Existence of weak solutions for degenerate semilinear elliptic equations in unbounded domains. (English) Zbl 1189.35132

Raghavendra, Venkataramanarao; Kar, Rasmita

Electron. J. Differ. Equ. 2009, Paper No. 160, 7 p. (2009).

Summary: We prove the existence of a weak solution for the degenerate semilinear elliptic Dirichlet boundary-value problem
\[ Lu-\mu ug_1+ h(u)g_2= f\quad \text{in }\Omega,\qquad u = 0\quad \text{on }\partial\Omega \]
in a suitable weighted Sobolev space. Here the domain \(\Omega\subset\mathbb R^n\), \(n\geq 3\), is not necessarily bounded, and \(h\) is a continuous bounded nonlinearity. The theory is also extended for \(h\) continuous and unbounded.

MSC:

35J70	Degenerate elliptic equations
35J61	Semilinear elliptic equations
35D30	Weak solutions to PDEs
35J25	Boundary value problems for second-order elliptic equations

Keywords:

degenerate equations; weighted Sobolev space; unbounded domain