Topological Methods in Nonlinear Analysis

Existence, multiplicity and concentration of positive solutions for a class of quasilinear problems

Claudianor O. Alves and Yanheng Ding

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Abstract

Using variational methods we establish existence and multiplicity of positive solutions for the following class of quasilinear problems $$ -\Delta_{p}u + \lambda V(x)|u|^{p-2}u= \mu |u|^{p-2}u+|u|^{p^{*}-2}u \quad\text{in } {\mathbb R}^{N} $$ where $\Delta_{p}u$ is the $p$-Laplacian operator, $2 \leq p < N$, $p^{*}={pN}/(N-p)$, $\lambda, \mu \in (0, \infty)$ and $V\colon {\mathbb R}^{N} \rightarrow {\mathbb R}$ is a continuous function verifying some hypothesis.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 29, Number 2 (2007), 265-278.

Dates
First available in Project Euclid: 13 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1463148717

Mathematical Reviews number (MathSciNet)
MR2345063

Zentralblatt MATH identifier
1132.35301

Citation

Alves, Claudianor O.; Ding, Yanheng. Existence, multiplicity and concentration of positive solutions for a class of quasilinear problems. Topol. Methods Nonlinear Anal. 29 (2007), no. 2, 265--278. https://projecteuclid.org/euclid.tmna/1463148717


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References

  • C. O. Alves, Existência de solução positiva de equações não-lineares variacionais em ${\Bbb R}^{N}$, Doct. Dissertation, Un. B. (1996) \ref\key 2
  • C. O. Alves, Existence of positive solutions for a problem with lack of compactness involving the $p$-Laplacian , Nonlinear Anal., 51 (2002), 1187–1206 \ref\key 3
  • C. O. Alves, P. C. Carrião and E. S. Medeiros, Multiplicity of solutions for a class of quasilinear problem in exterior domains with Neumann conditions , Abstr. Appl. Anal., 3 (2004), 251–268 \ref\key 4
  • C. O. Alves and Y. H. Ding, Multiplicity of positive solutions to a $p$-Laplacian equation involving critical nonlinearity , J. Math. Anal. Appl., 279 (2003), 508–521 \ref\key 5
  • T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems in ${\Bbb R}^{N}$ , Comm. Partial Differential Equations, 20 (1995), 1725–1741 \ref\key 6 ––––, Multiple positive solutions for a nonlinear Schrödinger equation , Z. Angew Math. Phys., 51 (2000), 366–384 \ref\key 7
  • V. Benci and G. Cerami, The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems , Arch. Rational Mech. Anal., 114 (1991), 79–83 \ref\key 8 ––––, Existence of positive solutions of the equation $-\Delta{u}+a(x)u=u^{({N+2})/({N-2})}$ in ${\Bbb R}^{N}$ , J. Funct. Anal., 88 (1990), 91–117 \ref\key 9
  • H. Brézis and E. Lieb, A relation between pointwise convergence of funtions and convergence of functionals , Proc. Amer. Math. Soc., 88 (1983), 486–490 \ref\key 10
  • H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations invovling critical Sobolev exponents , Comm. Pure Appl. Math., 36 (1983), 437–447 \ref\key 11
  • M. Clapp and Y. H. Ding, Positive solutions of a Schrodinger equation with critical nonlinearity , Z. Angew. Math. Phys, 55 (2004), 592–605 \ref\key 12
  • D. G. de Figueiredo and Y. H. Ding, Solutions of a nonlinear Schrodinger equation , Discr. Cont. Dynam. System, 08 (2002), 563–584 \ref\key 13
  • I. Ekeland, On the variational principle , J. Math. Anal. Appl., 47 (1974), 324–353 \ref\key 14
  • J. Garcia Azorero and I. Peral Alonso, Existence and non-uniqueness for the $p$-Laplacian: Nonlinear eigenvalues , Comm. Partial Differentil Equations, 12 (1987), 1389–1430 \ref\key 15 ––––, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term , Trans. Amer. Math. Soc., 323 (1991), 877–895 \ref\key 16
  • M. Gueda and L. Veron, Quasilinear ellipitc equations involving critical Sobolev exponents , Nonlinear Anal., 13 (1989), 879–902 \ref\key 17
  • P. L. Lions, The concentration-compactness principle in the calculus of variations: The limit case, , Rev. Mat. Iberoamericana, 1 (1985), 145–201 \ref\key 18
  • G. Talenti, Best constant in Sobolev inequality , Annali di Mat., 110 (1976), 353–372 \ref\key 19
  • N. S. Trudinger, On Harnack type imequalities and their applications to a quasilinear ellipitc equations , Comm. Pure Appl. Math., 20 (1967), 721–747 \ref\key 20
  • W. Willem, Minimax Theorems, Birkhäuser (1986)