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& = |u|^{2^{*}-2} u\big( \ln( \tau +|u|) \big)^{ |x|^{\beta} } & \ \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 provide necessary conditions on Euclidean domains for inclusions $$W^{1,p(\cdot )}(\Omega ) \hookrightarrow L^{q(\cdot $$ W 1 , p ( · ) Ω <mml:mo...

Moduli spaces M of self-dual SU(2) connections (“instantons”) over a compact Riemannian 4.manifold (hl, g) carry a natural L2 metric g, which is generally incomplete. For instantons of Pontryagin index 1 over a compact, simply connected, oriented, positive-definite base manifold, the completion M is Donaldson’s compactification; in fact the boundary of the completion is an isometric copy of (M,...

Abstract The problem of prescribing conformally the scalar curvature a closed Riemannian manifold as given Morse function reduces to solving an elliptic partial differential equation with critical Sobolev exponent. Two ways attacking this consist in subcritical approximations or negative pseudogradient flows. We show under mild nondegeneracy assumption equivalence both approaches respect zero w...

We study the existence of sign changing solutions to following problem(0.1){Δu+|u|p−1u=0inΩε;u=0on∂Ωε, where p=n+2n−2 is critical Sobolev exponent and Ωε a bounded smooth domain in Rn, n≥3, form Ωε=Ω\B(0,ε). Here Ω containing origin 0 B(0,ε) denotes ball centered at with radius ε>0. construct new type sign-changing high energy problem (0.1), when parameter ε small enough.