Existence of multiple positive solutions of semilinear elliptic equations in Esteban-Lions domains with holes (Q1026969)

Let \(\Omega\) be the union of an infinite cylinder in \({\mathbb{R}}^{N}\) with the axis parallel to \((0,0, \dots,0,1)\) and circular basis of radius \(r\) sitting on the plane orthogonal to \((0,0, \dots,0,1)\), and of a hemisphere of radius \(r\) attached to the basis, minus a hole. Let \(h\), \(K\) be real valued functions in \(\Omega\). Let \(\lambda\geq 0\) be a parameter. Let \(N\geq 2\), \(1<p<2^{*}-1\). Here \(2^{*}\) denotes the (Sobolev) conjugate exponent of \(2\). Under sutiable assumptions on \(h\), \(K\), the authors show that there exists \(\lambda^{*}\) such that the problem \[ -\Delta u(x)+u(x)=\lambda K(x) u^{p}+h(x)\qquad\forall x\in\Omega,\qquad u>0,\qquad u\in H^{1}_{0}(\Omega) \] admits a unique solution for \(\lambda=0\) and \(\lambda=\lambda^{*}\), two solutions for \(0<\lambda<\lambda^{*}\), no solution for \(\lambda>\lambda^{*}\). The authors also provide some estimate on \(\lambda^{*}\) and study the behaviour of the solutions corresponding to the parameter \(\lambda\) as \(\lambda\) tends of \(0\).