Abstract
This paper concerns the existence and multiplicity of solutions for the Schrődinger–Kirchhoff type problems involving the fractional p–Laplacian and critical exponent. As a particular case, we study the following degenerate Kirchhoff-type nonlocal problem:
$$\begin{align}& \left\| u \right\|_{\lambda }^{\left( \theta -1 \right)p}\left[ \lambda \left( -\Delta \right)_{p}^{s}u+V\left( x \right){{\left| u \right|}^{p-2}}u \right]={{\left| u \right|}^{p_{s}^{\star }-2}}u+f\left( x,u \right)\,in\,{{\mathbb{R}}^{N}}, \\ & {{\left\| u \right\|}_{\lambda }}={{\left( \lambda \int\limits_{\mathbb{R}}{\int\limits_{2N}{\frac{{{\left| u\left( x \right)-u\left( y \right) \right|}^{p}}}{{{\left| x-y \right|}^{N+ps}}}}dxdy+\int\limits_{{{\mathbb{R}}^{N}}}{V\left( x \right){{\left| u \right|}^{p}}dx}} \right)}^{{1}/{p}\;}} \\ \end{align}$$
where
$\left( -\Delta \right)_{p}^{s}$is the fractional p–Laplacian with 0 < s < 1 < p < N/s,
$p_{s}^{\star }={Np}/{\left( N-ps \right)}\;$is the critical fractional Sobolev exponent, λ > 0 is a real parameter,
$1<\theta \le {p_{s}^{\star }}/{p}\;,$and f : ℝN × ℝ → ℝ is a Carathéodory function satisfying superlinear growth conditions. For
$\theta \in \left( 1,{p_{s}^{\star }}/{p}\; \right),$by using the concentration compactness principle in fractional Sobolev spaces, we show that if f(x, t) is odd with respect to t, for any m ∈ ℕ+ there exists a Λm > 0 such that the above problem has m pairs of solutions for all λ ∈ (0, Λm]. For
$\theta ={p_{s}^{\star }}/{p}\;,$by using Krasnoselskii’s genus theory, we get the existence of infinitely many solutions for the above problem for λ large enough. The main features, as well as the main difficulties, of this paper are the facts that the Kirchhoff function is zero at zero and the potential function satisfies the critical frequency infx∈ℝ V(x) = 0. In particular, we also consider that the Kirchhoff term satisfies the critical assumption and the nonlinear term satisfies critical and superlinear growth conditions. To the best of our knowledge, our results are new even in p–Laplacian case.
Bound state solutions of Kirchhoff type problems with critical exponent
