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Nello studio dei problemi del tipo $- Δ u = f (x, u) + h (x)$ , si impongono generalmente delle condizione sul comportamento asintotico di $f (x, u)$ rispetto allo spettro di $- Δ$ . Avendo in vista dei problemi quasilineari del tipo $- Δ u = f (x, u, \nabla u) + h (x)$ , sembra naturale introdurre una nozione di spettro per $- Δ$ che tenga conto della dipendenza del membro di destra rispetto al gradiende $\nabla u$ . L'oggetto di questo lavoro è di definire, studiare e applicare questa nuova nozione di spettro.

Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity

Djairo Guedes de Figueiredo; Jean-Pierre Gossez; Pedro Ubilla — 2006

Journal of the European Mathematical Society

We study the existence, nonexistence and multiplicity of positive solutions for the family of problems $- Δ u = f_{λ} (x, u)$ , $u \in H_{0}^{1} (Ω)$ , where $Ω$ is a bounded domain in $ℝ^{N}$ , $N \geq 3$ and $λ > 0$ is a parameter. The results include the well-known nonlinearities of the Ambrosetti–Brezis–Cerami type in a more general form, namely $λ a (x) u^{q} + b (x) u^{p}$ , where $0 \leq q < 1 < p \leq 2^{*} - 1$ . The coefficient $a (x)$ is assumed to be nonnegative but $b (x)$ is allowed to change sign, even in the critical case. The notions of local superlinearity and local sublinearity introduced in [9] are essential in this...

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Spectre d'ordre supérieur et problèmes aux limites quasi-linéaires

Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity