On the existence of stationary solutions for higher-order \(p\)-Kirchhoff problems (Q2923451)

In this paper, the authors investigate the eigenvalue \(p\)-Kirchhoff Dirichlet problem NEWLINE\[NEWLINE \begin{cases} M(\| u\|^p)\Delta_p^L u = \lambda\{\gamma\| u\|_{p,w}^{p(\gamma-1)} w(x)| u|^{p-2}u+f(x,u)\} \quad\text{in }\Omega \\ D^\alpha u_k|_{\partial \Omega}=0 \quad\text{for all }\alpha\text{ with }|\alpha|\leq L-1\text{ and all } k =1,\dots, d\end{cases} \eqno{(1.1)} NEWLINE\]NEWLINE where \(\Omega\subseteq\mathbb R^n\), \(n\geq 1\), is a bounded domain, \(u=(u_1,\dots u_d)\), \(d\geq 1\), \(p >1\), \(L\in\mathbb N\), \(\lambda\in\mathbb R\), \(\alpha\) is a multi-index, \(\gamma\in[1,p_L^*/p)\), and \(p_L^*\) is the critical Sobolev exponent. The vectorial \(p\)-polyharmonic operator \(\Delta_p^L\) is defined by NEWLINENEWLINE\[NEWLINE \Delta_p^L \varphi=\begin{cases} \mathcal{D}_L(|\mathcal{D}_L\varphi|^{p-2}\mathcal{D}_L\varphi )\quad &\text{if }L=2j,\\ -\operatorname{div}\{\Delta^{j-1}(|\mathcal{D}_L\varphi|^{p-2}\mathcal{D}_L\varphi )\} \quad &\text{if }L=2j-1, \end{cases} j=1,2,\dots, NEWLINE\]NEWLINENEWLINE for all \(\varphi=(\varphi_1,\dots, \varphi_d)\in C^\infty_0(\Omega)\), where \(D_L\) denotes the vectorial operator NEWLINENEWLINE\[NEWLINED_L \varphi=\begin{cases} (\Delta^j \varphi_1,\dots, \Delta^j\varphi_d)\, \quad &\text{if }L=2j, \\ (D\Delta^{j-1} \varphi_1,\dots, D\Delta^{j-1}\varphi_d) \quad &\text{if }L=2j-1, \end{cases} \text{for }j=1,2,\dots. \eqno{(1.3)} NEWLINE\]NEWLINENEWLINE The weight function \(w\) is positive a.e. in \(\Omega\) and NEWLINE\[NEWLINE w\in L^\infty(\Omega),\quad \overline{w} > \frac{n}{n-\gamma[n-Lp]^+}.\eqno(1.4) NEWLINE\]NEWLINENEWLINE The Kirchhoff function \(M: \mathbb R_0^+\to \mathbb R_0^+\) is assumed to verify the general structural assumption \((\mathcal{M})\) \(M\) is continuous, non-decreasing and there exists \(s>0\) such that NEWLINENEWLINE\[NEWLINE s \gamma \tau^\gamma\leq \tau M(\tau)\quad \text{for all } \tau\in\mathbb R^+. NEWLINE\]NEWLINE We denote \(\mathcal{M}(\tau)=\int_0^\tau M(z)\, dz\) and let \(\lambda_1\), \(u_1\) be the first eigenvalue and the first eigenfunction, respectively, of the corresponding eigenvalue problem.NEWLINENEWLINENEWLINEThe nonlinearity \(f\) verifies the following condition.NEWLINENEWLINENEWLINE\((\mathcal{F})\) Let \(f : \Omega\times\mathbb R^d\to\mathbb R^d\), \(f=f(x,v)\not\equiv 0\), be a Carathéodory function, which admits a potential \(F : \Omega\times\mathbb R^d\to\mathbb R\), \(f= D_vF\), with \(F(x,0)=0\) a.e. in \(\Omega\), satisfying the following properties.NEWLINE{\parindent=0.5cm \begin{itemize}\item[(a)] There exist \(q\in(1,\gamma p)\) and \(C_f>0\) such that NEWLINE\[NEWLINE| f(x,v)|\leq C_f w(x) (1+| v|^{q-1})\quad\text{for a.a. }x\in\Omega \text{ and all } v\in\mathbb R^d. NEWLINE\]NEWLINENEWLINE\item[(b)] There exists \(\mathbf{p}^\ast\in (\gamma p, p_L^\ast/\overline{w}')\) such that NEWLINE\[NEWLINE\limsup_{| v| \to 0} \frac{| f(x,v)\cdot v|}{w(x)| v|^{\mathbf{p}^\ast}}< \infty, NEWLINE\]NEWLINE uniformly a.e. in \(\Omega\). \item[(c)] There holds NEWLINE\[NEWLINE \int_\Omega F(x,u_1)\,dx > \frac{1}{p}\left( \frac{\mathcal{M}(\lambda_1)}{s \lambda_1^\gamma}-1\right). NEWLINE\]NEWLINENEWLINENEWLINE\end{itemize}} Using these assumptions, the authors define in the preliminary part two numbers \(\lambda_\ast\), \(\lambda^\ast\) and then formulate the main result.NEWLINENEWLINENEWLINETheorem 2.1 Let \((\mathcal{F})\)-\((a), (b)\) hold. {\parindent=0.5cm \begin{itemize}\item[(i)] If \(\lambda\in[0,\lambda_\ast)\), then (1.1) has only the trivial solution. \item[(ii)] If furthermore \((\mathcal{F})\)-\((c)\) holds and \(q\in (1,p)\) in \((\mathcal{F})\)-\((a)\), then (1.1) admits at least two nontrivial solutions for every \(\lambda\in(\lambda^\ast,s \lambda_1^\gamma)\).NEWLINENEWLINE\end{itemize}} To prove this result, the author uses a three critical points theorem given in [\textit{F. Colasuonno} et al., Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75, No. 12, 4496--4512 (2012; Zbl 1251.35059)].