Radially symmetric solutions to a superlinear Dirichlet problem in a ball with jumping nonlinearities

Authors:
Alfonso Castro and Alexandra Kurepa

Journal:
Trans. Amer. Math. Soc. **315** (1989), 353-372

MSC:
Primary 35J65; Secondary 35B05

DOI:
https://doi.org/10.1090/S0002-9947-1989-0933323-8

MathSciNet review:
933323

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be bounded functions with . Let be a locally Lipschitzian function satisfying the superlinear jumping condition: (i) (ii) for some , and (iii) for some where is the primitive of . Here we prove that the number of solutions of the boundary value problem for with for tends to when tends to . The proofs are based on the "energy" and "phase plane" analysis.

**[1]**A. Ambrosetti and G. Prodi,*On the inversion of some differentiable mappings with singularities between Banach spaces*, Ann. Mat. Pura Appl.**93**(1972), 231-246. MR**0320844 (47:9377)****[2]**A. Castro and A. Kurepa,*Energy analysis of a nonlinear singular differential equation and applications*, Rev. Colombiana Mat.**21**(1987), 155-166. MR**968385 (89j:34083)****[3]**-,*Infinitely many radially symmetric solutions to a superlinear Dirichlet problem in a ball*, Proc. Amer. Math. Soc.**101**(1987), 57-64. MR**897070 (88j:35058)****[4]**A. Castro and R. Shivaji,*Multiple solutions for a Dirichlet problem with jumping nonlinearities*, Trends in Theory and Practice of Nonlinear Analysis, V. Lakshmikantham (ed.), North-Holland, 1985. MR**817476****[5]**-,*Multiple Solutions for a Dirichlet problem with jumping nonlinearities*II, J. Math. Anal. Appl.**133**(1988), 509-528. MR**954725 (89e:34031)****[6]**A. Castro and A. C. Lazer,*On periodic solutions of weakly coupled systems of differential equations*, Boll. Un. Mat. Ital. (5)**18-B**(1981), 733-742. MR**641732 (83b:34046)****[7]**D. G. Costa and D. G. De Figueiredo,*Radial solutions for a Dirichlet problem in a ball*, J. Differential Equations**60**(1985), 80-89. MR**808258 (87i:35069)****[8]**J. Kazdan and F. W. Warner,*Remarks on some quasilinear elliptic equations*, Comm. Pure Appl. Math.**28**(1975), 567-597. MR**0477445 (57:16972)****[9]**A. Lazer and P. J. McKenna,*On the number of solutions of nonlinear Dirichlet problem*, J. Math. Anal. Appl.**84**(1981), 282-294. MR**639539 (83e:35050)****[10]**-,*On a conjecture related to the number of solutions of a nonlinear Dirichlet problem*, Proc. Roy. Soc. Edinburgh Sect. A**95**(1983), 275-283. MR**726879 (86g:34028)****[11]**-,*Critical point theory and boundary value problems with nonlinearities crossing multiple eigenvalues*, Comm. Partial Differential Equations**10**(1985), 107-150. MR**777047 (86f:35025)****[12]**D. Lupo, S. Solimini and P. N. Srikanth,*Multiplicity results for an o.d.e. problem with even nonlinearity*, Quad. Matematici (II), no. 112, Dicembre 1985.**[13]**B. Ruf and S. Solimini,*On a class of superlinear Sturm-Liouville problems with arbitrarily many solutions*, SIAM J. Math. Anal.**17**(1986), 761-771. MR**846387 (87g:34023)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
35J65,
35B05

Retrieve articles in all journals with MSC: 35J65, 35B05

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1989-0933323-8

Keywords:
Dirichlet problem,
superlinear jumping nonlinearity,
singular differential equation,
radially symmetric solutions

Article copyright:
© Copyright 1989
American Mathematical Society