Shengbing Deng

Blow-up solutions for Paneitz–Branson type equations with critical growth

Let (M , g) be a smooth, compact Riemannian manifold of dimension n 7. We consider the Paneitz-Branson type equation Δ 2u − divg(A du) + au = |u| 2 � −2−e u in M , where Δg = − divg ∇ is the Laplace-Beltrami operator, A is a smooth symmetrical (2, 0)-tensor fields, a is a smooth function on M ,2 � = 2n n−4 is the critical exponent for the Sobolev embedding and e is a small positive parameter. Under suitable conditions on the TrgA, we construct solutions ue which blow up at one point of the manifold as e goes to zero.

Concentration on minimal submanifolds for a Yamabe-type problem

ABSTRACT We construct solutions to a Yamabe-type problem on a Riemannian manifold M without boundary and of dimension greater than 2, with nonlinearity close to higher critical Sobolev exponents. These solutions concentrate their mass around a nondegenerate minimal submanifold of M, provided a certain geometric condition involving the sectional curvatures is satisfied. A connection with the solution of a class of PDEs on the submanifold with a singular term of attractive or repulsive type is established.

New solutions for critical Neumann problems in ℝ2

Abstract We consider the elliptic equation - Δ ⁢ u + u = 0 {-\Delta u+u=0} in a bounded, smooth domain Ω in ℝ 2 {\mathbb{R}^{2}} subject to the nonlinear Neumann boundary condition ∂ ⁡ u ∂ ⁡ ν = λ ⁢ u ⁢ e u 2 {\frac{\partial u}{\partial\nu}=\lambda ue^{u^{2}}} , where ν denotes the outer normal vector of ∂ ⁡ Ω {\partial\Omega} . Here λ > 0 {\lambda>0} is a small parameter. For any λ small we construct positive solutions concentrating, as λ → 0 {\lambda\to 0} , around points of the boundary of Ω.