Infinitely many positive solutions for Kirchhoff-type problems.(English)Zbl 1157.35382

Summary: This paper is concerned with the existence of infinitely many positive solutions to a class of Kirchhoff-type problems $$-(a+b\int _{\varOmega}|\nabla u|^2\, dx)\varDelta u = \lambda f(x,u)$$ in $$\varOmega$$ and $$u=0$$ on $$\partial \varOmega$$, where $$\varOmega$$ is a smooth bounded domain of $$\mathbb R^N$$, $$a ,b> 0 , \lambda > 0$$ and $$f: \varOmega \times \mathbb R \to \mathbb R$$ is a Carathéodory function satisfying some further conditions. We obtain a sequence of a.e. positive weak solutions to the above problem tending to zero in $$L^\infty (\varOmega)$$ with $$f$$ being more general than that of [K. Perera and Z. Zhang, J. Differ. Equations 221, No. 1, 246–255 (2006; Zbl 1357.35131); J. Math. Anal. Appl. 317, No. 2, 456–463 (2006; Zbl 1100.35008)].

MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 35J20 Variational methods for second-order elliptic equations 35D05 Existence of generalized solutions of PDE (MSC2000)

Citations:

Zbl 1100.35008; Zbl 1357.35131
Full Text:

References:

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