Positive solutions for a system of nonlinear Hammerstein integral equations and applications

We are concerned with the existence and multiplicity of positive solutions for the system of nonlinear Hammerstein integral equationsu(t)=∫abk1(t,s)g1(s)f1(s,u(s),v(s))ds,v(t)=∫abk2(t,s)g2(s)f2(s,u(s),v(s))ds,where ki∈C[a,b]×[a,b],R+,fi∈C[a,b]×R+2,R+, and gi∈C[a,b]gi∈C[a,b] is almost everywhere positive on [a,b](i=1,2)[a,b](i=1,2). We overcome the difficulty arising from the difference between the two kernels k1(t,s)g1(s)k1(t,s)g1(s) and k2(t,s)g2(s)k2(t,s)g2(s) by defining certain integral constants, and use fixed point index theory to establish our main results, based on a priori estimates achieved by utilizing Jensen’s inequality for concave functions and nonnegative matrices. Our nonlinearities f and g cover the following three cases: the first with both superlinear, the second with both sublinear, and the last with one sublinear and the other superlinear. Our main results are applied to establish the existence and multiplicity of positive symmetric solutions for an elliptic system in an annulus.