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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 C1w(E) instead of C1(E), where C1w(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,u)) = f(x, u) in Ω with u (Ω). Here a satisfies |a(x,ξ)|  c0 h(x)(1 + |ξ|p-1) for any ξ in Rn, a.e. , where ).

 

 

 

 

 

 

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