We consider a parametric nonlinear Dirichlet problem driven by the p-Laplacian and exhibiting the combined effects of singular and superlinear terms. Using variational methods combined with truncation and comparison techniques, we prove a bifurcation-type theorem. More precisely, we show that there exists a critical parameter value λ * > 0 s.t. for all λ ∈ (0, λ *) (λ being the parameter) the problem has at least two positive smooth solutions, for λ = λ * the problem has at least one positive smooth solution and for λ > λ * the positive solutions disappear.

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We consider a (p, 2)-equation with a Carathéodory reaction f (z, x) which is resonant at ±∞ and has constant sign, z-dependent zeros. Using variational methods, together with truncation and comparison techniques and Morse theory, we establish the existence of five nontrivial smooth solutions (four of constant sign and the fifth nodal). If the reaction f (z, x) is C 1 in x ∈ ‫,ޒ‬ then we produce a second nodal solution for a total of six nontrivial smooth solutions.

We consider a parametric Neumann problem with an indefinite and unbounded potential. Using a combination of critical point theory with truncation and comparison techniques, with Morse theory and with invariance arguments for a suitable negative gradient flow, we prove two multiplicity theorems for certain values of the parameter. In the first theorem we produce three solutions and in the second five. For all solutions we provide sign information. Our work improves significantly results existing in the literature.

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