On a variational degenerate elliptic problem
It is considered the following degenerate elliptic boundary value problem \[ \text{-div}(a(x)\bigtriangledown u)=f(x,u)\quad \text{in }\Omega, \qquad u=0 \quad \text{on }\partial\Omega, \] where \(f\) is a given regular function, \(\Omega\) is a bounded or unbounded domain in \(\mathbb R^n, n\geq 2,\) and \(a(x)\) is a measurable and nonnegative weight on \(\Omega\), which can have at most a finite number of (essential) zeroes. A weighted Sobolev inequality which generalizes a result by \textit{L. Caffarelli, R. Kohn} and \textit{L. Nirenberg} [Compos. Math. 53, 259-275 (1984; Zbl 0563.46024)] is established. The existence of nonnegative weak solutions for problems under consideration is proved via variational methods.
scientific article; zbMATH DE number 1508201