## An existence result for a class of superlinear $$p$$-Laplacian semipositone systems.(English)Zbl 1046.34043

The authors consider the following system (*) $$-( r^{N-1}\phi (u^{\prime })) ^{\prime }=\lambda r^{N-1}f(v);$$ $$a<r<b;$$ $$-( r^{N-1}\phi (v^{\prime })) ^{\prime }=\lambda r^{N-1}g(u);$$ $$a<r<b;$$ $$u(a)=0=u(b);$$ $$v(a)=0=v(b),$$ where $$\phi (s)=\alpha (s^{2})s$$ is an odd, increasing homeomorphism of the real line, $$\lambda >0.$$ They consider the following conditions; (A1) For each $$c>0$$ there exists a constant $$A_{c}>0$$ such that $$\phi ^{-1}(cx)\geq A_{c}\phi ^{-1}(x)$$ $$\forall x\geq 0$$ and $$A_{c}\to \infty$$ as $$c\to \infty .$$ (A2) $$f,g:[ 0,\infty ) \to \mathbb{R}$$ are continuous, and there exists $$M>0$$ such that $$f(z)\geq -M/2$$ and $$g(z)\geq -M/2$$ for $$z\in [ 0,\infty ) .$$ (A3) $$\lim_{z\to \infty }h^{*}(z)/\phi (z)=\infty$$ and $$\lim_{z\to \infty }A_{h^{*}(z)}/z=\infty$$, where $$h^{*}(z)=\inf_{w\geq z}\{ \min (f(w),$$ $$g(w))\}$$ and $$A_{c}$$is defined as in (A1). Using Leray-Schauder degree theory, they prove that if the conditions (A1), (A2) and (A3) hold, than there exists $$\lambda ^{*}>0$$ such that for $$0<\lambda <\lambda ^{*},$$ system (*) has at least one positive solution. An interesting example is given.

### MSC:

 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations

### Keywords:

existence result; semipositone system; degree theory