By Lemmas 4.2 4.3, we have that the functional (mathcal{J}) satisfies ((mathit{PS})).
The functional (mathcal{F}) denoted by (1.6) satisfies the Palais-Smale condition.
Next, we will prove that the functional (mathcal{J}) defined by (4.1) satisfies ((mathit{PS})).
The functional (mathcal{F} u)) defined by (1.6) possesses a local minimum at (u=0).
Thus, we find a countable family of critical points of the functional (mathcal{F}) defined by (4.8) such that (c_{beta}=mathcal{F} u_{beta})) with (u_{beta}) a weak solution of problem (1.5).
Here we are interested in elliptic obstacle problems by minimizing the energy functional (mathcal{J}[u]=int_{Omega}( A x nabla ucdot nabla u + mathbf{f}cdotnabla u),dx) in the Sobolev spaces (uin W^{1,2}_{0}(Omega)) satisfying the admissible condition (uinmathcal {A}).
In fact, for every (uin E), we define the linear functional (mathcal{L}_{u}) by mathcal{L}_{u} v)= int_{mathbb {R}^{3}}K x u^{2}v,dx,quad forall vin mathcal{D}^{1,2}bigl(mathbb {R}^{3}bigr).
We further define the Lyapunov functional (mathcal{L}(t)) by mathcal{L}(t)=NE t)+N_{1}phi(t)+N_{2}psi(t)+N_{3} chi(t), where N, (N_{1}), (N_{2}), (N_{3}) are positive constants to be chosen later.
We define the functional (x^*) on (mathcal{V }) by begin{aligned} x^* mathcal{S }(P))=int _mathbb{T ^q}P mathbf{z}),dmu _q^* mathbf{z} -frac{1}{3m^q}sum _q^* mathbf{z} -frac{1}{3m^q}sum}_mathbf{k}le quad Pin m-1}P mathbf{yN. m-1}P mathbf{y
The form of functionals (1.9) and (1.10) is suggested by the functionals (1.7 - 1.8 1.7 - 1.8
