On a class of (p; q)-Laplacian problems involving the critical Sobolev-Hardy exponents in starshaped domain

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Abstract

Let Ω ⊂ ℝn be a bounded starshaped domain and consider the (p; q)-Laplacian problem

-∆pu - ∆pu = λ(x)|u|p*-2u + μ|u|r-2u

where μ is a positive parameter, 1 < q ≤ p < n, r ≥ p* and is the critical Sobolev exponent. In this short note we address the question of non-existence for non-trivial solutions to the (p; q)-Laplacian problem. In particular we show the non-existence of non-trivial solutions to the problem by using a method based on Pohozaev identity.

References

  • [1] V. Benci, G. Cerami: Existence of positive solutions of the equation ∆_u + a(x)u = u(N+2)=(N∆2) in RN. J. Funct. Anal. 88 (1990) 91{117.

  • [2] V. Benci, P. D’Avenia, D. Fortunato, L. Pisani: Solitons in several space dimensions: Derrick’s problem and in_nitely many solutions. Arch. Ration. Mech. Anal. 154 (4) (2000) 297{324.

  • [3] V. Benci, A. M. Micheletti, D. Visetti: An eigenvalue problem for a quasilinear elliptic _eld equation. J. Di_er. Equ. 184 (2) (2002) 299{320.

  • [4] P. Candito, S. A. Marano, K. Perera: On a class of critical (p; q)-Laplacian problems. Nonlinear Di_er. Equ. Appl. 22 (2015) 1959{1972.

  • [5] L. Cherfils, Y. Iåyasov: On the stationary solutions of generalized reaction di_usion equations with p&q-Laplacian. Commun. Pure Appl. Anal. 4 (1) (2005) 9{22.

  • [6] G. H. Derrick: Comments on nonlinear wave equations as models for elementary particles. J. Math. Phys. 5 (1964) 1252{1254.

  • [7] P. C. Fife: Mathematical aspects of reacting and di_using systems. Springer, Berlin (1979).

  • [8] G. M. Figueiredo: Existence of positive solutions for a class of p&q elliptic problems with critical growth on Rn. J. Math. Anal. Appl. 378 (2011) 507{518.

  • [9] R. Filippucci, P. Pucci, F. Robert: On a p-Laplace equation with multiple critical nonlinearitie. J. Math. Pures Appl 91 (2009) 156{177.

  • [10] N. Ghoussoub, C. Yuan: Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents. Trans. Amer. Math. Soc. 352 (2000) 5703{5743.

  • [11] M. Guedda, L. Véron: Quasilinear elliptic equations involving critical Sobolev exponents. Nonlinear Anal. 13 (1989) 879{902.

  • [12] Q. Guo, J. Han, P. Niu: Existence and multiplicity of solutions for critical elliptic equations with multi-polar potentials in symmetric domains. Nonlinear Analysis 75 (2012) 5765{5786.

  • [13] D. Kang: Solutions of the quasilinear elliptic problem with a critical Sobolev-Hardy exponent and a Hardy-type term. J. Math. Anal. Appl. 341 (2008) 764{782.

  • [14] G. B. Li, X. Liang: The existence of nontrivial solutions to nonlinear elliptic equation of p ∆ q-Laplacian type on RN. Nonlinear Anal. 71 (2009) 2316{2334.

  • [15] Y. Li, B. Ruf, Q. Guo, P. Niu: Quasilinear elliptic problems with combined critical Sobolev-Hardy terms. Annali di Matematica 192 (2013) 93{113.

  • [16] R. López: Constant Mean Curvature Surfaces with Boundary. Springer Monographs in Mathematics (2013).

  • [17] S. A. Marano, N. S. Papageorgiou: Constant-sign and nodal solutions of coercive (p; q)-Laplacian problems. Nonlinear Anal. 77 (2013) 118{129.

  • [18] M. S. Shahrokhi-Dehkordi, A. Taheri: Quasiconvexity and uniqueness of stationary points on a space of measure preserving maps. Journal of Convex Analysis 17 (1) (2010) 69{79.

  • [19] M. Sun: Multiplicity of solutions for a class of the quasilinear elliptic equations at resonance. J. Math. Anal. Appl. 386 (2) (2012) 661{668.

  • [20] H. Wilhelmsson: Explosive instabilities of reaction-di_usion equations. Phys. Rev. A 36 (2) (1987) 965{966.

  • [21] H. Yin, Z. Yang: A class of p ∆ q-Laplacian type equation with concave-convex nonlinearities in bounded domain. J. Math. Anal. Appl. 382 (2011) 843{855.

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researchers in the fields of: algebraic structures, calculus of variations, combinatorics, control and optimization, cryptography, differential equations, differential geometry, fuzzy logic and fuzzy set theory, global analysis, mathematical physics and number theory

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