Let us consider for $0<\alpha\leq V(x)\leq \beta$ and $0\leq K(x)<\gamma$ the equation \begin{equation}\label{\star} \Delta^2u+V(x)u=g(x, u)+K(x)u, \end{equation} where $|g(x,s)|\leq \varepsilon|s|^{1+\sigma}+c|s|^{p}$ with $1<\sigma<p<\frac{N+4}{N-4}$ and $s\geq 0.$

Is the solution $u$ of equation above in ${H}^4(\mathbb{R}^N)?$

  • $\begingroup$ the potential has: $0<a\leq V(x)\leq b$ $\endgroup$ – Pádua Jan 4 at 18:22
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    $\begingroup$ You should include assumptions about $V$ and $K$ in your problem. Including assumptions about $V$ is not okay. Also in the condition for $g$ you have $g(s)$ while in the problem $g$ depends on two variables. Correct it. $\endgroup$ – Piotr Hajlasz Jan 4 at 19:48
  • $\begingroup$ I included. Thank you Piotr Hajlasz. $\endgroup$ – Pádua Jan 5 at 3:20
  • $\begingroup$ If it is possible, it would also be good if you could add context of your question. $\endgroup$ – Piotr Hajlasz Jan 5 at 4:12
  • $\begingroup$ I'm searching by regularity of solutions for this nonlinear schrodinger equation in whole space $\endgroup$ – Pádua Jan 5 at 17:30

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