On spectral asymptotics and bifurcation for elliptic operators with odd superlinear term. (English) Zbl 0682.47032

Chiappinelli, Raffaele

Nonlinear Anal., Theory Methods Appl. 13, No. 7, 871-878 (1989).

The author studies the existence and asymptotic behaviour of eigenvalues \(\mu_ n(r)\), corresponding to eigenfunctions of \(L^ 2\)-norm \(\| u_ n\| =r\), of the nonlinear problem \[ (NL)\quad Lu+F(u)=\mu u \] on a bounded domain \(\Omega \subset {\mathbb{R}}^ N\), subject to Dirichlet boundary conditions. Here L is a uniformly elliptic operator with smooth coefficients, and F is a nonlinear Nemytskij operator from \(L_ p\) into \(L_ 1\), generated by some odd (in u) nonlinearity \(f=f(x,u)\). In the linearised problem \[ (L)\quad Lu=\mu u, \] the eigenvalues \(\mu^ 0_ n\) satisfy \(\mu^ 0_ n=cn^{2/N}+O(n^{1/N} \log n)\), as \(n\to \infty\) [see, e.g., J. Brüning, Math. Z. 137, 75-85 (1974; Zbl 0268.47053)]. One of the many interesting results in this paper states that \(\mu_ n(r)\) has the same asymptotic behaviour if \(1<p<1+2N\). In case \(p=1\), this result has been obtained by the same author in a previous paper [Boll. Unione Mat. Ital. 4(B), 867-882 (1985; Zbl 0597.35094)].

Reviewer: J.Appell

MSC:

47J05	Equations involving nonlinear operators (general)
35B32	Bifurcations in context of PDEs

Keywords:

existence and asymptotic behaviour of eigenvalues; nonlinear problem; Dirichlet boundary conditions; uniformly elliptic operator with smooth coefficients; nonlinear Nemytskij operator

Citations:

Zbl 0268.47053; Zbl 0597.35094

References:

[1]	Berger, M. S., Nonlinearity and Functional Analysis (1977), Academic Press: Academic Press New York
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[3]	Chiappinelli, R., On the eigenvalues and the spectrum for a class of semilinear elliptic operators, Boll. Un. mat. ital., 4B, 867-882 (1985) · Zbl 0597.35094
[4]	Métivier, G., Valeurs propres de problèmes aux limites elliptiques irréguliers, Bull. Soc. math. Fr., 51/52, 115-219 (1977) · Zbl 0401.35088
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