Nehari-type ground state solutions for asymptotically periodic bi-harmonic Kirchhoff-type problems in $\mathbb{R}^N$

TL;DR

This work studies Nehari-type ground state solutions for a nonlocal Kirchhoff-type biharmonic equation on $\mathbb{R}^N$ with $a>0$, $b\ge0$, $N\ge5$ and (asymptotically) periodic potentials $V$ and nonlinearities $f$. A variational framework on the Hilbert space $H^2(\mathbb{R}^N)$ is developed via the energy functional $J$, which incorporates the nonlocal Kirchhoff term and the quadratic form with $V$, and the problem is tackled by minimizing on the Nehari manifold $\mathcal N=\{u\neq0: J'(u)u=0\}$. The authors prove the existence of a nontrivial ground state with $J(u)=c_{\mathcal N}=\inf_{\mathcal N}J=\inf_{u\neq0}\max_{t\ge0}J(tu)$ in the periodic setting, using Palais–Smale sequences and concentration-compactness arguments to extract a nonzero limit solving $J'(u)=0$. In the asymptotically periodic case, a limiting problem $J_0$ is employed to compare minimax levels, yielding a nontrivial ground state for the original problem as well. Overall, the results extend Kirchhoff-type theory to the biharmonic operator in periodic and asymptotically periodic environments, under appropriate growth and monotonicity assumptions on $f$ and $V$.

Abstract

We investigate the following Kirchhoff-type biharmonic equation \begin{equation}\label{pr} \left\{ \begin{array}{ll} Δ^2 u+ \left(a+b\int_{\mathbb{R}^N}|\nabla u|^2d x\right)(-Δu+V(x)u)=f(x,u),\quad x\in \mathbb{R}^N,\\ u\in H^{2}(\mathbb{R}^N), \end{array} \right. \end{equation} where $a>0$, $b\geq 0$ and $V(x)$ and $f(x, u)$ are periodic or asymptotically periodic in $x$. We study the existence of Nehari-type ground state solutions of the problem just above with $f(x,u)u-4F(x,u)$ sign-changing, where $F(x,u):=\int_0^uf(x,s)d s$. We significantly extend some results from the previous literature.

Theorems & Definitions (23)

• Theorem 1.1
• Theorem 1.2
• Lemma 1.3
• proof
• Corollary 1.4
• Corollary 1.5
• Lemma 1.6
• proof
• Lemma 1.7
• proof