Symmetry of n-mode positive solutions for two-dimensional H\'enon type systems

We provide a symmetry result for n-mode positive solutions of a general class of semi-linear elliptic systems under cooperative conditions on the nonlinearities. Moreover, we apply the result to a class of H\'enon systems and provide the existence of multiple n-mode positive solutions.Comment: 8 page

Coron problem for fractional equations

We prove that the critical problem for the fractional Laplacian in an annular type domain admits a nontrivial solution provided that the inner hole is sufficiently small.Comment: 10 pages, revised versio

Nonlocal problems at critical growth in contractible domains

We prove the existence of a positive solution for nonlocal problems involving the fractional Laplacian and a critical growth power nonlinearity when the equation is set in a suitable contractible domain.Comment: 17 page

Compact embedding from W01,2(Ω) to Lq(x)(Ω) and its application to nonlinear elliptic boundary value problem with variable critical exponent

AbstractWe study the compact embedding from W01,2(Ω) to Lq(x)(Ω) with a variable critical exponent 1⩽q(x)⩽2N/(N−2), N⩾3 if there exist a point x0∈Ω, a small η>0, 0<l<1 and C0>0 such that q(x0)=2N/(N−2) and q(x)⩽2N/(N−2)−C0/(log(1/|x−x0|))l for |x−x0|⩽η. As an application, we show an existence of a positive solution to the nonlinear elliptic boundary value problem −Δu=uq(x)−1 in Ω, u(x)=0 on ∂Ω