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Multi-bump solutions for a Kirchhoff problem type (1507.07361v1)
Published 27 Jul 2015 in math.AP
Abstract: In this paper, we are going to study the existence of solution for the following Kirchhoff problem $$ \left{ \begin{array}{l} M\biggl(\displaystyle\int_{\mathbb{R}{3}}|\nabla u|{2} dx +\displaystyle\int_{\mathbb{R}{3}} \lambda a(x)+1)u{2} dx\biggl) \biggl(- \Delta u + (\lambda a(x)+1)u\biggl) = f(u) \mbox{ in } \,\,\, \mathbb{R}{3}, \ \mbox{}\ u \in H{1}(\mathbb{R}{3}). \end{array} \right. $$ Assuming that the nonnegative function $a(x)$ has a potential well with $int (a{-1}({0}))$ consisting of $k$ disjoint components $\Omega_1, \Omega_2, ....., \Omega_k$ and the nonlinearity $f(t)$ has a subcritical growth, we are able to establish the existence of positive multi-bump solutions by variational methods.