Local superlinearity for elliptic systems involving parameters
The authors consider the nonlinear boundary value problem \[ -u''= f(t,u,v,a,b)\text{ in }(0,1), \qquad -v''= g(t,u,v,a,b)\text{ in }(0,1),\tag{1} \] \[ u(0)=u(1)=0,\qquad v(0)=v(1),\tag{2} \] where \(f\) and \(g\) are superlinear at the origin as well as at infinity; \(a\), \(b\) are nonnegative constants. The authors study existence, nonexistence, and multiplicity of positive solutions to (1), (2). As a main application of their results, the authors prove the existence and multiplicity of positive radial solutions of the semilinear elliptic system in annular domains \[ -\Delta u= h(|x|,u,v)\text{ in }A(r_1,r_2),\qquad -\Delta v= k(|x|,u,v)\text{ in }A(r_1,r_2), \] \[ (u,v)=(0,0)\text{ on }|x|=r_1,\qquad (u,v)=(a,b)\text{ on }|x|=r_2, \] where \(a\), \(b\) are nonnegative parameters, \(A(r_1,r_2)=\{x\in \mathbb{R}^N: r_1<|x|<r_2\}\), with \(N\geq 3\).
scientific article; zbMATH DE number 2157581
scientific article