  This paper studies nonlinear eigenvalues problems with a double non
homogeneity governed by the $p(x)$-Laplacian operator, under the Dirichlet
boundary condition on a bounded domain of $\mathbb{R}^N(N\geq2)$. According to
the type of the nonlinear part (sublinear, superlinear) we use the Lagrange
multiplier's method, the Ekeland's variational principle and the Mountain-Pass
theorem to show that the spectrum includes a continuous set of eigenvalues,
which can in some contexts be all the set $\mathbb{R_+^{*}}$. Moreover, we show
that the smallest eigenvalue obtained from the Lagrange multipliers is exactly
the first eigenvalue in the Ljusternik-Schnirelman eigenvalues sequence.
  Key words: Nonlinear eigenvalue problems, $p(x)$-Laplacian, Lagrange
multipliers, Ekeland variational principle, Ljusternik-Schnirelman principle,
Mountain-Pass theorem.
