Brezis-Kamin type results involving locally integrable weights

Abstract

We study existence and asymptotic behavior of entire positive bounded solutions for the following class of semilinear elliptic problem \begin{equation*} \left\{ \begin{aligned} L(u) & = \varrho(x) f(x,u) & \text{ in } & \mathbb{R}^N,\ N \geq 3,\\ u & > 0, & \text{ in } & \mathbb{R}^N, \end{aligned} \right. \end{equation*}where$0 \leq \varrho \in L^p_{loc}(\mathbb{R}^N),$ for some$ N < p \leq \infty.$ Here,$L$ is a local uniform elliptic operator and$f(x,s)$ is a nonlinearity with sublinear behavior at zero and at$+\infty$. This type of result has already been studied in the celebrated work by H. Brezis and S. Kamin for the case when$L= -\Delta$ and$\varrho \in L^{\infty}_{loc}(\mathbb{R}^N)$. Our approach allows us to include for instance$ - \text{dive} \left ( (1+|x|^\mu )^\nu\nabla u \right ) = u^q(|x|^\alpha + |x| ^\beta)^{-1}$ with suitable$\alpha,\ \beta > 0,$$\mu,\ \nu \in \mathbb{R}$ and$0 < q < 1.$ Here, we include two local uniform elliptic situations:$\mu > 0$ with$\nu =1$ or$\nu =-1.$