A positive solution for an asymptotically linear elliptic problem on N autonomous at infinity
ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002), pp. 597-614.

In this paper we establish the existence of a positive solution for an asymptotically linear elliptic problem on N . The main difficulties to overcome are the lack of a priori bounds for Palais-Smale sequences and a lack of compactness as the domain is unbounded. For the first one we make use of techniques introduced by Lions in his work on concentration compactness. For the second we show how the fact that the “Problem at infinity” is autonomous, in contrast to just periodic, can be used in order to regain compactness.

DOI : 10.1051/cocv:2002068
Classification : 35J60, 58E05
Mots-clés : elliptic equations, asymptotically linear problems in $\mathbb {R}^N$, lack of compactness
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     author = {Jeanjean, Louis and Tanaka, Kazunaga},
     title = {A positive solution for an asymptotically linear elliptic problem on $\mathbb {R}^N$ autonomous at infinity},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {597--614},
     publisher = {EDP-Sciences},
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     mrnumber = {1925042},
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     url = {http://www.numdam.org/articles/10.1051/cocv:2002068/}
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Jeanjean, Louis; Tanaka, Kazunaga. A positive solution for an asymptotically linear elliptic problem on $\mathbb {R}^N$ autonomous at infinity. ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002), pp. 597-614. doi : 10.1051/cocv:2002068. http://www.numdam.org/articles/10.1051/cocv:2002068/

[1] A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications. J. Funct. Anal. 14 (1973) 349-381. | MR | Zbl

[2] H. Berestycki and P.L. Lions, Nonlinear scalar field equations I. Arch. Rational Mech. Anal. 82 (1983) 313-346. | MR | Zbl

[3] H. Berestycki, T. Gallouët and O. Kavian, Equations de Champs scalaires euclidiens non linéaires dans le plan. C. R. Acad. Sci. Paris Sér. I Math. 297 (1983) 307-310. | MR | Zbl

[4] H. Brezis, Analyse fonctionnelle. Masson (1983). | MR | Zbl

[5] V. Coti Zelati and P.H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on N . Comm. Pure Appl. Math. XIV (1992) 1217-1269. | MR | Zbl

[6] I. Ekeland, Convexity methods in Hamiltonian Mechanics. Springer (1990). | MR | Zbl

[7] L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on N . Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 787-809. | Zbl

[8] P.L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Parts I and II. Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984) 109-145 and 223-283. | Numdam | Zbl

[9] P.H. Rabinowitz, On a class of nonlinear Shrödinger equations. ZAMP 43 (1992) 270-291. | MR | Zbl

[10] C.A. Stuart, Bifurcation in L p ( N ) for a semilinear elliptic equation. Proc. London Math. Soc. 57 (1988) 511-541. | MR | Zbl

[11] C.A. Stuart and H.S. Zhou, A variational problem related to self-trapping of an electromagnetic field. Math. Meth. Appl. Sci. 19 (1996) 1397-1407. | MR | Zbl

[12] C.A. Stuart and H.S. Zhou, Applying the mountain-pass theorem to an asymtotically linear elliptic equation on N . Comm. Partial Differential Equations 24 (1999) 1731-1758. | MR | Zbl

[13] A. Szulkin and W. Zou, Homoclinic orbits for asymptotically linear Hamiltonian systems. J. Funct. Anal. 187 (2001) 25-41. | MR | Zbl

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