We consider the monomial weight |x1|1 · · · |xn|n in R, where Ai ≥ 0 is a real number for each i = 1, ..., n, and establish Sobolev, isoperimetric, Morrey, and Trudinger inequalities involving this weight. They are the analogue of the classical ones with the Lebesgue measure dx replaced by |x1|1 · · · |xn|ndx, and they contain the best or critical exponent (which depends on A1, ..., An). More i...

We consider the nonlinear eigenvalue problem −div ( |∇u|∇u ) = λ|u|u in Ω, u = 0 on ∂Ω, where Ω is a bounded open set in R with smooth boundary and p, q are continuous functions on Ω such that 1 < infΩ q < infΩ p < supΩ q, supΩ p < N , and q(x) < Np(x)/ (N − p(x)) for all x ∈ Ω. The main result of this paper establishes that any λ > 0 sufficiently small is an eigenvalue of the above nonhomogene...

We show the existence of nodal solutions to perturbed quasilinear elliptic equations with critical Sobolev exponent on compact Riemannian manifolds. A nonexistence result is also given.

We consider the nonlinear half-Laplacian heat equation $$\begin{aligned} u_t+(-\Delta )^{\frac{1}{2}} u-|u|^{p-1}u=0,\quad {\mathbb {R}}^n\times (0,T). \end{aligned}$$We prove that all blows-up are type I, provided \(n \le 4\) and \( 1<p<p_{*} (n)\) where p_{*} is an explicit exponent which below \(\frac{n+1}{n-1}\), critical Sobolev exponent. Central to our proof a Giga-Kohn monotonicity formu...

For bi-Lipschitz homeomorphisms of a compact manifold it is known that topological entropy always finite. manifolds dimension two or greater, we show that in the closure space homeomor- phisms, with respect to either H older or Sobolev topologies, entropy generically infinite. We also prove versions C1-Closing Lemma in these spaces. Finally, give examples infinite which are and/or every e...

We study a system of nonlinear partial differential equations describing the unsteady motions incompressible chemically reacting non-Newtonian fluids. The under consideration consists generalized Navier–Stokes with power-law-type stress-strain relation, where power-law index depends on concentration chemical, coupled to convection-diffusion equation for concentration. This arises in rheology sy...

We study the existence of sign-changing solutions for following supercritical problem with variable exponent and logarithmic nonlinearity$ \begin{equation*} \left\{ \begin{aligned} - \Delta u&amp; = |u|^{2^{*}-2} u\big( \ln( \tau +|u|) \big)^{ |x|^{\beta} } &amp; \ \mbox{in} B_{1}, \\ 0 \mbox{on} \partial \end{aligned} \right. \end{equation*} $where $ B_{1} is unit ball in \mathbb{R}^{N} $, N\g...

Abstract We propose a new variational model in Sobolev–Orlicz spaces with non-standard growth conditions of the objective functional and discuss its applications to image processing. The characteristic feature proposed is that variable exponent, which associated growth, unknown priori it depends on particular function belongs domain functional. So, we deal constrained minimization problem lives...