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Existence results and a priori bounds for higher order elliptic equations and systems. (English) Zbl 1180.35214
By using the theory of topological degree, the author proves the existence of positive solutions of the homogeneous Dirichlet problem for some semilinear elliptic systems of the form:
\[ -L_{i}u_{i}=f_{i}(x,u_{1},\dots,u_{n}) \quad \text{in }\Omega\subset\mathbb R^n, \quad i=1,2,\dots,n, \] where \(L_{i}\) is a second order uniformly elliptic linear operator with variable coefficients. As a consequence, existence results are obtained for some concrete higher order semilinear elliptic equations. Particularly, the following Navier boundary problem for a \(2m\), \(m\geq 1\), – order elliptic equation is investigated:
\[ \begin{cases} (-L)^m u=\sum^{m-1}_{i=1}\alpha_{i}(-L)^{m-i}u+ f(x,u)&\text{in } \Omega,\\ (-L)^{k}{u}= 0 &\text{on } \partial\Omega, \;k=0,1,\dots,\;m-1,\end{cases}\tag{N} \] where \(L\) is second order operation with positive first eigenvalue \(\lambda_{1}=\lambda_{1}(L,\Omega)\) of the corresponding homogeneous Dirichlet problem. The notion of sublinearity and superlinearity is extended and the existence of positive solution of problem (N) is proved:
a) in the sub-linear case, e.a., if:
\[ \infty\geq \liminf_{v\rightarrow 0}\frac{f(x,v)}{v}>\lambda^*:=\max \bigg\{0,\lambda ^{m}_{1}- \sum^{m-1}_{i=1}\alpha_{i}\lambda_{1}^{m-i} \bigg\}>\limsup_{v\rightarrow{\infty}} \frac{f(x,v)}{v}; \] b) in the super-linear case by an additional subcritical growth condition:
\[ \lim_{v\rightarrow{\infty}}f(x,v){v}^{-p}=b(x)<\infty \quad \forall{x}\in\Omega, \quad \forall{p}< p^*:=(n+2m)/n-2m. \]

MSC:
35J47 Second-order elliptic systems
35J61 Semilinear elliptic equations
35B45 A priori estimates in context of PDEs
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35B09 Positive solutions to PDEs
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