Nontrivial solutions of Kirchhoff type problems (Q427657)

The paper is concerned with the problem NEWLINE\[NEWLINE \left\{ \begin{aligned} -\left(a+b\int_\Omega|\nabla u|^2dx\right)\Delta u &=f(x,u)\quad \text{in }\Omega, \\ u &=0\quad\text{on }\partial\Omega, \end{aligned} \right. NEWLINE\]NEWLINE where \(\Omega\) is a smoothly bounded domain in \(\mathbb{R}^N\), \(N\leq 3\). The nonlinearity \(f\in C(\Omega\times \mathbb{R}, \mathbb{R})\) is subcritical. Using variational methods the existence of a solution is proved under various growth conditions on \(f\) near \(0\) and \(\infty\). In the first theorem, \(f\) is superlinear at \(0\) and satisfies \(f(x,u)\sim cu^3\) some \(c>0\). In the second theorem \(f\) is asymptotically linear near \(0\) and satisfies \(f(x,u)/u^3\to \infty\) as \(|u|\to\infty\).