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Variational elliptic problems which are nonquadratic at infinity. (English) Zbl 0820.35059

Let us consider the Dirichlet problem

-Δu=f(x,u)inΩ,u=0onΩ,(P)

where Ω is a bounded smooth domain in N (N1) and the nonlinearity f:Ω× is assumed to be a Caratheodory function with subcritical growth. Then, it is well known that the weak solutions uH 0 1 (Ω) of (P) are the critical points of the C 1 functional

J(u)=1 2 Ω |u| 2 dx- Ω F(x,u)dx,

where F(x,s)= 0 s f(x,t)dt. In this variational setting, the literature usually distinguishes between two situations for problem (P): the subquadratic situation, where the potential F satisfies lim sup |s| F(x,s)/s 2 c<, and the superquadratic situation, where F satisfies lim |s| F(x,s)/s 2 =. The main goal of this paper is to present a unified approach to both situations by means of a condition of nonquadraticity at infinity on F.

MSC:
35J65Nonlinear boundary value problems for linear elliptic equations
49K20Optimal control problems with PDE (optimality conditions)
35J20Second order elliptic equations, variational methods