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Some general multiplicity results for critical points of parameterized functionals on reflexive Banach spaces are established. In particular, one of them improves some aspects of a recent result by B. Ricceri. Applications to boundary value problems are also given.

We establish two existence results for elliptic boundary-value problems with discontinuous nonlinearities. One of them concerns implicit elliptic equations of the form ψ(-Δu) = f(x,u). We emphasize that our assumptions permit the nonlinear term f to be discontinuous with respect to the second variable at each point.

In this paper we shall establish a result concerning the covering dimension of a set of the type $\{x\in X:\Phi (x)\cap \Psi (x)\ne \varnothing \}$, where $\Phi$, $\Psi$ are two multifunctions from $X$ into $Y$ and $X$, $Y$ are real Banach spaces. Moreover, some applications to the differential inclusions will be given.

We present two results on existence of infinitely many positive solutions to the Neumann problem ⎧ $-{\Delta }_{p}u+\lambda \left(x\right){\left|u\right|}^{p-2}u=\mu f(x,u)$ in Ω, ⎨ ⎩ ∂u/∂ν = 0 on ∂Ω, where $\Omega \subset {ℝ}^N$ is a bounded open set with sufficiently smooth boundary ∂Ω, ν is the outer unit normal vector to ∂Ω, p > 1, μ > 0, $\lambda \in L^{\infty }\left(\Omega \right)$ with $essin{f}_{x\in \Omega }\lambda \left(x\right)>0$ and f: Ω × ℝ → ℝ is a Carathéodory function. Our results ensure the existence of a sequence of nonzero and nonnegative weak solutions to the above problem.

Let $p>1$, $q>p$, $\lambda >0$ and $s\in ]1,p[$. We study, for $s\to p^{-}$, the behavior of positive solutions of the problem $-{\Delta }_{p}u=\lambda u^{s-1}+u^{q-1}$ in $\Omega$, $u_{\mid \partial \Omega }=0$. In particular, we give a positive answer to an open question formulated in a recent paper of the first author.

We consider a multifunction $F:T\times X\to 2^E$, where T, X and E are separable metric spaces, with E complete. Assuming that F is jointly measurable in the product and a.e. lower semicontinuous in the second variable, we establish the existence of a selection for F which is measurable with respect to the first variable and a.e. continuous with respect to the second one. Our result is in the spirit of [11], where multifunctions of only one variable are considered.

We establish an existence theorem for a Dirichlet problem with homogeneous boundary conditions by using a general variational principle of Ricceri.

We deal with the implicit integral equation $$h\left(u\left(t\right)\right)=f(t,{\int }_{I}g(t,z)u\left(z\right)dz)\text{for}\text{a.a.}t\in I,$$ where $I:=[0,1]$ and where $f:I\times [0,\lambda ]\to ℝ$, $g:I\times I\to [0,+\infty [$ and $h:]0,+\infty [\to ℝ$. We prove an existence theorem for solutions $u\in L^s\left(I\right)$ where the contituity of $f$ with respect to the second variable is not assumed.