## Multiple nontrivial solutions of resonant and nonresonant asymptotically linear problems

HTML articles powered by AMS MathViewer

- by Shair Ahmad PDF
- Proc. Amer. Math. Soc.
**96**(1986), 405-409 Request permission

## Abstract:

We give simple conditions under which a second order semilinear elliptic boundary value problem with the zero solution has at least two nonzero solutions. Our conditions involve the change in the spectrum of the linearization of the problem going from zero to infinity.## References

- S. Ahmad, A. C. Lazer, and J. L. Paul,
*Elementary critical point theory and perturbations of elliptic boundary value problems at resonance*, Indiana Univ. Math. J.**25**(1976), no. 10, 933–944. MR**427825**, DOI 10.1512/iumj.1976.25.25074 - H. Amann and E. Zehnder,
*Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**7**(1980), no. 4, 539–603. MR**600524** - Antonio Ambrosetti and Paul H. Rabinowitz,
*Dual variational methods in critical point theory and applications*, J. Functional Analysis**14**(1973), 349–381. MR**0370183**, DOI 10.1016/0022-1236(73)90051-7 - Antonio Ambrosetti,
*Differential equations with multiple solutions and nonlinear functional analysis*, Equadiff 82 (Würzburg, 1982) Lecture Notes in Math., vol. 1017, Springer, Berlin, 1983, pp. 10–37. MR**726565**, DOI 10.1007/BFb0103232 - P. Bartolo, V. Benci, and D. Fortunato,
*Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity*, Nonlinear Anal.**7**(1983), no. 9, 981–1012. MR**713209**, DOI 10.1016/0362-546X(83)90115-3 - Kung Ching Chang,
*Solutions of asymptotically linear operator equations via Morse theory*, Comm. Pure Appl. Math.**34**(1981), no. 5, 693–712. MR**622618**, DOI 10.1002/cpa.3160340503 - Helmut Hofer,
*A note on the topological degree at a critical point of mountainpass-type*, Proc. Amer. Math. Soc.**90**(1984), no. 2, 309–315. MR**727256**, DOI 10.1090/S0002-9939-1984-0727256-0 - A. C. Lazer and P. J. McKenna,
*Critical point theory and boundary value problems with nonlinearities crossing multiple eigenvalues*, Comm. Partial Differential Equations**10**(1985), no. 2, 107–150. MR**777047**, DOI 10.1080/03605308508820374 - L. Nirenberg,
*Topics in nonlinear functional analysis*, Courant Institute of Mathematical Sciences, New York University, New York, 1974. With a chapter by E. Zehnder; Notes by R. A. Artino; Lecture Notes, 1973–1974. MR**0488102** - Paul H. Rabinowitz,
*Some minimax theorems and applications to nonlinear partial differential equations*, Nonlinear analysis (collection of papers in honor of Erich H. Rothe), Academic Press, New York, 1978, pp. 161–177. MR**0501092** - Paul H. Rabinowitz,
*The mountain pass theorem: theme and variations*, Differential equations (S ao Paulo, 1981) Lecture Notes in Math., vol. 957, Springer, Berlin-New York, 1982, pp. 237–271. MR**679149** - David H. Sattinger,
*Topics in stability and bifurcation theory*, Lecture Notes in Mathematics, Vol. 309, Springer-Verlag, Berlin-New York, 1973. MR**0463624**

## Additional Information

- © Copyright 1986 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**96**(1986), 405-409 - MSC: Primary 35J65; Secondary 35B32
- DOI: https://doi.org/10.1090/S0002-9939-1986-0822429-2
- MathSciNet review: 822429