1D logistic reaction and p-Laplacian diffusion as p goes to one

Abstract

This work discusses the existence of the limit as p goes to 1 of the nontrivial solutions to the one-dimensional problem: $$\begin{aligned} {\left\{ \begin{array}{ll} -\left( |u_x|^{p-2} u_x\right) _x = \lambda |{u}|^{{p}-2}{u} -|{u}|^{{q}-2}{u}&{} \quad 0< x < 1\\ u(0)=u(1)=0, &{} \end{array}\right. } \end{aligned}$$ where $$\lambda $$ is a positive parameter and the exponents p, q satisfy $$1< p < q$$ . We prove that solutions do converge to a limit function, which solves in a proper sense a Dirichlet problem involving the 1-Laplacian operator.