New definitions of a solution of a one-dimensional elliptic equation with a singular first order divergence term

Abstract

In this paper, which is a continuation of our recent paper [1], we give two new definitions of a weak solution of the one-dimensional, elliptic, nonlinear singular problem which is formally written as $$\{\begin{array}{ll}-\frac{d}{dx}\left(a(x)\frac{du}{dx}\right)=-\frac{d\varphi (u)}{dx}-\frac{dg(x)}{dx}& \text{in}(0,L),\\ & \\ u(0)=u(L)=0,& \end{array}$$ where $\varphi :\mathbb{R}\mapsto \mathbb{R}\cup \{+\infty \}$ is a singular function whose model is given by $\varphi (s)={\varphi }_{\gamma }(s)=\frac{1}{|s{|}^{\gamma }}$ with $\gamma >0$. In these two definitions the solutions belong respectively to the space $u\in W_0^{1,1}(0,L)$ with $\varphi (u)\in L^1(0,L)$, and to the space $u\in W_0^{1,m}(0,L)$ with $\varphi (u)\in L^m(0,L)$ for $m>1$. In these frameworks we state and prove results of non-existence, existence, and non-isolation of the solutions.