Remarks on bifurcation for elliptic operators with odd nonlinearity
Consider the semilinear eigenvalue problem (*) \(Lu+f(x,u)=\mu u\) in \(\Omega\), \(u=0\) on \(\partial \Omega(\Omega\) bounded smooth domain in \({\mathbb{R}}^ N)\), where L is a second-order, formally selfadjoint elliptic operator in divergence form and \(f=f(x,s)\) is a Caratheodory function which is odd in s. For \(r>0\), let \(\mu_ n(r)\) \((n=1,2,...)\) be the Lusternik-Schnirelmann eigenvalues of (*), corresponding to eigenfunctions \(u_ n(r)\) such that \(\int_{\Omega}u^ 2_ n(r)=r^ 2;\) such eigenvalues do exist if \(| f(x,s)| \leq a| s|^ p+b,\) \(a,b\geq 0\), \(1\leq p<1+4/N\), for in this case the corresponding functional is bounded below and satisfies (PS) on \(M_ r:=\{v\in H^ 1_ 0(\Omega):\) \(\int_{\Omega}v^ 2=r^ 2\}.\) Further, let \((\mu^ 0_ n)\) be the eigenvalues of L in \(\Omega\), subject to zero Dirichlet b.c. Theorem: if \(b=0\) and \(p>1\) in the growth assumption on f, then each \(\mu^ 0_ n\) is a bifurcation point for (*) and in fact \(\mu_ n(r)=\mu^ 0_ n+O(r^{p-1})\) and \(u_ n(r)\to 0\) in \(H^ 1_ 0(\Omega)\) as \(r\to 0\). Corollary: Assume there exists \(q\in L^{\infty}(\Omega)\) so that \(| f(x,u)-q(x)u| \leq a| u|^ p,\) \(a\geq 0\), \(1<p<1+4/N\). If \((\lambda_ n)\) are the eigenvalues of \(\tilde L:=L+q\), then \(\mu_ n(r)=\lambda_ n+O(r^{p- 1})\) as \(r\to 0\). This improves a recent result of \textit{T. Shibata} [Boll. Unione Math. Ital., VII. Ser., B 2, No.2, 411-425 (1988)].
scientific article; zbMATH DE number 4111138