Semilinear biharmonic problem

Semilinear biharmonic problem

A semilinear biharmonic problem with so-called Navier boundary conditionsis

where $\mathrm{\Omega }\subset {ℝ}^N(N\ge 1)$ is an open bounded domain,$f(x,v)\in {𝒞}^1(\overline{\mathrm{\Omega }}\times ℝ;ℝ)$ in $v\in ℝ$.

Numerical results for the case $\mathrm{\Omega }=(-2,2)\times (-2,2)$ and$f(x,v)=v^p$: p=0.1, 0.7, 1.3, 7. Note that $u=-\mathrm{\Delta }v$.