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

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
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