Positive Solutions for a Quasilinear Elliptic Problem Involving Sublinear and Superlinear Terms

Abstract

We deal with the existence and non-existence of positive solutions for the problem \begin{eqnarray*} \displaystyle \left \{ \begin{array}{@{\,}c} -\Delta_p u +m(x)u^{p-1} = a(x)f(u) + \lambda b(x)g(u)~~ \text{in}~~ \mathbf{R}^N\,,\\ \\ \displaystyle u > 0~~ \text{in}~~\mathbf{R}^N\,,~~~ u(x)\rightarrow 0 \ \text{when} \ |x|\rightarrow\infty\,, \end{array} \right. \end{eqnarray*} where $\Delta_p$ is the $p$-Laplacian operator, $1<p<N$, $\lambda>0$ is a real parameter, $ f, g: (0, \infty)\rightarrow (0,\infty)$ and $m, a, b: \mathbf{R}^N\rightarrow[0,\infty )$; $ a, b\neq 0$ are continuous functions. In this work we consider, for example, nonlinearities with combined effects of concave and convex terms, besides allowing the presence of singularities. For existence of solutions, we exploit the lower and upper solutions method, combined with a technique of monotone-regularization on the nonlinearities $f$ and $g$ and for non-existence we use a consequence of Picone identity.