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| In the present talk we consider the following quasilinear elliptic equation
$ -\Delta_p u=
|u|^{p^*-2}u+g(u), \hbox{ in } \Omega $ coupled with Dirichlet boundary condition, where $\Omega$ is a bounded domain of $\mathbb{R}^N$ with smooth boundary $\partial \Omega$, $g$ is a continuous function with suitable growth condition. We will prove the existence of a weak solution for problem by combining semicontinuity argument with direct methods of calculus of variations. The existence of a local minimum for the energy functional is ensured provided a suitable algebraic inequality is fulfilled. |
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