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

Keywords:

nonlinear; eigenvalues; critical; limits
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