4 research outputs found
Regularity theorems for a class of degenerate elliptic equations
In this paper we study the regularity of a class of degenerate elliptic equations with special lower order terms. By introducing a proper distance and applying the compactness method, we establish the Hölder type estimates for the weak solutions
Multiplicity of solutions for semilinear subelliptic Dirichlet problem
In this paper, we study the semilinear subelliptic equation \left\{
\begin{array}{cc}
-\triangle_{X} u=f(x,u)+g(x,u) & \mbox{in}~\Omega, \\[2mm]
u=0\hfill & \mbox{on}~\partial\Omega,
\end{array}
\right. where is the
self-adjoint H\"{o}rmander operator associated with vector fields
satisfying the H\"{o}rmander condition,
, is a
Carath\'{e}odory function on , and is an open
bounded domain in with smooth boundary. Combining the
perturbation from symmetry method with the approaches involving eigenvalue
estimate and Morse index in estimating the min-max values, we obtain two kinds
of existence results for multiple weak solutions to the problem above.
Furthermore, we discuss the difference between the eigenvalue estimate approach
and the Morse index approach in degenerate situations. Compared with the
classical elliptic cases, both approaches here have their own strengths in the
degenerate cases. This new phenomenon implies the results in general degenerate
cases would be quite different from the situations in classical elliptic cases.Comment: This paper has been accepted for publication in SCIENCE CHINA
Mathematic