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Reverse functional inequalities and their applications to nonlinear elliptic boundary value problems. (Russian, English) Zbl 0991.35033

Sib. Mat. Zh. 42, No. 4, 781-795 (2001); translation in Sib. Math. J. 42, No. 4, 656-667 (2001).
Considered is the elliptic problem \[ \begin{gathered} \sum_{|\alpha|\leq 2m}a_{\alpha}(x) D^{\alpha}u(x)=\lambda f(x,u,Du,\dots,D^{2m-1}u) + z(x)\quad (\lambda>0,\;x\in \Omega\subset {\mathbb R}^n), \tag{1} \\ B_ju(x)=\sum_{|\alpha|<m_j}b_{j,\alpha}D^{\alpha}u(x)=0, \quad x\in\partial\Omega. \tag{2} \end{gathered} \] Conventionally we have \[ (-1)^m\sum_{|\alpha|=2m}a_{\alpha}(x) \xi^{\alpha}\geq \mu |\xi|^{2m}\quad (\mu>0)\quad\text{~for all~}\xi \in {\mathbb R}^n,\;x\in \Omega. \] 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 \[ f(x,\xi_0,\xi_1,\dots,\xi_{2m-1})\geq M(|\xi_0|), \;\;M(t)/t\to \infty\;\text{as} \;t\to\infty, \] 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 \[ |A(u)(x)|\leq \sum_{i=0}^{k_0}c_i|D^iu(x)|^{q_i},\quad x\in\Omega, \] where \(c_i,q_i>0\) and \(k_0<2m\) are some constants.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35J40 Boundary value problems for higher-order elliptic equations
35J60 Nonlinear elliptic equations
39B62 Functional inequalities, including subadditivity, convexity, etc.
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