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Nontrivial solutions of Kirchhoff type problems. (English) Zbl 1251.35027
The paper is concerned with the problem \[ \left\{ \begin{aligned} -\left(a+b\int_\Omega|\nabla u|^2dx\right)\Delta u &=f(x,u)\quad \text{in }\Omega, \\ u &=0\quad\text{on }\partial\Omega, \end{aligned} \right. \] 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 \(|u|\to\infty\).

35J60 Nonlinear elliptic equations
47J30 Variational methods involving nonlinear operators
35A15 Variational methods applied to PDEs
Full Text: DOI
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