# Biharmonic equation

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

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

• the potential has: $0<a\leq V(x)\leq b$ – Pádua Jan 4 at 18:22
• 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. – Piotr Hajlasz Jan 4 at 19:48
• I included. Thank you Piotr Hajlasz. – Pádua Jan 5 at 3:20
• If it is possible, it would also be good if you could add context of your question. – Piotr Hajlasz Jan 5 at 4:12
• I'm searching by regularity of solutions for this nonlinear schrodinger equation in whole space – Pádua Jan 5 at 17:30