Truncation of Nonlinearities in Some Supercritical Elliptic Problems
نویسنده
چکیده
We consider truncations of nonlinearities of semilinear elliptic problems by H 1 , not necessarily bounded, sub and super solutions. We prove that their associated energy functionals are of class C 1 and satisfy the Palais{Smale condition. We apply this result to prove existence of innnitely many solutions of a supercritical elliptic problem involving concave and convex nonlinearities. Troncature des termes non{lin eaires dans certains probl emes elliptiques sur{critiques R esum e { On consid ere les troncatures des termes non{lin eaires de probl emes elliptiques semi{lin eaires par des sous{solutions et sur{solutions de classe H 1 qui ne sont pas n ecessairement born ees. On d emontre que les fonctionnelles d'energie as-soc ees a ces troncatures sont de classe C 1 et satisfont la condition de Palais{Smale. On applique ce r esultat pour etablir l'existence d'une innnit e de solutions d'un probl eme elliptique sur{critique dont la non{lin earit e comporte un terme concave et un terme convexe. Version frann caise abr eg ee { Soit un domaine born e r egulier de R N. On consid ere le probl eme (1) (?u = juj q?1 u + juj p?1 u dans u = 0 sur @ ; o u 0 < q < 1 < p et > 0. On s'int eresse essentiellement au cas sur{critique p > (N + 2)=(N ? 2), mais les r esultats obtenus restent valables pour tout p > 1. Une solution faible du probl eme (1) est une fonction u 2 H 1 0 (() = H 1 0 telle que juj p 2 L 1 loc (() = L 1 loc et satisfaisant l' equation (1) au sens des distributions. En ce qui concerne l'existence de solutions positives de (1) = (1) , on rappelle le r esultat suivant (voir 1] et 4]). Il existe une constante 0 < < 1 telle que (a) si 0 < < , alors (1) admet au moins une solution classique positive u , (b) si = , alors (1) admet au moins une solution faible positive u 2 H 1 0 \ L p+1 , et (c) si > , alors (1) n'admet pas de solution classique positive. On consid ere la fonctionnelle d' energie E associ ee a (1) qui est d eenie par E (u) = 1 2 Z jruj 2 ? q + 1 Z juj q+1 ? 1 p + …
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