We prove existence of a positive, radial solution for a semilinear elliptic problem with a discontinuous nonlinearity. We use an approximating argument which requires no monotonicity assumptions on the nonlinearity.
@article{119259,
author = {Silvia Cingolani and Monica Lazzo},
title = {Discontinuous elliptic problems in $\Bbb R^N$ without monotonicity assumptions},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {42},
year = {2001},
pages = {451-458},
zbl = {1115.35043},
mrnumber = {1859592},
language = {en},
url = {http://dml.mathdoc.fr/item/119259}
}
Cingolani, Silvia; Lazzo, Monica. Discontinuous elliptic problems in $\Bbb R^N$ without monotonicity assumptions. Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) pp. 451-458. http://gdmltest.u-ga.fr/item/119259/
Elliptic equations with discontinuous nonlinearities, T.M.N.A. 2 (1993), 233-251. (1993) | MR 1251937 | Zbl 0827.35134
The dual variational principle and elliptic problems with discontinuous nonlinearities, J. Math. Anal. Appl. 140.2 (1989), 363-373. (1989) | MR 1001862 | Zbl 0687.35033
Remarks on the Grad-Shafranov equation, Appl. Math. Letters 3 (1990), 9-11. (1990) | MR 1077864 | Zbl 0772.35085
Global branching for discontinuous problems, Comment. Math. Univ. Carolinae 31 (1990), 213-222. (1990) | MR 1077891 | Zbl 0732.35101
Existence of steady vortex rings in an ideal fluid, Arch. Rational Mech. Anal. 108 (1989), 97-109. (1989) | MR 1011553 | Zbl 0709.35081
Some discontinuous variational problems, Differential Integral Equations 1 (1988), 341-349. (1988) | MR 0929921 | Zbl 0728.35037
A global branch of steady vortex rings, J. Rein. Angew. Math. 384 (1988), 1-23. (1988) | MR 0929976 | Zbl 0628.76032
Positive solutions for semilinear Dirichlet problems in an annulus, J. Differential Equations 94 (1991), 217-227. (1991) | MR 1137613 | Zbl 0768.35029
Some discontinuous problems with a quasilinear operator, J. Math. Anal. Appl. 187 (1994), 1059-1072. (1994) | MR 1298837 | Zbl 0815.35018
Global branching for discontinuous problems in the exterior domain of a ball, Boll. U.M.I. (7) 10A (1996), 641-652. (1996) | MR 1420946 | Zbl 0881.35041
Semilinear elliptic problems in $\Bbb R^N$ with discontinuous nonlinearities, Atti Sem. Fis. Univ. Modena 153 (1995), 293-305. (1995) | MR 1366063
Existence and multiplicity results for elliptic problems with critical growth and discontinuous nonlinearities, Nonlinear Anal. T.M.A. 29 (1997), 639-677. (1997) | MR 1452751 | Zbl 0877.35046
Nonlinear scalar field equations, I - Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), 313-375. (1983) | MR 0695535 | Zbl 0533.35029
Elliptic equations with discontinuous nonlinearities in $\Bbb R^N$, Nonlinear Anal. T.M.A. 31 (1998), 217-227. (1998) | MR 1487542
Soluzioni positive di problemi con parte non lineare discontinua e applicazione a un problema di frontiera libera, Boll. U.M.I. (6) 2B (1983), 321-338. (1983) | MR 0698497 | Zbl 0515.35025
Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), 102-129. (1981) | MR 0614246
A nonlinear elliptic eigenvalue problem for the Elenbaas equation, Boll. U.M.I. (5) 16B (1979), 555-565. (1979) | MR 0546475 | Zbl 0412.35090
Periodic solutions of Hamiltonian inclusions, J. Differential Equations 40 (1981), 1-6; C.R.A.S. 128 (1978), 951-952. (1981) | MR 0614215
Radial symmetry and decay rate of variational ground states in the zero mass case, SIAM J. Math. Anal. 29 (1998), 712-719. (1998) | MR 1617704
A global theory of steady vortex rings in an ideal fluid, Acta Math. 132 (1974), 13-51. (1974) | MR 0422916 | Zbl 0282.76014
Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1983. | MR 0737190 | Zbl 1042.35002
First order discontinuous implicit differential equations with discontinuous boundary conditions, Nonlinear Anal. T.M.A. 30 (1997), 1753-1761. (1997) | MR 1490090
On the method of upper and lower solutions for discontinuous nonlinearities, Nonlinear Anal. T.M.A. 23 (1994), 265-273. (1994) | MR 1289132
Generalized subharmonic functions: monotonic approximations and an improved maximum principle, Ann. Scuola Norm. Sup. Pisa 17 (1963), 207-222. (1963) | MR 0177186 | Zbl 0123.29104
Existence theorems for a semilinear elliptic boundary value problem, Ann. Polonici Math. LX (1994), 57-67. (1994) | MR 1295108 | Zbl 0826.35146
Elliptic boundary-value problems with discontinuous nonlinearities, Set-Valued Anal. 3 (1995), 167-180. (1995) | MR 1343482 | Zbl 0843.35034
Positive eigenvalues for elliptic equations with discontinuous nonlinearities, Boll. U.M.I. (5) 15B (1978), 814-827. (1978) | MR 0524102
Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149-162. (1977) | MR 0454365 | Zbl 0356.35028
Differential equations with discontinuous nonlinearities, Arch. Rational Mech. Anal. 63 (1976), 59-75. (1976) | MR 0481227
Maximal and minimal solutions of elliptic differential equations with discontinuous nonlinearities, Math. Z. 163 (1978), 239-249. (1978) | MR 0513729
A variational approach for boundary value problems with discontinuous nonlinearities, J. London Math. Soc. 21 (1980), 319-328. (1980) | MR 0575391