Geodesic
On the uniqueness and simplicity of the principal eigenvalue
Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 16 (2005) no. 2, pp. 133-142.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

Given an open set $\Omega$ of $\mathbb{R}^{N}$$(N > 2)$, bounded or unbounded, and a function $w \in L^{\frac{N}{2}} (\Omega)$ with $w^{+}\neq 0$ but allowed to change sign, we give a short proof that the positive principal eigenvalue of the problem $$ - \triangle u = \lambda w (x) u, \qquad u \in \mathcal{D}^{1,2}_{0} (\Omega)$$ is unique and simple. We apply this result to study unbounded Palais-Smale sequences as well as to give a priori estimates on the set of critical points of functionals of the type $$I(u) = \frac{1}{2}\int_{\Omega} |\nabla u|^{2} \, dx - \int_{\Omega} G(x,u) \, dx, \quad u \in \mathcal{D}^{1,2}_{0} (\Omega),$$ when $G$ has a subquadratic growth at infinity.
Dato un aperto connesso $\Omega$ di $\mathbb{R}^{N}$$(N > 2)$, limitato o illimitato, e una funzione $w \in L^{\frac{N}{2}} (\Omega)$ con $w^{+}\neq 0$ cui è consentito cambiare segno, si dimostra che l'autovalore principale positivo del problema $$ - \triangle u = \lambda w (x) u, \qquad u \in \mathcal{D}^{1,2}_{0} (\Omega)$$ è unico e semplice. Tale risultato viene applicato allo studio delle successioni di Palais-Smale illimitate ed utilizzato per costruire stime a priori sull'insieme dei punti critici di funzionali del tipo $$I(u) = \frac{1}{2}\int_{\Omega} |\nabla u|^{2} \, dx - \int_{\Omega} G(x,u) \, dx, \quad u \in \mathcal{D}^{1,2}_{0} (\Omega),$$ dove $G$ ha un andamento subquadratico all'infinito. 
@article{RLIN_2005_9_16_2_a5,
     author = {Lucia, Marcello},
     title = {On the uniqueness and simplicity of the principal eigenvalue},
     journal = {Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni},
     pages = {133--142},
     publisher = {mathdoc},
     volume = {Ser. 9, 16},
     number = {2},
     year = {2005},
     zbl = {1225.35159},
     mrnumber = {1098396},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RLIN_2005_9_16_2_a5/}
}
TY  - JOUR
AU  - Lucia, Marcello
TI  - On the uniqueness and simplicity of the principal eigenvalue
JO  - Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni
PY  - 2005
SP  - 133
EP  - 142
VL  - 16
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/RLIN_2005_9_16_2_a5/
LA  - en
ID  - RLIN_2005_9_16_2_a5
ER  - 
%0 Journal Article
%A Lucia, Marcello
%T On the uniqueness and simplicity of the principal eigenvalue
%J Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni
%D 2005
%P 133-142
%V 16
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/RLIN_2005_9_16_2_a5/
%G en
%F RLIN_2005_9_16_2_a5
Lucia, Marcello. On the uniqueness and simplicity of the principal eigenvalue. Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 16 (2005) no. 2, pp. 133-142. http://geodesic.mathdoc.fr/item/RLIN_2005_9_16_2_a5/

[1] W. Allegretto, Principal eigenvalues for indefinite-weight elliptic problems on $\mathbb{R}^{N}$. Proc. Am. Math. Monthly, 116, 1992, 701-706. | DOI | MR | Zbl

[2] A. Ancona, Une propriété d'invariance des ensembles absorbants par perturbation d'un opérateur elliptique. Comm. PDE, 4, 1979, 321-337. | DOI | MR | Zbl

[3] H. Berestycki - L. Nirenberg - S.R.S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains. Comm. Pure Appl. math., 47, 1994, 47-92. | DOI | MR | Zbl

[4] H. Brezis - A. Ponce, Remarks on the strong maximum principle. Differential Integral Equations, 16, 2003, 1-12. | MR | Zbl

[5] K.J. Brown - C. Cosner - J. Fleckinger, Principal eigenvalues for problems with indefinite weight function on $\mathbb{R}^{N}$. Proc. Amer. Math. Soc., 109, 1990, 147-155. | DOI | MR | Zbl

[6] K.J. Brown - N. Stavrakakis, Global Bifurcation results for a semilinear elliptic equation on all of $\mathbb{R}^{N}$. Duke Math. J., 85, 1996, 77-94. | fulltext mini-dml | DOI | MR | Zbl

[7] M. Cuesta, Eigenvalue problems for the $p$-Laplacian with indefinite weights. Electron. J. Differential Equations, 33, 2001, 1-9. | fulltext EuDML | MR | Zbl

[8] D.G. De Figueiredo, Positive solutions of semilinear elliptic problems, Differential equations. Lecture Notes in Math., 957, Springer-Verlag, Berlin 1982, 34-87. | MR | Zbl

[9] L.C. Evans - R.F. Gariepy, Measure Theory and fine properties of functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, 1992. | MR | Zbl

[10] D. Gilbarg - N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin 1983. | MR | Zbl

[11] P. Hess - T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function. Comm. PDE, 5, 1980, 999-1030. | DOI | MR | Zbl

[12] M. Lucia - P. Magrone - H.S. Zhou, A Dirichlet problem with asymptotically linear and changing sign nonlinearity. Rev. Mat. Comput., 16, 2003, 465-481. | fulltext EuDML | MR | Zbl

[13] A. Manes - A.M. Micheletti, Un'estensione della teoria variazionale classica degli autovalori per operatori ellittici del secondo ordine. Bollettino U.M.I., 7, 1973, 285-301. | MR | Zbl

[14] G. Stampacchia, Équations elliptiques du second ordre à coefficients discontinus. Les Presses de l'Université de Montréal, Montréal 1966. | fulltext mini-dml | MR | Zbl

[15] A. Szulkin - M. Willem, Eigenvalue problems with indefinite weight. Studia Math., 135, 1999, 191-201. | fulltext EuDML | fulltext mini-dml | MR | Zbl

[16] A. Tertikas, Uniqueness and nonuniqueness of positive solutions for a semilinear elliptic equation in $\mathbb{R}^{N}$. Diff. and Int. Eqns., 8, 1995, 829-848. | MR | Zbl

[17] H.S. Zhou, An application of a mountain pass theorem. Acta Math. Sinica (N.S.), 18, 2002, 27-36. | DOI | MR | Zbl