Monotone solutions of second-order nonlinear differential equations

Necessary and sufficient conditions are obtained for existence of monotone solutions of a nonlinear differential equation. As applications, several existence criteria and comparison theorems are derived.     

Positive solutions for second-order nonlinear differential equations

We prove the existence of positive solutions to the scalar equation y^″(x)+F(x,y,y^′)=0$y^{″}(x)+F(x,y,y^{\prime })=0$. Applications to semilinear elliptic equations in exterior domains are considered.     

Monotone positive solutions of second-order nonlinear differential equations

We obtain an existence theorem for monotone positive solutions of nonlinear second-order ordinary differential equations by using the Schauder–Tikhonov fixed point theorem. The result can also be applied to prove the existence of positive solutions of certain semilinear elliptic equations in $\text{R}{}^{\mathrm{n}}\text{(n⩾3)}$.     

Existence and uniqueness of solutions of nonlinear elliptic equations without growth conditions at infinity

In this paper, we consider the nonlinear elliptic problem $$ - \Delta u + {\left| u \right|^{p - 1}}u + {\left| {\nabla u} \right|^q} = f$$ in ? ^N , where p > 1 and q > 0. We show that if f ?? L _loc ^r (? ^N ) for suitable r ?? 1, then there exists a distributional solution of the equation, independently of the behavior of f at infinity. We also analyze the uniqueness of this solution in some cases.     

Oscillation of solutions of second-order nonlinear self-adjoint differential equations

We are concerned with the oscillation problem for the nonlinear self-adjoint differential equation (a(t)x′)′+b(t)g(x)=0. Here g(x) satisfied the signum condition xg(x)>0 if x≠0, but is not imposed such monotonicity as superlinear or sublinear. We show that certain growth conditions on g(x) play an essential role in a decision whether all nontrivial solutions are oscillatory or not. Our main theorems extend recent results in a serious of papers and are best possible for the oscillation of solutions in a sense. To accomplish our results, we use Sturm's comparison method and phase plane analysis of systems of Liénard type. We also explain an analogy between our results and an oscillation criterion of Kneser-Hille type for linear differential equations.     

On maximum principles for a class of nonlinear second-order elliptic equations

In a recent paper [3] the authors derived maximum principles which involved $\text{u(x)}\text{and}\text{q = ¦}\text{grad}\text{u¦}$, where u(x) is a classical solution of an alliptic differential equation of the form ($\text{g(q}{}^2\text{)u}{}_{\mathrm{,i}}\text{)}{}_{\mathrm{,i}}\text{+ ?(u) ?(q}{}^2\text{) = 0}$. In this paper these results are extended to the more general case in which $\text{g = g(u, q}{}^2\text{)}\text{and}\text{?(u) ?(q}{}^2\text{)}$ is replaced by h(u, q²).