Existence of compact support solutions for a quasilinear and singular problem. (Q454545)

This paper deals with the quasilinear problem in a bounded smooth domain \(\Omega \) involving the \(p\)-Laplacian. The nonlinearity is of the form \(K(x)(\lambda u^q - u^r),\) where \( -1<r<q<p-1\), \(K\) is a positive function having the singular behavior near the boundary of \(\Omega \) and \(\lambda \) is a parameter. The singularity of \(K\) behaves as a negative power of the distance from the boundary. Besides non-existence results for \(\lambda \) small, the authors prove also the existence of positive or compact support (positive) solutions for \(\lambda \) large, depending on the assumptions on parameters \(p, q\) and \(r\). This work extends similar results for semilinear problem \((p=2)\) due to \textit{Y. Haitao} [J. Math. Anal. Appl. 319, No. 2, 830--840 (2006; Zbl 1155.35363)].