Palais-Smale Condition of a functional

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Palais-Smale condition: Defined as in the strong formulation wiki link: https://en.wikipedia.org/wiki/Palais%E2%80%93Smale_compactness_condition



I am unable to check how the following functional $J$ satisfies the PS condition:
Let $OmegasubsetmathbbR^N$, $Ngeq 1$ be a bounded open set and $J:H_0^1(Omega)tomathbbR$ defined for a given function $vin H_0^1(Omega)$ by
$$
J(u)=int_Omeganabla u
$$
where for $u>0$,
$$
F(x,u)=G(x,u+v)-G(x,v)
$$
such that
$$
|G(x,u)|leq c+eta|u|^2,text on OmegatimesmathbbR
$$
Then $J$ satisfies the PS condition, i.e. for any sequence $u_nin H_0^1(Omega)$ such that $J(u_n)$ is uniformly bounded and $J'(u_n)to 0$ has a strongly convergent sub-sequence in $H_0^1(Omega)$.



Can you please give a brief idea how to prove the above line.



Thank you very much.




pde sobolev-spaces calculus-of-variations fractional-sobolev-spaces




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    up vote
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    Palais-Smale condition: Defined as in the strong formulation wiki link: https://en.wikipedia.org/wiki/Palais%E2%80%93Smale_compactness_condition



    I am unable to check how the following functional $J$ satisfies the PS condition:
    Let $OmegasubsetmathbbR^N$, $Ngeq 1$ be a bounded open set and $J:H_0^1(Omega)tomathbbR$ defined for a given function $vin H_0^1(Omega)$ by
    $$
    J(u)=int_Omeganabla u
    $$
    where for $u>0$,
    $$
    F(x,u)=G(x,u+v)-G(x,v)
    $$
    such that
    $$
    |G(x,u)|leq c+eta|u|^2,text on OmegatimesmathbbR
    $$
    Then $J$ satisfies the PS condition, i.e. for any sequence $u_nin H_0^1(Omega)$ such that $J(u_n)$ is uniformly bounded and $J'(u_n)to 0$ has a strongly convergent sub-sequence in $H_0^1(Omega)$.



    Can you please give a brief idea how to prove the above line.



    Thank you very much.




    pde sobolev-spaces calculus-of-variations fractional-sobolev-spaces




    share|cite|improve this question





















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      Palais-Smale condition: Defined as in the strong formulation wiki link: https://en.wikipedia.org/wiki/Palais%E2%80%93Smale_compactness_condition



      I am unable to check how the following functional $J$ satisfies the PS condition:
      Let $OmegasubsetmathbbR^N$, $Ngeq 1$ be a bounded open set and $J:H_0^1(Omega)tomathbbR$ defined for a given function $vin H_0^1(Omega)$ by
      $$
      J(u)=int_Omeganabla u
      $$
      where for $u>0$,
      $$
      F(x,u)=G(x,u+v)-G(x,v)
      $$
      such that
      $$
      |G(x,u)|leq c+eta|u|^2,text on OmegatimesmathbbR
      $$
      Then $J$ satisfies the PS condition, i.e. for any sequence $u_nin H_0^1(Omega)$ such that $J(u_n)$ is uniformly bounded and $J'(u_n)to 0$ has a strongly convergent sub-sequence in $H_0^1(Omega)$.



      Can you please give a brief idea how to prove the above line.



      Thank you very much.




      pde sobolev-spaces calculus-of-variations fractional-sobolev-spaces




      share|cite|improve this question











      Palais-Smale condition: Defined as in the strong formulation wiki link: https://en.wikipedia.org/wiki/Palais%E2%80%93Smale_compactness_condition



      I am unable to check how the following functional $J$ satisfies the PS condition:
      Let $OmegasubsetmathbbR^N$, $Ngeq 1$ be a bounded open set and $J:H_0^1(Omega)tomathbbR$ defined for a given function $vin H_0^1(Omega)$ by
      $$
      J(u)=int_Omeganabla u
      $$
      where for $u>0$,
      $$
      F(x,u)=G(x,u+v)-G(x,v)
      $$
      such that
      $$
      |G(x,u)|leq c+eta|u|^2,text on OmegatimesmathbbR
      $$
      Then $J$ satisfies the PS condition, i.e. for any sequence $u_nin H_0^1(Omega)$ such that $J(u_n)$ is uniformly bounded and $J'(u_n)to 0$ has a strongly convergent sub-sequence in $H_0^1(Omega)$.



      Can you please give a brief idea how to prove the above line.



      Thank you very much.





      pde sobolev-spaces calculus-of-variations fractional-sobolev-spaces





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