In the present paper the Dirichlet problem for semilinear elliptic and parabolic equations in general form is considered. New condition guaranteeing the global solvability of this problem for a wide class of superlinear sources, including e u and |u|p−1u , p > 1 , is formulated. For sublinear case (for example ln(1+ |u|) or |u|p−1u , p < 1) this condition is automatically fulfilled. Our approac...

This paper deals with optimal control problems of semilinear parabolic equations with pointwise state constraints and coupled integral state-control constraints. We obtain necessary optimality conditions in the form of a Pontryagin’s minimum principle for local solutions in the sense of Lp, p ≤ +∞.

Our purpose is to find positive solutions u ∈ D(R ) of the semilinear elliptic problem −∆u = h(x)u for 2 < p. The function h may have an indefinite sign. Key ingredients are a h-dependent concentration-compactness Lemma and a characterization of compact embeddings of D(R ) into weighted Lebesgue spaces.

Via nonsmooth critical point theory we prove the existence of at least two solutions inW 1,p 0 (Ω) for a jumping problem involving the Euler equation of multiple integrals of calculus of variations under natural growth conditions. Some new difficulties arise in comparison with the study of the semilinear and also the quasilinear case.

In this paper we study semilinear sets over arbitrary ordered fields. In particular we show that open (resp. closed) semilinear sets are convex if and only if they are basic, that is, a finite intersection of open (resp. closed) halfspaces. As a corollary we get that any bounded convex closed semilinear set is the convex hull of a finite family of points and we obtain a separation result for co...

We investigate the descriptional complexity of operations on semilinear sets. Roughly speaking, a semilinear set is the finite union of linear sets, which are built by constant and period vectors. The interesting parameters of a semilinear set are: (i) the maximal value that appears in the vectors of periods and constants and (ii) the number of such sets of periods and constants necessary to de...

The design problem of a high-gain observer is considered for some 2 × and 3 semilinear reaction-diffusion systems, with possibly distinct diffusivities, considering distributed measurement part the state. Due to limitations imposed by parabolic operator, such an observer, infinite-dimensional state transformation first applied map system into more suitable set partial differential equations. th...

We consider the semilinear problem −∆u + λu = |u|p−2u in Ω, u = 0 on ∂Ω where Ω ⊂ R is a bounded smooth domain and 2 < p < 2∗ = 2N/(N − 2). We show that if Ω is invariant by a nontrivial orthogonal involution then, for λ > 0 sufficiently large, the equivariant topology of Ω is related with the number of solutions which change sign exactly once.

Let P be a polar space induced by a quasipolarity on a projective space G. In this article, we give sufficient conditions for an embedding (i.e. an incidence-preserving map which is injective on points and lines) of P into a projective space H to be extendable to an incidencepreserving map from G to H. As a consequence, we obtain conditions for such an embedding to be induced by a semilinear ma...

In this paper we show existence of finite energy solutions for the Cauchy problem associated with a semilinear wave equation with interior damping and supercritical source terms. The main contribution consists in dealing with super-supercritical source terms (terms of the order of |u|p with p ≥ 5 in n = 3 dimensions), an open and highly recognized problem in the literature on nonlinear wave equ...