On an elliptic equation of $p$-Laplacian type with nonlinear boundary condition - doi: 10.5269/bspm.v29i2.11965

Abstract

We consider elliptic equations of $p$-Laplacian type with the nonlinear boundary condition of the form
\begin{equation*}
\begin{cases}
\begin{array}{rlll}
-\Delta_p u +|u|^{p-2}u & = & \lambda f_1(u) +\mu g_1(u)& \text{ in } \Omega,\\
|\nabla u|^{p-2}\frac{\partial u}{\partial n} &= & \lambda f_2(u)+\mu g_2(u) & \text{ in } \partial\Omega,
\end{array}
\end{cases}
\end{equation*}
where $\Omega \subset \R^N$ ($N \geqq 3$) is a bounded domain with smooth boundary $\partial\Omega$, $\frac{\partial}{\partial n}$ is the outer unit normal derivative, $\lambda, \mu$ are parameters. The functions $f_i$, $i=1, 2$, are assumed to be $(p-1)$-sublinear while $g_i$, $i=1,2$, are $(p-1)$-assymptotically linear at infinity. Using variational techniques, an existence result is given.