzo, R. A priori estimates and existence of positive solutions of a superlinear polyharmonic equation. Dynamic Systems and Applications 3 (01 1994). |
en_US |
| dc.identifier.citation |
Struwe, M. Variational Methods. Springer Berlin Heidelberg, Berlin/Heidelberg, 1990. |
en_US |
| dc.identifier.citation |
Thomson, B. S. Elementary real analysis, 2nd edition. ed. Www.classicalrealanalysis.com, S.l, 2008. |
en_US |
| dc.identifier.citation |
van der Vorst, R. Variational identities and applications to differential systems. Archive for Rational Mechanics and Analysis 116, 4 (1992), 375–398. |
en_US |
| dc.identifier.citation |
VAN DER VORST, R. C. A. M. Best constant for the embedding of the space h2 ∩ h10 (ω) into l 2n n−4 (ω). Differ. Integral Equ. Appl. 6 (1993), 259–276. |
en_US |
| dc.identifier.citation |
von Wahl, W., and McLeod, J. Semilinear elliptic and parabolic equations of arbitrary order. Proceedings of the Royal Society of Edinburgh Section A: Mathematics 78, 3-4 (1978), 193–207. |
en_US |
| dc.identifier.uri |
http://hdl.handle.net/1993/35009 |
|
| dc.description.abstract |
n this thesis some Lane-Emden problems of different order are studied. Tackling the issue of existence of a positive solution and regularity of the solutions are of paramount importance for each instance. In addition to discussing the general Lane-Emden equation, the cases of having an advection term to the original problem and investigating some fourth order nonlinear Dirichlet and Navier problems are of considerable interest. While the well-studied general equation points out that for p≥N+2/N−2 and Ω a star-shaped domain in RN there would be no non-trivial solution, some advantageous results regarding the existence of a positive solution and regularity of the solutions on a general bounded domain inRNare addressed for the equations where an advection is involved, as well as some nonlinear fourth order problem with given Dirichlet and Navier boundary conditions. |
en_US |
| dc.rights |
info:eu-repo/semantics/openAccess |
|
| dc.subject |
Elliptic partial differential equations |
en_US |
| dc.title |
On Lane-Emden equation and some variations |
en_US |
| dc.type |
info:eu-repo/semantics/masterThesis |
|
| dc.degree.discipline |
Mathematics |
en_US |
| dc.contributor.examiningcommittee |
Lui, Shaun (Mathematics) |
en_US |
| dc.contributor.examiningcommittee |
Slevinsky, Richard (Mathematics) |
en_US |
| dc.degree.level |
Master of Science (M.Sc.) |
en_US |
| dc.description.note |
October 2020 |
en_US |
