Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities

Journal of Differential Equations

Volumn 244, Issue 12, 2008, Pages 3031-3059

  Bonanno, Gabriele ^a   Candito, Pasquale ^b  

^a UNIVERSITY OF MESSINA   (Italy)

^b UNIVERSITY MEDITERRANEA OF REGGIO CALABRIA   (Italy)

Author keywords

Critical points; Dirichlet problem with discontinuous nonlinearities; Multiple solutions; Variational hemivariational inequalities

Indexed keywords

EID: 43049140189     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2008.02.025     Document Type: Article

References (42)

1. Ambrosetti A., and Rabinowitz P.H. Dual variational methods in critical point theory and applications. J. Funct. Anal. 14 (1973) 349-381
2. Averna D., and Bonanno G. A three critical points theorem and its applications to the ordinary Dirichlet problem. Topol. Methods Nonlinear Anal. 22 (2003) 93-103
3. D. Averna, G. Bonanno, A mountain pass theorem for a suitable class of functions, Rocky Mountain J. Math., in press
4. Averna D., and Salvati R. Three solutions for a mixed boundary value problem involving the one-dimensional p-Laplacian. J. Math. Anal. Appl. 298 (2004) 245-260
5. Bonanno G. A minimax inequality and its applications to ordinary differential equations. J. Math. Anal. Appl. 270 (2002) 210-229
6. Bonanno G. Some remarks on a three critical points theorem. Nonlinear Anal. 54 (2003) 651-665
7. Bonanno G. A critical points theorem and nonlinear differential problems. J. Global Optim. 28 (2004) 249-258
8. Bonanno G. Multiple critical points theorems without the Palais-Smale condition. J. Math. Anal. Appl. 299 (2004) 600-614
9. Bonanno G., and Candito P. Three solutions to a Neumann problem for elliptic equations involving the p-Laplacian. Arch. Math. (Basel) 80 (2003) 424-429
10. Bonanno G., and Candito P. On a class of nonlinear variational-hemivariational inequalities. Appl. Anal. 83 (2004) 1229-1244
11. Bonanno G., and Giovannelli N. An eigenvalue Dirichlet problem involving the p-Laplacian with discontinuous nonlinearities. J. Math. Anal. Appl. 308 (2005) 596-604
12. Bonanno G., and Livrea R. Periodic solutions for a class of second order Hamiltonian systems. Electron. J. Differential Equations 2005 (2005) 1-13
13. Bonanno G., and O'Regan D. A boundary value problem on the half-line via critical point methods. Dynam. Systems Appl. 15 (2006) 395-408
14. Bonanno G., and Marano S.A. Positive solutions of elliptic equations with discontinuous nonlinearities. Topol. Methods Nonlinear Anal. 8 (1996) 263-273
15. Brézis H. Analyse Fonctionelle - Théorie et Applications (1983), Masson, Paris
16. Chang K.C. Variational methods for non-differentiable functionals and their applications to partial differential equations. J. Math. Anal. Appl. 80 (1981) 102-129
17. Clarke F.H. Optimization and Nonsmooth Analysis. Classics Appl. Math. vol. 5 (1990), SIAM, Philadelphia
18. De Giorgi E., Buttazzo G., and Dal Maso G. On the lower semicontinuity of certain integral functionals. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (8) Mat. Appl. 74 (1983) 274-282
19. García-Melían J., and Sabina de Lis J. Maximum and comparison principles for operators involving the p-Laplacian. J. Math. Anal. Appl. 218 (1998) 49-65
20. Gasiński L., and Papageorgiou N.S. Nonlinear hemivariational inequalities at resonance. J. Math. Anal. Appl. 244 (2000) 200-213
21. Gilbarg D., and Trudinger N.S. Elliptic Partial Differential Equations of Second Order. second ed. (1983), Springer-Verlag, Berlin
22. Hu S., Kourogenis N.C., and Papageorgiou N.S. Nonlinear elliptic eigenvalue problems with discontinuities. J. Math. Anal. Appl. 233 (1999) 406-424
23. N. Math. Nachr. 278 (2005) 1756-1765
24. Kristály A., Lisei H., and Varga C. Multiple solutions for p-laplacian type equations. Nonlinear Anal. 68 (2008) 1375-1381
25. Kufner A., John O., and Fučik S. Function Spaces (1977), Kluwer, Leiden
26. Li S., and Liu Z. Perturbations from symmetric elliptic boundary value problems. J. Differential Equations 185 (2002) 271-280
27. Li S., Wu S., and Zhou H.-S. Solutions to semilinear elliptic problems with combined nonlinearities. J. Differential Equations 185 (2002) 200-224
28. Marano S.A. Elliptic boundary value problems with discontinuous nonlinearities. Set-Valued Anal. 3 (1995) 167-180
29. Marano S.A., and Motreanu D. On a three critical points theorem for non-differentiable functions and applications to nonlinear boundary value problems. Nonlinear Anal. 48 (2002) 37-52
30. Marano S.A., and Motreanu D. Infinitely many critical points of non-differentiable functions and applications to a Neumann type problem involving the p-Laplacian. J. Differential Equations 182 (2002) 108-120
31. Motreanu D., and Panagiotopoulos P.D. Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities. Nonconvex Optim. Appl. vol. 29 (1998), Kluwer, Dordrecht
32. Motreanu D., and Rǎdulescu V. Variational and Non-variational Methods in Nonlinear Analysis and Boundary Value Problems. Nonconvex Optim. Appl. (2003), Kluwer
33. Panagiotopoulos P.D. Hemivariational Inequalities. Applications in Mechanics and Engineering (1993), Springer-Verlag, Berlin
34. Rabinowitz P.H. Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Reg. Conf. Ser. Math. vol. 65 (1985), Amer. Math. Soc., Providence, RI
35. Ricceri B. A general variational principle and some of its applications. J. Comput. Appl. Math. 113 (2000) 401-410
36. Ricceri B. On a three critical points theorem. Arch. Math. (Basel) 75 (2000) 220-226
37. Salvati R. Multiple solutions for a mixed boundary value problem. Math. Sci. Res. J. 7 7 (2003) 275-983
38. Struwe M. Variational Methods (1996), Springer-Verlag, Berlin
39. Szulkin A. Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems. Ann. Inst. H. Poincaré 3 (1986) 77-109
40. Talenti G. Some inequalities of Sobolev type on two-dimensional spheres. In: Walter W. (Ed). General Inequalities, vol. 5. Internat. Ser. Numer. Math. vol. 80 (1987), Birkhäuser (Basel) 401-408
41. Y. Tian, W. Ge, Second-order Sturm-Liouville boundary value problem involving the one-dimensional p-Laplacian, Rocky Mountain J. Math., in press
42. Tian Y., and Ge W. Multiple solutions for second-order Sturm-Liouville boundary value problem. Taiwanese J. Math. 11 (2007) 975-988