Comparison theorems for variational problems and their application to elliptic

I. A. Kuzin

Izvestiya. Mathematics , Tome 43 (1994) no. 2, pp. 331-346.

Résumé

The behavior of PS-sequences for problems of the form $$ \begin{cases} -\sum\limits_{i=1}^N \nabla_ia_i(\mathbf x,u,\nabla u)+b(\mathbf x,u,\nabla u)=0 \quad\text{in}\quad \mathbf R^N, \\ u\to 0 \quad\text{as}\quad |\mathbf x|\to\infty \end{cases} $$ with functions $a_i,b\colon(\mathbf x, \mu,\xi)\mapsto c$ odd with respect to $\mu$, $\xi$ and such that $a_i(\mathbf x, \mu,\xi)\to\bar a_i(\mu,\xi)$, $b(\mathbf x, \mu,\xi)\to\bar b(\mu,\xi)$ as $|\mathbf x|\to\infty$, is studied. On the basis of this, theorems are proved on the existence of $l$ distinct pairs of nontrivial solutions of this problem.

Bibliographie

[1] Krasnoselskii M. A., Topologicheskie metody v teorii nelineinykh integralnykh uravnenii, Gostekhizdat, M., 1956, 392 pp. | MR

[2] Rabinowitz P., Minimax methods in critical point theory with applications to differential equations, Published for the Conference Board of the Mathematical Sciences, Washington, DC, CBMS Regional Conference Series in Mathematics, 65, American Mathematical Society, Providence, RI, 1986 | MR

[3] Pokhozhaev S. I., “Ob odnom podkhode k nelineinym uravneniyam”, DAN SSSR, 247 (1979), 1327–1331 | MR | Zbl

[4] Lions P. L., “The concentration-compactness principle in the calculus of variations. The locally compact case, I”, Analyse non lineaire, 1984, no. 1, 109–145 | MR | Zbl

[5] Lions P. L., “The concentration-compactness principle in the calculus of variations. The locally compact case, II”, Analyse non lineaire, 1984, no. 1, 223–283 | MR | Zbl

[6] Wei-Yue Ding, Wei-Ming Ni, “On the existence of positive entire solutions of a semilinear elliptic equations”, Arch. Rat. Mech. Anal., 91 (1986), 283–308 | DOI | MR | Zbl

[7] Zhu Xiping, Cao Daomin, “The concentration-compactness principle in nonlinear elliptic equations”, Acta Math. Sci., 9 (1989), 307–328 | MR | Zbl

[8] Benchi V., Cerami G., “Positive solutions of some nonlinear elliptic problems in exterior domains”, Arch. Rat. Mech. Anal., 99 (1987), 283–300 | DOI | MR

[9] Yang Jianfu, Zhu Xiping, “On the existence of nontrivial solution of a quasilinear elliptic boundary value problem for unbounded domains. 1: Positive mass case”, Acta Math. Sci., 7 (1987), 341–359 | MR | Zbl

[10] Yang Jianfu, Zhu Xiping, “On the existence of nontrivial solution of quasilinear elliptic boundary value problem for unbounded domains. 2: Zero mass case”, Acta Math. Sci., 7 (1987), 447–459 | MR | Zbl

[11] Struwe M., “A global compactness result for elliptic boundary value problems involving limiting nonlinearities”, Math. Z., 187 (1984), 511–517 | DOI | MR | Zbl

[12] Pokhozhaev S. I., “O sobstvennykh funktsiyakh kvazilineinykh ellipticheskikh zadach”, Matem. sb., 82 (1970), 192–212 | MR

[13] Kuzin I. L., “Suschestvovanie osnovnogo sostoyaniya dlya nekotorykh nelineinykh skalyarnykh polei”, DAN SSSR, 308 (1989), 34–38 | MR

[14] DiBendetto E., “$C^{1+\alpha}$ local regularity of weak solution of degenerate elliptic equations”, Nonl. Anal. TMA, 7 (1983), 837–850