Concentrating bound states for Kirchhoff type problems in \({\mathbb R}^3\) involving critical Sobolev exponents (Q2923149)

The following problem is studied in the paper NEWLINE\[NEWLINE -\left(\epsilon^2a+\epsilon b\int_{\mathbb{R}^3}\left|\nabla u\right|^2\right)\Delta u +V(z)u =f(u)+u^5, \;\;z\in\mathbb{R}^3, u\in H^1(\mathbb{R}^3), u>0, NEWLINE\]NEWLINE where \(\epsilon>0\) is a small parameter and \(a,b>0\) are constants. It is assumed that \(V\) is a locally Hölder continuous function which is bounded from below by a positive constant and \(\inf_\Lambda<\min_{\partial \Lambda}V\) for some open bounded set \(\Lambda\) in \(\mathbb{R}^3\). Here, the nonlinearity \(f\in C^1(\mathbb{R}^+,\mathbb{R})\) is superlinear and subcritical. More precisely, it is assumed that \(f(s)=o(s^3)\) as \(s\rightarrow0^+\), \(f(s)/s^3\) is strictly increasing for \(s>0\) and \(\lambda s^{q_1}\leq f(s)\leq C\left(1+\left|s\right|^{q-1}\right)\) for some constants \(\lambda,C>0\), \(3\leq q_1<5\) and \(4<q<6\).NEWLINENEWLINEThe first main result states that for \(\epsilon>0\) sufficiently small there is a weak soution \(u_\epsilon\) such that \(u_\epsilon\leq\alpha\exp(-\beta\left|z-z_\epsilon\right|/\epsilon)\) where \(\alpha\) and \(\beta\) are positive constants and \(z_\epsilon\) is a maximum \(z_\epsilon\in\Lambda\) of \(u_\epsilon\) which in addition satisfies \(V(z_\epsilon)\rightarrow\inf_\Lambda V\) as \(\epsilon\rightarrow0\). The second main result is about the multiplicity of weak solutions under the additional assumption that \(V\) attains a global minimum on \(\Lambda\). The proof here is mainly based on Ljusternik-Shnirelmann theory.