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Multiple solutions for Dirichlet problems which are superlinear at \(+\infty\) and (sub)linear at \(-\infty\). (English) Zbl 1181.35053

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.

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
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