de Figueiredo, Djairo G. On the superlinear Ambrosetti-Prodi problem. (English) Zbl 0554.35045 Nonlinear Anal., Theory Methods Appl. 8, 655-665 (1984). The author considers the problem \(-\Delta u=g(x,u)+t\phi_ 1+h\) in \(\Omega\), \(u=0\) on \(\partial \Omega\), where \(\Omega\) is a smooth bounded domain in \(\mathbb R^ N\), \(\liminf_{s\to \infty} s^{-1}g(x,0)>\lambda_ 1\limsup_{s\to -\infty} g'\!_ 2(x,s)<\lambda_ 1\) and \(\phi_ 1\) is the positive eigenfunction of -\(\Delta\). If \(g\) grows more slowly than the Sobolev exponent at \(+\infty\) and if a technical condition is satisfied, the author shows that the problem has at least two solutions for \(t\) large negative. This partially generalizes a result of the reviewer and Amann and Hess. [More recently, the author and S. Solimini [MRC report 2568 (Wisconsin)] have improved the main result.] (Note that the properness question he raises can be proved if \((\ln s)^ Bs^{- \alpha}g(s)>C>0\) as \(s\to \infty\) where \(C\) is finite and \(\alpha \in [1,(N-2)^{-1}(N+2))\) by a blowing up argument. Here, for simplicity, we assume that \(g\) is independent of \(x\).) Reviewer: E. N. Dancer Cited in 37 Documents MSC: 35J65 Nonlinear boundary value problems for linear elliptic equations 35A15 Variational methods applied to PDEs 35A01 Existence problems for PDEs: global existence, local existence, non-existence 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces Keywords:superlinear Ambrosetti-Prodi problem; semiliner elliptic boundary value problem; jumping nonlinearity; multiplicity of solutions; Sobolev exponent PDFBibTeX XMLCite \textit{D. G. de Figueiredo}, Nonlinear Anal., Theory Methods Appl. 8, 655--665 (1984; Zbl 0554.35045) Full Text: DOI References: [1] Ambrosetti, A.; Prodi, G., On the inversion of some differentiable mappings with singularities between Banach spaces, Annali. Math. pura appl. Ser. IV, 93, 231-247 (1972) · Zbl 0288.35020 [2] Berger, M. S.; Podolak, E., On the solutions of a nonlinear Dirichlet problem, Indiana Univ. Math. J., 14, 837-846 (1975) · Zbl 0329.35026 [3] Amann, H.; Hess, P., A multiplicity result for a class of elliptic boundary value problems, Proc. R. Soc. Edinb., 84A, 145-151 (1979) · Zbl 0416.35029 [4] Fucik, S., Remarks on a result by A. Ambrosetti and G. Prodi, Boll. Un. mat. Ital., 11, 259-267 (1975) · Zbl 0303.35037 [5] Kazdan, J. L.; Warner, F. W., Remarks on some quasilinear elliptic equations, Communs pure appl. Math., XXVIII, 567-597 (1975) · Zbl 0325.35038 [6] Dancer, E. N., On the ranges of certain weakly nonlinear elliptic partial differential equations, J. Math. pures appl., 57, 351-366 (1978) · Zbl 0394.35040 [7] Brézis, H.; Turner, R. L., On a class of superlinear elliptic problems, Communs P.D.E., 2, 601-614 (1977) · Zbl 0358.35032 [8] Ambrosetti, A.; Rabinowitz, P., Dual variational methods in critical point theory and applications, J. funct. Analysis, 14, 349-381 (1973) · Zbl 0273.49063 [9] De Figueiredo, D. G., Lectures on boundary value problems of the Ambrosetti-Prodi type, Atas do 12° Seminário Brasileiro de Análise (October 1980), Sāo Paulo [10] De Figueiredo, D. G.; Lions, P.-L.; Nussbaum, R., A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. pures appl., 61, 41-63 (1982) · Zbl 0452.35030 [11] Brézis, H.; Kato, T., Remarks on the Schrödinger operator with singular complex potentials, J. Math. pures appl., 58, 137-151 (1979) · Zbl 0408.35025 [12] Amann, H., On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana Univ. Math. J., 21, 125-146 (1971) · Zbl 0219.35037 [13] Sattinger, D. H., Topics in Stability and Bifurcation Theory, Springer Lecture Notes in Mathematics, Vol. 309 (1973) · Zbl 0268.35042 [17] Berestycki, H.; Lions, P. L., Sharp existence results for a class of semilinear elliptic problems, Bol. Soc. Bras. Mat., 12, 9-20 (1981) · Zbl 0571.35038 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.