Cone Sobolev inequality and Dirichlet problem for nonlinear elliptic equations on a manifold with conical singularities

Abstract

In present work, we first establish the corresponding Sobolev inequality and Poincaré inequality on the cone Sobolev spaces, and then, as an application of such inequalities, we prove the existence of non-trivial weak solution for Dirichlet boundary value problem for a class of non-linear elliptic equation on manifolds with conical singularities. Mathematics Subject Classification (2000) 35J20 · 58J05

Lower and upper bounds of Dirichlet eigenvalues for totally characteristic degenerate elliptic operators

“…Next, we introduce the definition of the cone Sobolev spaces H m,γ p (B) on manifolds with conical singularities (cf. [2][3][4][5]). …”

Existence theorem for a class of semilinear totally characteristic elliptic equations with critical cone Sobolev exponents

“…More details on the properties of the spaces H m,γ p,0 (B) and H m,γ p (B) can be seen in the recent article [3] of the authors.…”

“…Other properties of the Mellin transform can be found in [11] or [3]. In our study, we will always use the so-called weighted Mellin transform…”

“…Yamabe problem on manifolds with conical singularities. In [3], by using these inequalities we already got the existence theorem for a class of semilinear degenerate equations on manifolds with conical singularites, that is, for the following Dirichlet problem…”

“…[11]) , they introduced the so-called L p -theory for the cone Sobolev spaces. Recently, the authors established the so-called cone Sobolev inequality (see Proposition 3.1) and Poincaré inequality (see Proposition 3.2) for the weighted Sobolev spaces (2.6) (see [3]) for details). Such kind of inequalities seem to be of fundamental importance to prove the existence of the solutions for such nonlinear problems with totally characteristic degeneracy, also they are expected to be very useful in solving some geometry problem, e.g.…”