×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Perera, K.; Zhang, Z., Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. differential equations, 221, 1, 246-255, (2006) · Zbl 1357.35131
[2] Zhang, Z.; Perera, K., Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. math. anal. appl., 317, 2, 456-463, (2006) · Zbl 1100.35008
[3] Mao, A.; Zhang, Z., Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition, Nonlinear anal., 70, 3, 1275-1287, (2009) · Zbl 1160.35421
[4] He, X.-m.; Zou, W.-m., Multiplicity of solutions for a class of Kirchhoff type problems, Acta math. appl. sin. engl. ser., 26, 3, 387-394, (2010) · Zbl 1196.35077
[5] Perera, K., Nontrivial critical groups in \(p\)-Laplacian problems via the Yang index, Topol. methods nonlinear anal., 21, 2, 301-309, (2003) · Zbl 1039.47041
[6] Liu, S.; Li, S., Existence of solutions for asymptotically ‘linear’ \(p\)-Laplacian equations, Bull. London math. soc., 36, 1, 81-87, (2004) · Zbl 1088.35025
[7] Bartsch, T.; Li, S., Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear anal., 28, 3, 419-441, (1997) · Zbl 0872.58018
[8] Jiu, Q.; Su, J., Existence and multiplicity results for Dirichlet problems with \(p\)-Laplacian, J. math. anal. appl., 281, 2, 587-601, (2003) · Zbl 1146.35358
[9] Wang, Z.Q., On a superlinear elliptic equation, Ann. inst. H. Poincaré anal. non linéaire, 8, 1, 43-57, (1991) · Zbl 0733.35043
[10] Liu, S., Existence of solutions to a superlinear \(p\)-Laplacian equation, Electron. J. differential equations, No. 66, 6 pp, (2001), electronic
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.