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Critical eigenvalues for a nonlinear problem. (English) Zbl 1207.35138

Summary: We study the mapping \(\lambda^*:]1,\infty[\to\mathbb R_+\), where \(\lambda^*(\alpha)\) is the supremum of positive \(\lambda\)’s such that the problem
\[ (P_\lambda^\alpha)\begin{cases} \Delta u +\lambda (1+ u)^\alpha=0 &\text{in }B_1,\\ u>0 &\text{in }B_1,\\ u=0 &\text{on }\partial B_1, \end{cases} \]
admits a solution. Where \(B_1\) is the unit ball in \(\mathbb R^n\), \(n\geq 3\). We show that \(\lambda^*\) is a decreasing function, with
\[ \lim_{\alpha\to\infty} \lambda^*(\alpha)=0 \quad\text{and}\quad \lim_{\alpha\to \infty}u_{\lambda^*(\alpha)}(0)=+\infty, \]
where \(u_{\lambda^*(\alpha)}\) is the unique solution of the problem \((P_{\lambda^*(\alpha)}^\alpha)\).
We also give the explicit solutions of the problem \((P_\lambda^\alpha)\), when \(\alpha=1\), \(n=3\) and show that \(\lambda^*(1)=\pi^2\). We show that the problem \((P_{\pi^2}^1)\) does not admit a solution.
In the end, we give a numerical approximation of \(\lambda^*(2)\), when \(n=3\).

MSC:

35J60 Nonlinear elliptic equations
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
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