Solutions for a quasilinear elliptic problem with indefinite nonlinearity with critical growth

Abstract

We are interested in nonhomogeneous problems with a nonlinearity that changes sign and may possess a critical growth as follows \begin{equation*} \begin{cases} -\operatorname{div}\left(a(|\nabla u|^p) |\nabla u|^{p-2} \nabla u \right) =\lambda|u|^{q-2}u+W(x)|u|^{r-2}u&\text{in}~\Omega,\\ u=0&\text{on}~\partial\Omega, \end{cases} \end{equation*} where $\Omega\subset\mathbb{R}^N$ is a bounded domain with smooth boundary $\partial\Omega$, $N\geq 2$, $1<p\leq q<N$, $q<r\leq q^*$, $\lambda\in\mathbb{R}$ and function $W$ is a weight function which changes sign in $\Omega$. Using variational methods, we prove the existence of four solutions: two solutions which do not change sign and two solutions which change sign exactly once in $\Omega$.

Keywords

• subcritical and critical exponents
• $p\&q$ laplacian operator
• indefinite problems