On entire solutions of second order semilinear elliptic equations (Q1078403)

The authors investigate the existence of the solutions \(u\in C^ 2({\mathbb{R}}^ n)\) of the equation \[ (1)\quad \Delta u+f(x,u,\nabla u)=0. \] The function f is locally Hölder continuous and such that \(| f(x,u,p)| \leq \rho (\Omega,M)(1+| p|^ 2)\) for any bounded domain \(\Omega \subset {\mathbb{R}}^ n\) and \(0\leq u\leq M\), \(p\in {\mathbb{R}}^ n\). The main result is the following theorem. Suppose that there exist a locally Hölder continuous function \(\phi\) (r) and continuously differentiable function F(u,t) such that \(| f(x,u,p)| \leq \phi (| x|)F(u,| p|),\quad \int^{\infty}_{0}r \phi (r) dr<\infty\) and one of the following conditions is satisfied: 1) F(u,t) is nondecreasing in u and t, and \(\lim_{u\to 0}F(u,t)/u=0\); 2) F(u,t) is nondecreasing in u and t, and \(\lim_{u\to \infty}F(u,t)/u=0\); 3) F(u,t) is nonincreasing in u and nondecreasing in t, but F(u,u) is nonincreasing. Then (1) possesses infinitely many positive solutions which are bounded and bounded away from zero. Furthermore, if f(x,u,p)\(\geq 0\) (or f(x,u,p)\(\leq 0)\) then (1) possesses infinitely many positive solutions each of which is bounded and tends to a positive constant as \(| x| \to \infty\).