In this paper we will study the
existence of signed solutions for problems of the type
$-L u=\lambda h(x)|x|^{\delta}(u_{+})^q-|x|^{\gamma}(u_{-})^p, \quad $ in $\Omega$,
$u_{\pm}$ ≠0, $\quad u\in E,$ $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$
(P)
where $\Omega$ is either a whole space $\mathbb R^N$
or a bounded smooth domain, $Lu =:$ div$(|x|^{\alpha}|\nabla u|^{m-2}\nabla u), $ $\lambda >0, \quad0 < q < m-1 < p \leq m$*$-1,$ $\alpha, $ $\delta $ and $\gamma $
are real numbers, $ N> m-\alpha, $ $m$*$=\frac{(\gamma+N)m}{(\alpha+N-m)}$, $h:\Omega \rightarrow \mathbb R$ is a positive continuous function, $u_{\pm}=\max \{\pm
u,0\}$ and $E$ is a Banach space that will be defined later on. We
will show that (P) has a solution that changes sign in several
situations. The proof of the main results are done by using
variational methods applied to the energy functional associated to
$(P)$.
{{journal.titleEn}} 
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