### An existence theorem for an implicit integral equation with discontinuous right-hand side.

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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 $\Omega \subset {\mathbb{R}}^{N}$ 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.

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.

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 deal with the implicit integral equation $$h\left(u\left(t\right)\right)=f(\phantom{\rule{0.166667em}{0ex}}t\phantom{\rule{0.166667em}{0ex}},{\int}_{I}g(t,z)\phantom{\rule{0.166667em}{0ex}}u\left(z\right)\phantom{\rule{0.166667em}{0ex}}dz)\phantom{\rule{4.0pt}{0ex}}\text{for}\phantom{\rule{4.0pt}{0ex}}\text{a.a.}\phantom{\rule{4.0pt}{0ex}}t\in I,$$ where $I:=[0,1]$ and where $f:I\times [0,\lambda ]\to \mathbb{R}$, $g:I\times I\to [0,+\infty [$ and $h:\phantom{\rule{0.166667em}{0ex}}]\phantom{\rule{0.166667em}{0ex}}0,+\infty \phantom{\rule{0.166667em}{0ex}}[\phantom{\rule{0.166667em}{0ex}}\to \mathbb{R}$. 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.

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