Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term
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- by J. García Azorero and I. Peral Alonso PDF
- Trans. Amer. Math. Soc. 323 (1991), 877-895 Request permission
Abstract:
We study the existence of solutions for the following nonlinear degenerate elliptic problems in a bounded domain $\Omega \subset {{\mathbf {R}}^N}$ \[ - \operatorname {div} (|\nabla u{|^{p - 2}}\nabla u) = |u{|^{{p^{\ast }} - 2}}u + \lambda |u{|^{q - 2}}u,\qquad \lambda > 0,\] where ${p^{\ast }}$ is the critical Sobolev exponent, and $u{|_{\delta \Omega }} \equiv 0$. By using critical point methods we obtain the existence of solutions in the following cases: If $p < q < {p^{\ast }}$, there exists ${\lambda _0} > 0$ such that for all $\lambda > {\lambda _0}$ there exists a nontrivial solution. If $\max (p,{p^{\ast }} - p/(p - 1)) < q < {p^{\ast }}$, there exists nontrivial solution for all $\lambda > 0$. If $1 < q < p$ there exists ${\lambda _1}$ such that, for $0 < \lambda < {\lambda _1}$, there exist infinitely many solutions. Finally, we obtain a multiplicity result in a noncritical problem when the associated functional is not symmetric.References
- Aomar Anane, Simplicité et isolation de la première valeur propre du $p$-laplacien avec poids, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 16, 725–728 (French, with English summary). MR 920052
- Jean-Pierre Aubin and Ivar Ekeland, Applied nonlinear analysis, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. A Wiley-Interscience Publication. MR 749753 F. V. Atkinson, L. A. Pelletier, and J. Serrin, unpublished work. H. Brézis, Some variational problems with lack of compactness, Proc. Berkeley Sympos. Nonlinear Functional Analysis, 1985. G. Barles, Remarks on the uniqueness results of the first eigenvalue of the $p$-Laplacian, Ann. de Toulouse (to appear).
- Vieri Benci, On critical point theory for indefinite functionals in the presence of symmetries, Trans. Amer. Math. Soc. 274 (1982), no. 2, 533–572. MR 675067, DOI 10.1090/S0002-9947-1982-0675067-X
- V. Benci and D. Fortunato, Bifurcation from the essential spectrum for odd variational operators, Confer. Sem. Mat. Univ. Bari 178 (1981), 26. MR 641108 T. Bhattacharya, Radial symmetry of the first eigenfunction for the $p$-laplacian in the ball, Preprint.
- Haïm Brézis and Louis Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), no. 4, 437–477. MR 709644, DOI 10.1002/cpa.3160360405 H. Brézis and L. Oswald, Remarks on sublinear elliptic equations, Inst. Math. Appl. Preprint Ser. 112 (1984).
- E. DiBenedetto, $C^{1+\alpha }$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983), no. 8, 827–850. MR 709038, DOI 10.1016/0362-546X(83)90061-5 I. Diaz and J. E. Saa, Uniqueness of nonnegative solutions for elliptic nonlinear diffusion equations with a general perturbation term, Proc. VII CEDYA, Santander, 1985.
- Henrik Egnell, Existence and nonexistence results for $m$-Laplace equations involving critical Sobolev exponents, Arch. Rational Mech. Anal. 104 (1988), no. 1, 57–77. MR 956567, DOI 10.1007/BF00256932
- J. P. García Azorero and I. Peral Alonso, Existence and nonuniqueness for the $p$-Laplacian: nonlinear eigenvalues, Comm. Partial Differential Equations 12 (1987), no. 12, 1389–1430. MR 912211, DOI 10.1080/03605308708820534
- Jesús García Azorero and Ireneo Peral Alonso, Comportement asymptotique des valeurs propres du $p$-laplacien, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), no. 2, 75–78 (French, with English summary). MR 954263
- Mohammed Guedda and Laurent Véron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal. 13 (1989), no. 8, 879–902. MR 1009077, DOI 10.1016/0362-546X(89)90020-5
- P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana 1 (1985), no. 1, 145–201. MR 834360, DOI 10.4171/RMI/6
- P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. II, Rev. Mat. Iberoamericana 1 (1985), no. 2, 45–121. MR 850686, DOI 10.4171/RMI/12
- Mitsuharu Ôtani, Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations, J. Funct. Anal. 76 (1988), no. 1, 140–159. MR 923049, DOI 10.1016/0022-1236(88)90053-5
- Patrizia Pucci and James Serrin, A general variational identity, Indiana Univ. Math. J. 35 (1986), no. 3, 681–703. MR 855181, DOI 10.1512/iumj.1986.35.35036
- Paul H. Rabinowitz, Multiple critical points of perturbed symmetric functionals, Trans. Amer. Math. Soc. 272 (1982), no. 2, 753–769. MR 662065, DOI 10.1090/S0002-9947-1982-0662065-5
- Paul H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, vol. 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. MR 845785, DOI 10.1090/cbms/065
- Peter Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), no. 1, 126–150. MR 727034, DOI 10.1016/0022-0396(84)90105-0
Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 323 (1991), 877-895
- MSC: Primary 35J65; Secondary 35B30
- DOI: https://doi.org/10.1090/S0002-9947-1991-1083144-2
- MathSciNet review: 1083144