On the solutions of the -Laplacian problem at resonance

Some recent results on the Dirichlet problem for $(p, q)$-Laplace equations

“…there exists ε > 0 such that for every (α, β) ∈ ]λ 1 (p), Analogous results on the whole space, which causes further difficulties, are established in [8,17]. Finally, we refer to the survey paper [25] for non-variational problems involving the (p, q)-Laplace operator.…”

“…Further papers on the same subject are [8,22,27,32]; see also the references therein. In particular, [8] treats the case Ω := R N , [22,27] deal with asymmetric nonlinearities crossing an eigenvalue, while [32] assumes q := 2.…”

“…Eigenvalue problems are treated in Section 3, where, accordingly, g(x, t) := α|t| p−2 t + β|t| q−2 t ∀ (x, t) ∈ Ω × R, with (α, β) ∈ R 2 . The results we present are chiefly taken from [9,12,23,35,36] and concern bounded domains; for the whole space see, e.g., [7,8,17]. Section 4 collects existence and multiplicity theorems, sometimes with a precise sign information.…”

“…Finally, Section 4.3 deals with asymmetric nonlinearities, meaning that the asymptotic behavior at −∞ and +∞ is different. Due to limited space, we relied only on [10,12,16], [21]- [24], [26]- [29], [31,32], [38]- [40], where bounded domains are considered, and refer to [8,11,15,17] for the case Ω := R N .…”

Existence and non-existence of positive solution for (p, q)-Laplacian with singular weights

“…However, there are few results one the eigenvalue problems for the (p, q)-Laplacian. In [4,5], the authors established the existence of the principal eigenvalue and of a continuous family of eigenvalues for problem…”