Motreanu, D.; Motreanu, V. V.; Papageorgiou, N. S. Multiple solutions for Dirichlet problems which are superlinear at \(+\infty\) and (sub)linear at \(-\infty\). (English) Zbl 1181.35053 Commun. Appl. Anal. 13, No. 3, 341-357 (2009). Summary: We consider a semilinear Dirichlet elliptic problem with a right-hand side nonlinearity which exhibits an asymmetric grow near \(+\infty\) and near \(-\infty\). Namely, it is (sub-)linear near \(-\infty\) and superlinear near \(+\infty\). However, it does not need to satisfy the Ambrosetti-Robinowitz condition on the positive semiaxis. Combining variational methods with Morse theory, we show that the problem has at least two nontrivial solutions, one of which is negative. Cited in 12 Documents MSC: 35J25 Boundary value problems for second-order elliptic equations 35J61 Semilinear elliptic equations 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 35J20 Variational methods for second-order elliptic equations Keywords:Dirichlet problem; Ambrosetti-Rabinowitz condition; critical groups; Morse theory PDFBibTeX XMLCite \textit{D. Motreanu} et al., Commun. Appl. Anal. 13, No. 3, 341--357 (2009; Zbl 1181.35053)