### Abstract.

The main result of this work is a Dancer-type bifurcation result for the quasilinear elliptic problem
(P)
$$ \left\{\begin{aligned} -\Delta_p u &= \lambda\vert u \vert^{p-2}u + h\left(x,u(x);\lambda\right)\,\,\hbox{ in }\,\,\Omega;\\ u&= 0\,\,\hbox{on}\,\,\partial\Omega.\\ \end{aligned}\right. $$
Here, Ω is a bounded domain in
$${\mathbb{R}}^N (N \geq 1), \Delta_p u\,\, {\mathop = \limits^{\rm def} }\,\, {\rm div}(\mid \nabla u\mid^{p-2}\nabla u)$$
denotes the Dirichlet *p*-Laplacian on
$$W^{1,p}_0(\Omega), 1 < p < \infty$$
, and
$$\lambda \in {\mathbb{R}}$$
is a spectral parameter. Let μ_{1} denote the first (smallest) eigenvalue of −Δ_{p}. Under some natural hypotheses on the perturbation function
$$h : \Omega \times {\mathbb{R}}\times
{\mathbb{R}} \rightarrow {\mathbb{R}}$$
, we show that the trivial solution
$$(0, \mu_1) \in E = W^{1,p}_0 (\Omega)\times {\mathbb{R}}$$
is a bifurcation point for problem (P) and, moreover, there are *two distinct continua*,
$$\mathcal{Z}^+_{\mu_1}$$
and
$$\mathcal{Z}^-_{\mu_1}$$
, consisting of nontrivial solutions
$$(u,\lambda) \in E$$
to problem (P) which bifurcate from the set of trivial solutions at the bifurcation point (0, μ_{1}). The continua
$$\mathcal{Z}^+_{\mu_1}$$
and
$$\mathcal{Z}^-_{\mu_1}$$
are *either* both unbounded in *E*, *or else* their intersection
$$\mathcal{Z}^+_{\mu_1} \cap \mathcal{Z}^-_{\mu_1}$$
contains also a point other than (0, μ_{1}). For the semilinear problem (P) (i.e., for *p* = 2) this is a classical result due to E. N. Dancer from 1974. We also provide an example of how the union
$$\mathcal{Z}^+_{\mu_1} \cap
\mathcal{Z}^-_{\mu_1}$$
looks like (for *p* > 2) in an interesting particular case.

Our proofs are based on very precise, local asymptotic analysis for λ near μ_{1} (for any 1 < *p* < ∞) which is combined with standard topological degree arguments from global bifurcation theory used in Dancer’s original work.