×

A variational approach to superlinear elliptic problems. (English) Zbl 0552.35030

The authors consider the following Dirichlet-problem in a bounded smooth domain \(\Omega\) in \({\mathbb{R}}^ N\) (”Ambrosetti-Prodi problem”): \(-\Delta u=g(x,u)+t\Phi +h,\quad u|_{\partial \Omega}=0.\) Here \(\Phi\) denotes the eigenfunction belonging to the first eigenvalue \(\lambda_ 1\) of -\(\Delta\) in \(\Omega\), t is a real parameter, h is orthogonal to \(\Phi\) and the nonlinearity satisfies \[ \limsup_{s\to - \infty}(x,s)/s<\lambda_ 1<\lim \inf_{s\to +\infty}g(x,s)/s. \] Under further growth and structural conditions on g which guarantee the Palais- Smale condition for the corresponding variational functional it is proved that there exists a value \(\bar t\in {\mathbb{R}}\) depending on h such that the above problem has at least two solutions for \(t<\bar t\) and no solution for \(t>\bar t\). An additional condition is given which secures the existence of a solution also for \(t=\bar t\). These results are deduced by variational methods in an abstract setting. The existence of the second solution is proved by means of the mountain pass lemma.
Reviewer: F.Tomi

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ekeland I., Bull.AMS 1 pp 443– (1979) · Zbl 0441.49011 · doi:10.1090/S0273-0979-1979-14595-6
[2] Hofer H., Math.Ann. 261 pp 493– (1982) · Zbl 0488.47034 · doi:10.1007/BF01457453
[3] Rabinowitz P., MRC Tech.Rep. 2465 (1983)
[4] Dancer E. N., J.Math.Pures et Appl. 57 pp 351– (1978)
[5] Chang K. C., personal communication
[6] Figueiredo D.G., MRC Tech.Rep. 2522 (1983)
[7] Gidas B., Comm.PDE 6 pp 883– (1981) · Zbl 0462.35041 · doi:10.1080/03605308108820196
[8] Br\(eacute;zis H., Comm.PDE 2 pp 601-- (1977)\) · Zbl 0358.35032 · doi:10.1080/03605307708820041
[9] Br\(eacute;zis H., J.Math.Pures et Appl. 58 pp 137-- (1979)\)
[10] Kazdan J., Comm. Pure Appl.Math. pp 567– (1975) · Zbl 0325.35038 · doi:10.1002/cpa.3160280502
[11] Amann H., Proc.Royal Soc.Edinburgh 84 pp 145– (1979) · Zbl 0416.35029 · doi:10.1017/S0308210500017017
[12] Berestycki H., J.Fct.Anal. 84 (1979)
[13] Ambrosetti A., J.Fctl.Anal. 14 pp 349– (1973)
[14] Pucci P., To appear 14 (1973)
[15] Willem M., Trabalho de Mathem\(aacute;tica no199\)
[16] Gilbarg, D. and Trudinger, N. S. ”Elliptic Partial Differential Equations”. · Zbl 0691.35001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.