### Existence of positive solutions for a semilinear elliptic system

#### Abstract

In this paper, we are concerned with the existence of

(component-wise) positive solutions for a semilinear elliptic

system, where the nonlinear term is superlinear in one equation and

sublinear in the other equation. By constructing a cone $K_1 \times

K_2$ which is the Cartesian product of two cones in space

$C(\overline{\Omega})$ and computing the fixed point index in $K_1

\times K_2$, we establish the existence of positive solutions for

the system. It is remarkable that we deal with our problem on the

Cartesian product of two cones, in which the features of two

equations can be exploited better.

(component-wise) positive solutions for a semilinear elliptic

system, where the nonlinear term is superlinear in one equation and

sublinear in the other equation. By constructing a cone $K_1 \times

K_2$ which is the Cartesian product of two cones in space

$C(\overline{\Omega})$ and computing the fixed point index in $K_1

\times K_2$, we establish the existence of positive solutions for

the system. It is remarkable that we deal with our problem on the

Cartesian product of two cones, in which the features of two

equations can be exploited better.

#### Keywords

Positive solutions; elliptic system; fixed point index

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