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Superlinear indefinite elliptic problems and nonlinear Liouville theorems. (English) Zbl 0816.35030
The authors study the boundary value problem \begin{aligned} \sum_{i,j=1}^ N a_{ij} (x) {{\partial^ 2 u} \over {\partial x_ i \partial x_ j}}+ \sum_{i=1}^ N b_ i(x) {{\partial u} \over {\partial x_ i}}+ a(x) g(u) &= 0\quad \text{in }\Omega,\\ \sum_{j,k=1}^ N \nu_ j a_{jk} {{\partial u} \over {\partial x_ k}}+ \alpha(x)u &= 0 \quad \text{on } \partial\Omega. \end{aligned} Here the above differential operator is uniformly elliptic, $$\alpha(x) \geq 0$$ on $$\partial\Omega$$, but the coefficient $$a(x)$$ may change sign. The nonlinearity $$g$$ is assumed to be $$C^ 1$$ with $$g(0)= g'(0) =0$$, $$g(s)>0$$ for large $$s>0$$, and such that the limit $$s^{-p} g(s)$$ exists, as $$s\to\infty$$, for some $$p>1$$. The main existence result then states that the above boundary value problem has a solution if $$1<p< (N+2)/ (N-1)$$. Many interesting additional statements are given, mainly for the model equation $$\Delta u- m(x)u+ a(x) g(u)=0$$.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations
##### Keywords:
nonlinear Liouville theorems
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