Branches of radial solutions for semipositone problems
Let \(\Omega\) be the unit ball in \(\mathbb{R}^N\) \((N> 1)\) centered at the origin, \(f: \mathbb{R}\to \mathbb{R}\) a differentiable, monotone function such that \(f(0)< 0\), \(\lim_{d\to \infty} {f(x)\over d}= \infty\), \(F(d)- {N- 2\over 2N} df(d)\geq M\) for all \(d\in \mathbb{R}\), where \(M\in \mathbb{R}\), \(F(d)= \int^d_0 f(s) ds\) and \(\lim_{d\to \infty} ({d\over f(d)})^{N/2} (F(kd)- {N-2\over 2N} df(d))= \infty\) for some \(k\in ]0,1[\). Theorem 1. Let \(S\subset \mathbb{R}_+\times C(\overline\Omega)\) be a connected component of radially symmetric solutions to \(-\Delta u= \lambda f\circ u\) in \(\Omega\), \(u= 0\) on \(\partial\Omega\) with \(u(0)> 0\). If \(S\) is nonempty then there exists a \(k\in \mathbb{N}\) such that if \((\lambda, u)\in S\) then \(u\) has \(2k\) or \(2k+ 1\) nodal hypersurfaces in \(\Omega\). If \((\lambda_0, u_0)\in S\), \(u_0\) has \(2k\) nodal hypersurfaces in \(\Omega\) and \(\nabla u_0(x)\neq 0\) for \(x\in \partial\Omega\) iff there exists \((\lambda_1, u_1)\in S\) satisfying \(u_0(0)= u_1(0)\) and \(u_1\) has \(2k+ 1\) nodal hypersurfaces in \(\Omega\). There exists \(\widetilde\lambda> 0\) such that if \((\lambda, u)\) verifies \(-\Delta u= \lambda f\circ u\) in \(\Omega\), \(u= 0\) on \(\partial\Omega\) with \(\lambda< \widetilde\lambda\) then the connected component containing \((\lambda, u)\) is unbounded, moreover \(\nabla u(x)\neq 0\) for \(x\in \partial\Omega\). For any \(k\in \mathbb{N}\) there exists a unique unbounded connected component \(S_k\) of radially symmetric solutions to \(-\Delta u= \lambda f\circ u\) in \(\Omega\), \(u= 0\) on \(\partial\Omega\) with \(u(0)> 0\), \(u\) having \(2k\) or \(2k+ 1\) nodal hypersurfaces in \(\Omega\). Theorem 2. If, in addition, \(f\) is convex and \(t\mapsto {f(t)\over tf'(t)- t}\) is nondecreasing function then for each \(\lambda> 0\) there exists at most one positive \(u\) such that \((\lambda, u)\in S_0\). Theorem 3. If \(f(t)= t^p- k\), \(k> 0\), \(1< p< {N+ 2\over N- 2}\), if \((\lambda, u)\) verifies \(-\Delta u= \lambda f\circ u\) in \(\Omega\), \(u= 0\) on \(\partial\Omega\) and \(u> 0\) then that \((\lambda, u)\in S_0\).
scientific article; zbMATH DE number 790882
scientific article