Topological Methods in Nonlinear Analysis

Positive solutions of nonlinear elliptic problems approximating degenerate equations

Monica Musso and Donato Passaseo

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Topol. Methods Nonlinear Anal., Volume 6, Number 2 (1995), 371-397.

First available in Project Euclid: 16 November 2016

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Musso, Monica; Passaseo, Donato. Positive solutions of nonlinear elliptic problems approximating degenerate equations. Topol. Methods Nonlinear Anal. 6 (1995), no. 2, 371--397.

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  • \ref\no1A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349–381
  • \ref\no2A. Bahri and J. M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent. The effect of the topology of the domain, Comm. Pure Appl. Math., 41 (1988), 253–294
  • \ref\no3V. Benci and G. Cerami, The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems, Arch. Rational Mech. Anal., 144(1991), 79–93 \ref\no4––––, Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Calc. Var. Partial Differential Equations, 2 (1994), 29–48
  • \ref\no5V. Benci, G. Cerami and D. Passaseo, On the number of positive solutions of some nonlinear elliptic problems , Nonlinear Analysis. A Tribute in Honour of G. Prodi (A. Ambrosetti and A. Marino, eds.), Quaderni della Scuola Norm. Sup., Pisa (1991), 93–107
  • \ref\no6H. Brezis, Elliptic equations with limiting Sobolev exponents–-the impact of topology, Proc. 50th Anniv. Courant Inst., Comm. Pure Appl. Math., 39 (1986), 17–39
  • \ref\no7H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437–477
  • \ref\no8G. Cerami and D. Passaseo, Existence and multiplicity of positive solutions for nonlinear elliptic problems in exterior domains with rich topology, Nonlinear Anal., 18 (1992), 109–119 \ref\no9––––, Existence and multiplicity results for semilinear elliptic Dirichlet problems in exterior domains, Nonlinear Anal., to appear
  • \ref\no10J. M. Coron, Topologie et cas limite des injections de Sobolev, C. R. Acad. Sci. Paris Sér. I, 299 (1984), 209–212
  • \ref\no11E. N. Dancer, A note on an equation with critical exponent, Bull. London Math. Soc., 20 (1988), 600–602
  • \ref\no12W. Ding, Positive solutions of $\D u+u^{n+2\over n-2}=0$ on contractible domains, J. Partial Differential Equations, 2 (1989), 83–88
  • \ref\no13E. Fabes, C. Kenig and R. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations, 7 (1982), 77–116
  • \ref\no14D. Fortunato, E. Jannelli and S. Solimini, (in preparation)
  • \ref\no15P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, Rev. Mat. Iberoamericana, 1 (1985), 45–121, 145–201
  • \ref\no16F. Merle and L. A. Peletier, Asymptotic behaviour of positive solutions of elliptic equations with critical and supercritical growth. I. The radial case, Arch. Rational Mech. Anal., 112 (1990), 1–19
  • \ref\no17M. K. V. Murthy and G. Stampacchia, Boundary value problems for some degenerate elliptic operators, Ann. Mat. Pura Appl., 80 (1968), 1–122
  • \ref\no18D. Passaseo, Multiplicity of positive solutions of nonlinear elliptic equations with critical Sobolev exponent in some contractible domains, Manuscripta Math., 65 (1989), 147–166 \ref\no19––––, The effect of the domain shape on the existence of positive solutions of the equation $\Delta u+u^{2^{\star } -1} =0$, Topol. Methods Nonlinear Anal., 3 (1994), 27–54 \ref\no20––––, Su alcune successioni di soluzioni positive di problemi ellittici con esponente critico, Rend. Mat. Acc. Lincei (9), 3 (1992), 15–21 \ref\no21––––, Nonexistence results for elliptic problems with supercritical nonlinearity in nontrivial domains, J. Funct. Anal., 114(1993), 97–105 \ref\no22––––, Some sufficient conditions for the existence of positive solutions to the equations $-\Delta u+a(x) u=u^{2^{\star } -1}$ in bounded domains , Ann. Inst. H. Poincaré Anal. Non Linéaire, to appear \ref\no23––––, Esistenza e molteplicità di soluzioni positive per equazioni ellittiche con nonlinearitá sopracritica , preprint no. 619, Dip. Mat. Pisa 1992 \ref\no24––––, Some concentration phenomena in degenerate semilinear elliptic problems, Nonlinear Anal., to appear
  • \ref\no25S. I. Pokhozhaev, Eigenfunctions of the equation $\D u+\la f(u)=0$, Soviet Math. Dokl., 6 (1965), 1408–1411
  • \ref\no26O. Rey, Sur un problème variationnel non compact : l'effet de petits trous dans le domaine, C. R. Acad. Sci. Paris Sér. I, 308(1989), 349–352
  • \ref\no27E. W. Stredulinsky, Weighted Inequalities and Degenerate Elliptic Partial Differential Equations, Lecture Notes in Math., 1074, Springer-Verlag (1984)
  • \ref\no28M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517