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Vietnam Journal of Mathematics 33:4 (2005) 391-408  Mountain Pass Theorem and Nonuniformly Elliptic Equations Nguyen Thanh Vu Abstract.  In this paper we improve Mountain Pass Theorem and Saddle Point Theorem. Our results only require that the functionals belong to $C^1_w(E)$ instead of $C^1(E)$, where $C^1_w(E)$ is the set of functionals that are weakly continuously differentiable on the Banach space $E$. An application is the existence of infinitely many generalized solutions to a nonuniformly nonlinear elliptic equation of the form $-div(a(x,\nabla u))=f(x,u)$ in $ \Omega$ with $u\in W^{1,p}_0 (\Omega)$. Here $a$ satisfies $|a(x,\xi)|\leqslant c_0 h(x)(1 + |\xi|^{p-1})$ for any $\xi$ in $\mathbb R^{n}$, a.e. $x\in\Omega$, where $h\in L^{\frac{p}{p-1}}(\Omega)$.

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