35 research outputs found
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
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
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
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