Existence of large solutions for a quasilinear elliptic problem via explosive sub-supersolutions

Abstract We consider the boundary blow-up quasilinear elliptic problems, div ( | ∇ u | m - 2 ∇ u ) ± λ | ∇ u | q ( m - 1 ) = k ( x ) g ( u ) in a C 2 bounded domain with boundary condition u | ∂ Ω = + ∞ , where m > 1 , q ∈ [ 0 , m / ( m - 1 ) ] and λ ⩾ 0 . Under suitable growth assumptions on k near the boundary and on g both at zero and at infinity, we show the existence of at least one solution in C 1 ( Ω ) . Our proof is based on the method of explosive sub-supersolutions, which permits positive weights k ( x ) which are unbounded and/or oscillatory near the boundary.

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