Existence of solutions for a nonhomogeneous biharmonic problem with critical Sobolev exponent (Q1812198)

Let \(\Omega\) be a bounded smooth domain in \(\mathbb R^N\), \(N\geq 5\), and \(p- \frac{2N}{N-4}\). The author shows that, the biharmonic problem \[ \Delta^2u= |u|^{p-2}u+ f(u)\quad \text{in }\Omega, \qquad \Delta u=u=0\quad \text{on }\partial\Omega \tag{1} \] possesses at least two solutions in \(H=H^2(\Omega)\cap H_0^1(\Omega)\) if \(f\in H^*\) and is not identically zero, and satisfies some natural conditions.