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A remark on the nodal regions of the solutions of some superlinear elliptic equations*

Published online by Cambridge University Press:  14 November 2011

Vieri Benci
Affiliation:
Istituto di Matematiche Applicate, Università, 56100 Pisa, Italy
Donato Fortunato
Affiliation:
Dipartimento di Matematica, Università, 70125 Bari, Italy

Synopsis

We study the problem

where Ω is a bounded domain in ℝn(N≧3), 2 <p ≦2N/N – 2, λ ∈R. We prove the existence of nontrivial solutions of (*) for which we can estimate the number of nodal regions dependent on the eigenvalues of −Δ less than λ.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1989

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References

1Ambrosetti, A. and Rabinowitz, P. H.. Dual variational methods in critical point theory. J. Fund. Anal. 14 (1973), 349381.CrossRefGoogle Scholar
2Benci, V.. Some Applications of the Generalized Morse-Conley index. Conf. Sem. Mat. Univ. Ban 218 (1987), 132.Google Scholar
3Benci, V.. A new approach to the Morse-Conley Theory and some applications (preprint).Google Scholar
4Benci, V. and Fortunato, D.. Subharmonic solutions of prescribed minimal period for nonautonomous differential equations. In Recent Advances in Hamiltonian Systems, eds. Dell'Antonio, G. F. and D'Onofrio, B., pp. 8396 (Singapore: World Science, 1987).Google Scholar
5Brezis, H. and Kato, T.. Remarks on the Schrödinger operator with singular complex potential. J. Math. PuresAppl. 58 (1979), 137151.Google Scholar
6Brezis, H. and Nirenberg, L.. Positive solutions of Nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math. 36 (1983), 437477.CrossRefGoogle Scholar
7Capozzi, A., Fortunato, D. and Palmieri, G.. An existence result for nonlinear elliptic problems involving critical Sobolev exponent. Ann. Inst. H. Poincaré Anal. Non linéaire 2 (1985), 463470.CrossRefGoogle Scholar
8Cerami, G., Fortunato, D. and Struwe, M.. Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents. Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 341350.CrossRefGoogle Scholar
9Hempel, J. A.. Multiple solutions for a class of nonlinear boundary value problems. Indiana Univ. Math. J. 20 (1971), 983996.CrossRefGoogle Scholar
10Kato, T.. Perturbation theory for linear operators (New-York: Springer, 1966).Google Scholar
11Lions, P. L.. On the existence of positive solutions of semilinear elliptic equations. SIAM Rev. 24 (1982), 441467.CrossRefGoogle Scholar
12Luckhaus, S.. Existence and regularity of weak solutions to the Dirichlet problem for semilinear elliptic systems of high order. J. Reine Angew. Math. 306 (1979), 192207.Google Scholar