In this paper, we study a nonlinear double phase problem with variable exponent and critical growth on the boundary. The has in reaction combined effects of Carathéodory perturbation defined only locally term. presence term does not permit to apply results point theory corresponding energy functional. Thus, use appropriate cut-off functions truncation techniques work an auxiliary coercive probl...

In this paper we prove an existence and uniqueness result for the double phase Dirichlet problem when lowest exponent is equal to 1. Our solution a function of bounded variation that simultaneously lies in suitable weighted Sobolev space found as limit sequence solutions intermediate problems whose $p$ goes As that, our approach involves study some relevant properties generalized Orlicz-Sobolev...

Isometric Sobolev spaces on finite graphs are characterized. The characterization implies that the following analogue of the Banach-Stone theorem is valid: if two Sobolev spaces on 3-connected graphs, with the exponent which is not an even integer, are isometric, then the corresponding graphs are isomorphic. As a corollary it is shown that for each finite group G and each p which is not an even...

Let s ∈ R and p ≥ 1; the Sobolev space Lp,s(Rd) is the space of tempered distributions f such that (Id−∆)s/2 f ∈ Lp, where (Id−∆)s/2 is the Fourier multiplier by (1 + |ξ|2)s/2. If s > d/p, then Lp,s is composed of continuous functions; more precisely, the Sobolev embeddings state that Lp,s↩Cs−d/p, see [24, Chapter 11]. In order to state in which sense this embedding is sharp, we need to recall ...

We use variational methods to study the existence and multiplicity of solutions for the following quasi-linear partial differential equation: ( −4pu = λ|u|r−2u+ μ |u| q−2 |x|s u in Ω, u|∂Ω = 0, where λ and μ are two positive parameters and Ω is a smooth bounded domain in Rn containing 0 in its interior. The variational approach requires that 1 < p < n, p ≤ q ≤ p∗(s) ≡ n−s n−pp and p ≤ r ≤ p ∗ ≡...

In this article a new method for moving from local to global results in variable exponent function spaces is presented. Several applications of the method are also given: Sobolev and trace embeddings; variable Riesz potential estimates; and maximal function inequalities in Morrey spaces are derived for unbounded domains.

Available online xxxx Keywords: Nonlinear heat equation Blow up Sobolev spaces with variable exponents a b s t r a c t In this paper we consider a nonlinear heat equation with nonlinearities of variable-exponent type. We show that any solution with nontrivial initial datum blows up in finite time. We also give a two-dimension numerical example to illustrate our result.

We consider the exponential reaction–diffusion equation in space-dimension n ∈ (2, 10). We show that for any integer k ≥ 2 there is a backward selfsimilar solution which crosses the singular steady state k-times. The sameholds for the power nonlinearity if the exponent is supercritical in the Sobolev sense and subcritical in the Joseph–Lundgren sense. © 2008 Elsevier Ltd. All rights reserved.