Elliptic eigenvalue problems

with highly discontinuous nonlinearities

Author:
Salvatore A. Marano

Journal:
Proc. Amer. Math. Soc. **125** (1997), 2953-2961

MSC (1991):
Primary 35J65, 35B30, 35R70

DOI:
https://doi.org/10.1090/S0002-9939-97-03969-5

MathSciNet review:
1402873

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a family of elliptic eigenvalue problems with highly discontinuous nonlinearities, the existence of unbounded continua of positive solutions containing (0,0) is established by using techniques and results from set-valued analysis. Some special cases are then presented and discussed.

**[1]**R.A. Adams:*Sobolev Spaces*, Academic Press, New York, 1975. MR**56:9247****[2]**H. Amann,*Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces*, SIAM Rev.**18**(1976), 620-709. MR**54:3519****[3]**A. Ambrosetti, M. Calahorrano, and F. Dobarro,*Global branching for discontinuous problems*, Comment. Math. Univ. Carolin.**31**(1990), 213-222. MR**91m:35229****[4]**J. Appell, E. De Pascale, H.T. Nguyêñ, and P.P. Zabreiko,*Multi-valued superpositions*, Dissertationes Math.**345**(1995), 1-97. MR**96h:47061****[5]**M. Badiale and G. Tarantello,*Existence and multiplicity results for elliptic problems with critical growth and discontinuous nonlinearities*, Nonlinear Anal., to appear.**[6]**G. Bonanno and S.A. Marano,*Positive solutions of elliptic equations with discontinuous nonlinearities*, Topol. Methods Nonlinear Anal.**8**(1996), to appear.**[7]**K.C. Chang,*Free boundary problems and the set-valued mappings*, J. Differential Equations**49**(1983), 1-28. MR**84h:35172****[8]**F. Chiarenza, M. Frasca, and P. Longo,*-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients*, Trans. Amer. Math. Soc.**336**(1993), 841-853. MR**93f:35232****[9]**E. De Giorgi, G. Buttazzo, and G. Dal Maso,*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. MR**87a:49019****[10]**D. Gilbarg and N.S. Trudinger,*Elliptic Partial Differential Equations of Second Order*, 2nd edn., Springer-Verlag, Berlin, 1983. MR**86c:35035****[11]**S. Heikkilä and V. Lakshmikantham,*Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations*, Marcel Dekker, New York, 1994. MR**95d:34002****[12]**C.J. Himmelberg,*Measurable relations*, Fund. Math.**87**(1975), 53-72. MR**51:3384****[13]**H.J. Kuiper,*On positive solutions of nonlinear elliptic eigenvalue problems*, Rend. Circ. Mat. Palermo (2)**20**(1971), 113-138. MR**48:6680****[14]**D. Lupo,*A bifurcation result for a Dirichlet problem with discontinuous nonlinearity*, Rend. Circ. Mat. Palermo (2)**38**(1989), 305-318. MR**91a:35026****[15]**S.A. Marano,*Elliptic boundary-value problems with discontinuous nonlinearities*, Set-Valued Anal.**3**(1995), 167-180. MR**96j:35079****[16]**S.A. Marano,*Implicit elliptic boundary-value problems with discontinuous nonlinearities*, Set-Valued Anal.**4**(1996), 287-300. CMP**97:02****[17]**I. Massabò and C.A. Stuart,*Elliptic eigenvalue problems with discontinuous nonlinearities*, J. Math. Anal. Appl.**66**(1978), 261-281. MR**80f:35102****[18]**I. Massabò,*Positive eigenvalues for elliptic equations with discontinuous nonlinearities*, Boll. Un. Mat. Ital. B (5)**15**(1978), 814-827. MR**80a:35051****[19]**D.K. Palagachev,*Quasilinear elliptic equations with VMO coefficients*, Trans. Amer. Math. Soc.**347**(1995), 2481-2493. MR**95k:35083****[20]**D.K. Palagachev, private communication.**[21]**S.I. Pohozaev,*Eigenfunctions of the equation*, Soviet Math. Dokl.**6**(1965), 1408-1411.**[22]**T. Pruszko,*Topological degree methods in multi-valued boundary value problems*, Nonlinear Anal.**5**(1981), 959-973. MR**83d:34034****[23]**P.H. Rabinowitz,*Some global results for nonlinear eigenvalue problems*, J. Funct. Anal.**7**(1971), 487-513. MR**46:745****[24]**P.H. Rabinowitz,*A global theorem for nonlinear eigenvalue problems and applications*, in E. Zarantonello (ed.),*Contributions to Nonlinear Functional Analysis*, Academic Press, New York, 1971, pp. 11-36. MR**52:11681**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
35J65,
35B30,
35R70

Retrieve articles in all journals with MSC (1991): 35J65, 35B30, 35R70

Additional Information

**Salvatore A. Marano**

Affiliation:
Department of Mathematics, University of Catania, Viale A. Doria 6, 95125 Catania, Italy

Email:
marano@dipmat.unict.it

DOI:
https://doi.org/10.1090/S0002-9939-97-03969-5

Keywords:
Elliptic eigenvalue problems,
discontinuous nonlinearities,
elliptic differential inclusions,
unbounded continuum of solutions

Received by editor(s):
March 15, 1996

Received by editor(s) in revised form:
May 7, 1996

Additional Notes:
Work performed under the auspices of G.N.A.F.A. of C.N.R. and partially supported by M.U.R.S.T. of Italy (40%, 1994).

Communicated by:
Palle E. T. Jorgensen

Article copyright:
© Copyright 1997
American Mathematical Society