| Abstract: |
| In this paper, we combine variational methods and truncation techniques to study the existence of a positive weak solution for a quasilinear elliptic problem driven by the $p$-Laplacian operator involving a reaction term which might or might not have a singularity at $0$. Furthermore, provided that solutions belong to $C^1(\overline{\Omega})$, uniqueness is achieved using a D\`{i}az-Sa\`{a} type argument, which relies on a Br\`{e}zis-Oswald assumption on the nonlinearity. Additionally, in the superlinear case, we give a multiplicity result that applies when an Ambrosetti-Rabinowitz type condition is fulfilled, e.g. in the concave-convex context. |
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