0
$\begingroup$

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 ${H}^4(\mathbb{R}^N)?$

$\endgroup$
  • $\begingroup$ the potential has: $0<a\leq V(x)\leq b$ $\endgroup$ – Pádua Jan 4 at 18:22
  • 1
    $\begingroup$ 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

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.