An existence result for $(p,q)$-Laplacian BVP with falling zeros.

Abstract

We show the existence of a positive solution to the $(p,q)$-Laplacian problem \begin{equation*} \begin{cases} -\Delta _{p}u-a\Delta _{q}u=\lambda f(u)-h(x)\quad \text{in }\Omega, \\ u=0\quad \text{on }\partial \Omega, \end{cases} \end{equation*} for $\lambda$ large, where $\Omega $ is a bounded domain in $\mathbb{R}^{n}$ with smooth boundary $\partial \Omega, a$ is a nonnegative constant, $h\in L^{\infty}(\Omega )$, $p>q>1$, and $f$ satisfies $f(0)=f(r)=0$ with $f>0$ on $(0,r)$ for some $r>0$.

Keywords

• positive solutions
• $(p
• q)$ laplacian
• semipositone