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The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions. (English) Zbl 1214.35077
Let $\Omega\subset {\mathbb R}^N$ be a bounded domain with smooth boundary and assume that $h:\overline\Omega\times {\mathbb R}\rightarrow{\mathbb R}$ is a continuous function. Assume $a$ and $b$ are positive numbers and denote $M(t):=at+b$. This paper is concerned with the qualitative analysis of solutions to the nonlinear stationary problem $$-M\left(\int_\Omega |\nabla u|^2dx\right)\Delta u=h(x,u)\qquad x\in\Omega\,,$$ subject to the Dirichlet boundary condition $u=0$ on $\partial\Omega$. The main objective of the present paper is to establish the existence of multiple solutions to this class of Kirchhoff-type problems. This is done by means of variational methods which combine the qualitative analysis on Nehari manifolds with the fibering map method.

35R09Integro-partial differential equations
35J60Nonlinear elliptic equations
35J20Second order elliptic equations, variational methods
35A01Existence problems for PDE: global existence, local existence, non-existence
Full Text: DOI
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