Positive solutions for slightly subcritical elliptic problems via Orlicz spaces (Q2149591)

The paper is concerned with semilinear equations with indefinite nonlinearities in a bounded open connected st \(\Omega\subset \mathbb R^N\) for \(N>2\). More precisely, the authors investigate existence and multiplicity of positive solutions of the Dirichlet problem \[ -\Delta u=\lambda u + a(x)f(u) \text{ in }\Omega;\quad u=0\text{ on } \partial\Omega, \] where \(\lambda \in \mathbb R\), the nonlinearity \(f\) satisfies a suitable growth of subcritical type and the weight \(a(x)\) does not have constant sign. Let \(\lambda_1\) be the first eigenvalue of the Laplace-Dirichlet operator on \(\Omega\). The main result states that the above Dirichlet problem admits at least a positive solution for \(\lambda \le \lambda_1\) and there exists a turning point \(\Lambda>\lambda_1\) such that the Dirichlet problem still admits at least a positive solution for \(\lambda=\Lambda\), while at least two ordered positive solutions can be found in the range \((\lambda_1,\Lambda)\). The work is inspired by \textit{S. Alama} and \textit{G. Tarantello} [Calc. Var. Partial Differ. Equ. 1, No. 4, 439--475 (1993; Zbl 0809.35022)]. The argument of the proof is based on the Crandall-Rabinowitz bifurcation theory and variational techniques, using Orlicz-Sobolev embeddings, Palais-Smale sequences and the Mountain Pass Theorem.