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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 $$|u|\to\infty$$.

MSC:
 35J60 Nonlinear elliptic equations 47J30 Variational methods involving nonlinear operators 35A15 Variational methods applied to PDEs
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References:
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