In this paper we study the existence of critical points for noncoercive functionals, whose principal part has a degenerate coerciveness. A bifurcation result at zero for the associated differential operator is established.

We deal with a bifurcation result for the Dirichlet problem
⎧$-{\Delta}_{p}u=\mu /{\left|x\right|}^{p}{\left|u\right|}^{p-2}u+\lambda f(x,u)$ a.e. in Ω,
⎨
⎩${u}_{|\partial \Omega}=0$.
Starting from a weak lower semicontinuity result by E. Montefusco, which allows us to apply a general variational principle by B. Ricceri, we prove that, for μ close to zero, there exists a positive number $\lambda {*}_{\mu}$ such that for every $\lambda \in ]0,\lambda {*}_{\mu}[$ the above problem admits a nonzero weak solution ${u}_{\lambda}$ in $W{\u2080}^{1,p}\left(\Omega \right)$ satisfying $li{m}_{\lambda \to 0\u207a}\left|\right|{u}_{\lambda}\left|\right|=0$.

A recent multiplicity result by Ricceri, stated for equations in Hilbert spaces, is extended to a wider class of Banach spaces. Applications to nonlinear boundary value problems involving the p-Laplacian are presented.

We deal with some Dirichlet problems involving a nonlocal term. The existence of two nonzero, nonnegative solutions is achieved by applying a recent result by Ricceri.

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