In this paper, we study the regularizing effect due to the interaction between the coefficient of the zero-order term and the datum for the following type of elliptic problem \(\begin{aligned} u\in W_{0}^{1,p}(\Omega ):\, -\operatorname {div}(M(x)|\nabla u|^{p-2}\nabla u)+b(x)h(u)=f(x), \end{aligned}\) where \(\Omega \) is a bounded open subset of \({\mathbb {R}}^{N},\, N>2,\, \, M\) is a bounded elliptic matrix, \(0\le b(x)\in L^{1}(\Omega )\) and \(h\) is a continuous odd-increasing function. Even if \(f(x)\) only belongs to \(L^{1}(\Omega ),\) the assumption \(\begin{aligned} \text{ there } \text{ exists }\, L\in \left( 0, \lim \limits _{s\rightarrow \infty }h(s)\right) \,\, \text{ such } \text{ that }\,\, |f(x)|\le L b(x) \end{aligned}\) implies the existence of a weak solution belonging to \(W_{0}^{1,p}(\Omega )\) and to \(L^{\infty }(\Omega )\) . Using the strong maximum principle we prove that such a solution \(u\) is strictly positive a.e. in the domain \(\Omega .\) In the second part, we continue to study the previous problem, we add a term having a superlinear growth depending on the gradient of the solution, and we prove that this problem admits a weak bounded solution and from the weak maximum principle we prove that each solution of the problem is positive. Finally, we study the existence and summability of solutions to problems featuring Hardy-type potentials.
