Discontinuous elliptic problems in ℝ N without monotonicity assumptions

Silvia Cingolani; Monica Lazzo

Commentationes Mathematicae Universitatis Carolinae (2001)

  • Volume: 42, Issue: 3, page 451-458
  • ISSN: 0010-2628

Abstract

top
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.

How to cite

top

Cingolani, Silvia, and Lazzo, Monica. "Discontinuous elliptic problems in $\mathbb {R}^N$ without monotonicity assumptions." Commentationes Mathematicae Universitatis Carolinae 42.3 (2001): 451-458. <http://eudml.org/doc/248767>.

@article{Cingolani2001,
abstract = {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.},
author = {Cingolani, Silvia, Lazzo, Monica},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {free boundary problem; plasma physics; free boundary problem; plasma physics},
language = {eng},
number = {3},
pages = {451-458},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Discontinuous elliptic problems in $\mathbb \{R\}^N$ without monotonicity assumptions},
url = {http://eudml.org/doc/248767},
volume = {42},
year = {2001},
}

TY - JOUR
AU - Cingolani, Silvia
AU - Lazzo, Monica
TI - Discontinuous elliptic problems in $\mathbb {R}^N$ without monotonicity assumptions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2001
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 42
IS - 3
SP - 451
EP - 458
AB - 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.
LA - eng
KW - free boundary problem; plasma physics; free boundary problem; plasma physics
UR - http://eudml.org/doc/248767
ER -

References

top
  1. Allegretto W., Nistri P., Elliptic equations with discontinuous nonlinearities, T.M.N.A. 2 (1993), 233-251. (1993) Zbl0827.35134MR1251937
  2. Ambrosetti A., Badiale M., The dual variational principle and elliptic problems with discontinuous nonlinearities, J. Math. Anal. Appl. 140.2 (1989), 363-373. (1989) Zbl0687.35033MR1001862
  3. Ambrosetti A., Calahorrano M., Dobarro F., Remarks on the Grad-Shafranov equation, Appl. Math. Letters 3 (1990), 9-11. (1990) Zbl0772.35085MR1077864
  4. Ambrosetti A., Calahorrano M., Dobarro F., Global branching for discontinuous problems, Comment. Math. Univ. Carolinae 31 (1990), 213-222. (1990) Zbl0732.35101MR1077891
  5. Ambrosetti A., Struwe M., Existence of steady vortex rings in an ideal fluid, Arch. Rational Mech. Anal. 108 (1989), 97-109. (1989) Zbl0709.35081MR1011553
  6. Ambrosetti A., Turner R.E.L., Some discontinuous variational problems, Differential Integral Equations 1 (1988), 341-349. (1988) Zbl0728.35037MR0929921
  7. Amick C.J., Turner R.E.L., A global branch of steady vortex rings, J. Rein. Angew. Math. 384 (1988), 1-23. (1988) Zbl0628.76032MR0929976
  8. Arcoya D., Positive solutions for semilinear Dirichlet problems in an annulus, J. Differential Equations 94 (1991), 217-227. (1991) Zbl0768.35029MR1137613
  9. Arcoya D., Calahorrano M., Some discontinuous problems with a quasilinear operator, J. Math. Anal. Appl. 187 (1994), 1059-1072. (1994) Zbl0815.35018MR1298837
  10. Arcoya D., Zertiti A., Global branching for discontinuous problems in the exterior domain of a ball, Boll. U.M.I. (7) 10A (1996), 641-652. (1996) Zbl0881.35041MR1420946
  11. Badiale M., Semilinear elliptic problems in ℝ N with discontinuous nonlinearities, Atti Sem. Fis. Univ. Modena 153 (1995), 293-305. (1995) MR1366063
  12. Badiale M., Tarantello G., Existence and multiplicity results for elliptic problems with critical growth and discontinuous nonlinearities, Nonlinear Anal. T.M.A. 29 (1997), 639-677. (1997) Zbl0877.35046MR1452751
  13. Berestycki H., Lions P.L., Nonlinear scalar field equations, I - Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), 313-375. (1983) Zbl0533.35029MR0695535
  14. Carl S., Heikkilä S., Elliptic equations with discontinuous nonlinearities in ℝ N , Nonlinear Anal. T.M.A. 31 (1998), 217-227. (1998) MR1487542
  15. Cerami G., 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) Zbl0515.35025MR0698497
  16. Chang K.C., Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), 102-129. (1981) MR0614246
  17. Cimatti G., A nonlinear elliptic eigenvalue problem for the Elenbaas equation, Boll. U.M.I. (5) 16B (1979), 555-565. (1979) Zbl0412.35090MR0546475
  18. Clarke F.H., Periodic solutions of Hamiltonian inclusions, J. Differential Equations 40 (1981), 1-6; C.R.A.S. 128 (1978), 951-952. (1981) MR0614215
  19. Flucher M., Müller S., Radial symmetry and decay rate of variational ground states in the zero mass case, SIAM J. Math. Anal. 29 (1998), 712-719. (1998) MR1617704
  20. Fraenkel L.E., Berger M.S., A global theory of steady vortex rings in an ideal fluid, Acta Math. 132 (1974), 13-51. (1974) Zbl0282.76014MR0422916
  21. Gilbarg D., Trudinger N.S., Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1983. Zbl1042.35002MR0737190
  22. Heikkilä S., First order discontinuous implicit differential equations with discontinuous boundary conditions, Nonlinear Anal. T.M.A. 30 (1997), 1753-1761. (1997) MR1490090
  23. Heikkilä S., Lakshmikantham V., On the method of upper and lower solutions for discontinuous nonlinearities, Nonlinear Anal. T.M.A. 23 (1994), 265-273. (1994) MR1289132
  24. Littman W., Generalized subharmonic functions: monotonic approximations and an improved maximum principle, Ann. Scuola Norm. Sup. Pisa 17 (1963), 207-222. (1963) Zbl0123.29104MR0177186
  25. Marano S., Existence theorems for a semilinear elliptic boundary value problem, Ann. Polonici Math. LX (1994), 57-67. (1994) Zbl0826.35146MR1295108
  26. Marano S., Elliptic boundary-value problems with discontinuous nonlinearities, Set-Valued Anal. 3 (1995), 167-180. (1995) Zbl0843.35034MR1343482
  27. Massabò I., Positive eigenvalues for elliptic equations with discontinuous nonlinearities, Boll. U.M.I. (5) 15B (1978), 814-827. (1978) MR0524102
  28. Strauss W.A., Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149-162. (1977) Zbl0356.35028MR0454365
  29. Stuart C.A., Differential equations with discontinuous nonlinearities, Arch. Rational Mech. Anal. 63 (1976), 59-75. (1976) MR0481227
  30. Stuart C.A., Maximal and minimal solutions of elliptic differential equations with discontinuous nonlinearities, Math. Z. 163 (1978), 239-249. (1978) MR0513729
  31. Stuart C.A., Toland J.F., A variational approach for boundary value problems with discontinuous nonlinearities, J. London Math. Soc. 21 (1980), 319-328. (1980) MR0575391

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.