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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