Variational method for precise asymptotic formulas for nonlinear eigenvalue problems
The following nonlinear eigenvalue problem is considered \[ \begin{alignedat}{2} -&u''(t)+ f(u(t))=\lambda u(t)^p, && \quad t\in I:=(-T,T),\\ & u(t)>0, && \quad t\in I,\\ & u(\pm T)=0, \end{alignedat} \] where \(\lambda\in\mathbb{R}\) is a parameter and \(p>1\). The main hypotheses on the nonlinearity \(f\) are the following: (i). \(f\in C^1(\mathbb{R}),\;f(-u)=-f(u)\). (ii). \(| f(u)| /u\rightarrow 0\) as \(u\rightarrow\infty\). The author investigates the relationship between \(\lambda\), the \(L^\infty\) norm of the solutions, their slope on the boundary and their \(L^{p+1}\) norm, especially when the later is large enough. To achieve this goal an associated variational problem with the equation is considered on \[ M_\alpha=\left\{ v\in H^1_0(I):\;| | v| | _{L^{p+1}}=\alpha\right\}, \] where \(\alpha >0\) is a parameter. After the existence of a solution pair \((\lambda(\alpha), u_\alpha)\in \mathbb{R}\times M_\alpha\) for this problem has been established, precise asymptotic formulas are obtained for \(| | u_\alpha| | _\infty,\;u'_\alpha(\pm T)^2\) and \(\lambda(\alpha)\) as \(\alpha\rightarrow\infty\).
scientific article; zbMATH DE number 2133442