On the asymptotic properties and necessary conditions for existence of solutions of nonlinear second order elliptic equations

In this paper functions $u(x)$ satisfying the inequality $L(u)+k(x)f(u)\leqslant0$ in a domain $\Omega$ are studied. Here $L(u)$ is a linear second order elliptic operator with positive definite characteristic form, $k(x)\geqslant0$, and $f(u)$ is defined in an interval $u^-<u<u^+$, in which $f(u)>0$, $f'(u)\geqslant0$ and $\int_u^{u^+}\frac{ds}{f(s)}<\infty$.
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