We show how a change of variable and peak solution methods can be used to prove that a number of nonlinear elliptic partial differential equations have many solutions.
In questo lavoro, si mostra come i cambi di variabile unitamente ai metodi utilizzati per trovare soluzioni ad uno o più picchi, possono essere usati per provare che varie equazioni alle derivate parziali non lineari hanno molte soluzioni.
@article{BUMI_2007_8_10B_3_1013_0,
author = {Edward N. Dancer and Shusen Yan},
title = {Remarks on the Existence of Many Solutions of Certain Nonlinear Elliptic Equations},
journal = {Bollettino dell'Unione Matematica Italiana},
volume = {10-A},
year = {2007},
pages = {1013-1023},
zbl = {1182.35106},
mrnumber = {2507911},
language = {en},
url = {http://dml.mathdoc.fr/item/BUMI_2007_8_10B_3_1013_0}
}
Dancer, Edward N.; Yan, Shusen. Remarks on the Existence of Many Solutions of Certain Nonlinear Elliptic Equations. Bollettino dell'Unione Matematica Italiana, Tome 10-A (2007) pp. 1013-1023. http://gdmltest.u-ga.fr/item/BUMI_2007_8_10B_3_1013_0/
[1] , Infinite dimensional Morse theory and multiple solution problems, Birkhauser, Boston, 1993. | MR 1196690 | Zbl 0779.58005
[2] , On the uniqueness of the positive solution of a singularity perturbed problem, Rocky Mtn. J. Math., 25 (1995), 957-975. | MR 1357103 | Zbl 0846.35046
[3] , Stable and finite Morse index solutions on or on bounded domains with small diffusion, Trans. Amer Math., Soc. 357 (2005), 1225-1243. | MR 2110438 | Zbl 1145.35369
[4] , A note on asymptotic uniqueness for some nonlinearities which change sign, Bull Austral Math. Soc., 61 (2000), 305-312. | MR 1748710 | Zbl 0945.35031
[5] - , On the profile of solutions with two sharp layers to a singularly perturbed semilinear Dirichlet problem, Proc. Royal Soc. (Edinburgh) 127A (1997), 691-701. | MR 1465415 | Zbl 0882.35052
[6] - , On the location of spikes of solutions with two sharp layers for a singularly perturbed semilinear Dirichlet problem, J. Diff. Eqns., 157 (1999), 82-101. | MR 1710015 | Zbl 1087.35507
[7] - , On the superlinear Lazer-McKenna conjecture, J. Diff Eqns., 21 (2005), 317-351. | MR 2119987 | Zbl 1190.35082
[8] - , On the superlinear Lazer-McKenna conjecture Part II, Comm. Partial Diff. Eqns., 30 (2005), 1331-1358. | MR 2180307 | Zbl 1330.35147
[9] - , On the profile of the changing sign mountain pass solution for an elliptic problem, Trans. Amer. Math. Soc., 354 (2002), 3573-3600. | MR 1911512 | Zbl 1109.35041
[10] - , Global and local behaviour of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598. | MR 615628 | Zbl 0465.35003
[11] - , A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Diff. Eqns., 6 (1981), 883-901. | MR 619749 | Zbl 0462.35041
[12] - , Multiple interior peak solutions for some singularly perturbed Neumann problems, J. Diff. Eqns., 158 (1999), 1-27. | MR 1721719 | Zbl 1061.35502
[13] - , Uniqueness of the positive solutions of in an annulus, Diff. Integral Eqns., 4 (1991), 583-599. | MR 1097920 | Zbl 0724.34023
[14] - , Generic properties of nonlinear boundary value problems, Comm. Partial Diff. Eqns., 4 (1979), 293-319. | MR 522714 | Zbl 0462.35016
[15] , Variational methods, Springer, Berlin, 1996. | MR 1411681
[16] , On the construction of single peaked solutions to a singularly perturbed semilinear Dirichlet problem, J. Diff. Eqns., 129 (1996), 315-333. | MR 1404386 | Zbl 0865.35011
[17] , On the number of interior multipeak solutions for singularly perturbed Neumann problems, Topological Methods in Nonlinear Analysis, 12 (1999), 61-78. | MR 1677747