We consider Kirchhoff type problems of the form ⎧ -M(ρ(u))(div(a(|∇u|)∇u) - a(|u|)u) = K(x)f(u) in Ω ⎨ ⎩ ∂u/∂ν = 0 on ∂Ω where $\Omega \subset {ℝ}^N$, N ≥ 3, is a smooth bounded domain, ν is the outward unit normal to ∂Ω, $\rho \left(u\right)={\int }_{\Omega }\left(\Phi \right(\left|\nabla u\right|)+\Phi \left(\right|u\left|\right))dx$, M: [0,∞) → ℝ is a continuous function, $K\in L^{\infty }\left(\Omega \right)$, and f: ℝ → ℝ is a continuous function not satisfying the Ambrosetti-Rabinowitz type condition. Using variational methods, we obtain some existence and multiplicity results.

The paper is dedicated to the existence of local solutions of strongly nonlinear equations in RN and the Orlicz spaces framework is used.

We prove the existence of weak solutions for steady flows of electrorheological fluids with homogeneous Navier-slip type boundary conditions provided $p\left(x\right)>2n/(n+2)$. To prove this, we show Poincaré- and Korn-type inequalities, and then construct Lipschitz truncation functions preserving the zero normal component in variable exponent Sobolev spaces.

We present two existence results for the Dirichlet elliptic inclusion with an upper semicontinuous multivalued right-hand side in exponential-type Orlicz spaces involving a vector Laplacian, subject to Dirichlet boundary conditions on a domain Ω⊂ ℝ². The first result is obtained via the multivalued version of the Leray-Schauder principle together with the Nakano-Dieudonné sequential weak compactness criterion. The second result is obtained by using the nonsmooth variational technique together with...

We deal with projective limits of classes of functions and prove that: (a) the Chebyshev polynomials constitute an absolute Schauder basis of the nuclear Fréchet spaces ${}_{\left(\right)}\left({[-1,1]}^r\right)$; (b) there is no continuous linear extension map from ${\Lambda }_{\left(\right)}^{\left(r\right)}$ into ${}_{\left(\right)}\left({ℝ}^r\right)$; (c) under some additional assumption on , there is an explicit extension map from ${}_{\left(\right)}\left({[-1,1]}^r\right)$ into ${}_{\left(\right)}\left({[-2,2]}^r\right)$ by use of a modification of the Chebyshev polynomials. These results extend the corresponding ones obtained by Beaugendre in [1] and [2].

We study imbeddings of the Sobolev space $W^{m,\varrho }\left(\Omega \right)$: = u: Ω → ℝ with $\varrho ({\partial }^{\alpha }u/\partial x^{\alpha })$ < ∞ when |α| ≤ m, in which Ω is a bounded Lipschitz domain in ℝⁿ, ϱ is a rearrangement-invariant (r.i.) norm and 1 ≤ m ≤ n - 1. For such a space we have shown there exist r.i. norms, ${\tau }_{\varrho }$ and ${\sigma }_{\varrho }$, that are optimal with respect to the inclusions $W^{m,\varrho }\left(\Omega \right)\subset W^{m,{\tau }_{\varrho }}\left(\Omega \right)\subset L_{{\sigma }_{\varrho }}\left(\Omega \right)$. General formulas for ${\tau }_{\varrho }$ and ${\sigma }_{\varrho }$ are obtained using the -method of interpolation. These lead to explicit expressions when ϱ is a Lorentz Gamma norm or an Orlicz norm.

Given a subset A of a topological space X, a locally convex space Y, and a family ℂ of subsets of Y we study the problem of the existence of a linear ℂ-extender $u:C_{\infty }(A,Y)\to C_{\infty }(X,Y)$, which is a linear operator extending bounded continuous functions f: A → C ⊂ Y, C ∈ ℂ, to bounded continuous functions f̅ = u(f): X → C ⊂ Y. Two necessary conditions for the existence of such an extender are found in terms of a topological game, which is a modification of the classical strong Choquet game. The results obtained allow us...

We study a rigidity property, at the vertex of some plane sector, for Gevrey classes of holomorphic functions in the sector. For this purpose, we prove a linear continuous version of Borel-Ritt's theorem with Gevrey conditions

The problem of the existence of extension maps from 0 to ℝ in the setting of the classical ultradifferentiable function spaces has been solved by Petzsche [9] by proving a generalization of the Borel and Mityagin theorems for $C^{\infty }$-spaces. We get a Ritt type improvement, i.e. from 0 to sectors of the Riemann surface of the function log for spaces of ultraholomorphic functions, by first establishing a generalization to some nonclassical ultradifferentiable function spaces.

It is proved that every operator from a weak*-closed subspace of ${\ell }_1$ into a space C(K) of continuous functions on a compact Hausdorff space K can be extended to an operator from ${\ell }_1$ to C(K).

Let M be a separable $C^{\infty }$ Finsler manifold of infinite dimension. Then it is proved, amongst other results, that under suitable conditions of local extensibility the germ of a $C^{\infty }$ function, or of a $C^{\infty }$ section of a vector bundle, on the union of a closed submanifold and a closed locally compact set in M, extends to a $C^{\infty }$ function on the whole of M.