Reverse functional inequalities and their applications to nonlinear elliptic boundary value problems (Q2773668)

Considered is the elliptic problem NEWLINE\[NEWLINE \begin{gathered} \sum_{|\alpha|\leq 2m}a_{\alpha}(x) D^{\alpha}u(x)=\lambda f(x,u,Du,\dots,D^{2m-1}u) + z(x)\quad (\lambda>0,\;x\in \Omega\subset {\mathbb R}^n), \tag{1} \\ B_ju(x)=\sum_{|\alpha|<m_j}b_{j,\alpha}D^{\alpha}u(x)=0, \quad x\in\partial\Omega. \tag{2} \end{gathered} NEWLINE\]NEWLINE Conventionally we have NEWLINE\[NEWLINE (-1)^m\sum_{|\alpha|=2m}a_{\alpha}(x) \xi^{\alpha}\geq \mu |\xi|^{2m}\quad (\mu>0)\quad\text{~for all~}\xi \in {\mathbb R}^n,\;x\in \Omega. NEWLINE\]NEWLINE The main conditions for the function \(f(x,\xi)\) (\(\xi=(\xi_0,\xi_1,\dots,\xi_{2m-1})\)) are the growth conditions and the inequality NEWLINE\[NEWLINE f(x,\xi_0,\xi_1,\dots,\xi_{2m-1})\geq M(|\xi_0|), \;\;M(t)/t\to \infty\;\text{as} \;t\to\infty, NEWLINE\]NEWLINE which means that the function \(f\) is positive whenever the parameter \(\xi_0\) is sufficiently large. Under some additional requirements, the right-hand side of this inequality may also have the form \(\psi(x)M(|\xi_0|)\), with \(\psi\) a nonnegative function. Let the problem (1), (2) with \(\lambda=0\) have a unique solution \(u\in W_p^{2m}(\Omega)\) provided that \(z(x)\in L_p(\Omega)\) (\(p>1\)). The functions occurring in (1) and (2), and the boundary of \(\Omega\) are assumed to be sufficiently smooth. The authors demonstrate that, on some interval \(\lambda\in (0,\lambda_0)\), the problem (1), (2), where \(z(x)\in L_p(\Omega)\) (\(p>1\)), has two solutions \(u_{\lambda}, U_{\lambda}\in W_p^{2m}(\Omega)\) such that \(\|u_{\lambda}\|_{W_p^{2m}(\Omega)}\to 0\) and \(\|U_{\lambda}\|_{W_p^{2m}(\Omega)}\to \infty\) as \(\lambda\to 0\). It is also shown that if \(f(x,0)=0\) and \(\frac{\partial f}{\partial \xi_i}(x,0)=0\) then the problem (1), (2) is solvable for every \(\lambda>0\) and the number of solutions is even. The proof is based on the Leray-Schauder degree theory and some estimates for solutions of elliptic inequalities. A typical inequality of this type is the inequality NEWLINE\[NEWLINE |A(u)(x)|\leq \sum_{i=0}^{k_0}c_i|D^iu(x)|^{q_i},\quad x\in\Omega, NEWLINE\]NEWLINE where \(c_i,q_i>0\) and \(k_0<2m\) are some constants.