Existence and non-existence of positive solution for (p, q)-Laplacian with singular weights

Abdellah Ahmed Zerouali; Belhadj Karim

doi:10.5269/bspm.v34i2.25229

Abdellah Ahmed Zerouali Centre Régional des Métiers de l'Education et de la Formation- Fès
Belhadj Karim Faculté des Sciences et Techniques ERRACHIDIA

DOI: https://doi.org/10.5269/bspm.v34i2.25229

Resumen

We use the Hardy-Sobolev inequality to study existence and non-existence results for a positive solution of the quasilinear elliptic problem -\Delta{p}u − \mu \Delta{q}u = \limda[mp(x)|u|p−2u + \mu mq(x)|u|q−2u] in \Omega driven by nonhomogeneous operator (p, q)-Laplacian with singular weights under the Dirichlet boundary condition. We also prove that in the case where μ > 0 and with 1 < q < p < \infinity the results are completely different from those for the usual eigenvalue for the problem p-Laplacian with singular weight under the Dirichlet boundary condition, which is retrieved when μ = 0. Precisely, we show that when μ > 0 there exists an interval of eigenvalues for our eigenvalue problem.

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Citas

A. Anane, A. Anane, Simplicity and isolation of first eigenvalue of the p-Laplacian with weight, Comptes rendus de l’Académie des sciences, Série 1, Mathématique, 305, No. 16 (1987), 725–728.

R. Aris, Mathematical Modelling Techniques, Research Notes in Mathematics, Pitman, London, 1978.

V. , A. M.Micheletti, D. Visetti, An eigenvalue problem for a quasilinear elliptic field equation, J. Differential Equations, 184, No.2 (2002), 299–320.

N. Benouhiba, Z. Belyacine, A class of eigenvalue problems for the (p, q)-Laplacian on RN,International JournalBenci of Pure and Applied Mathematics, Volume 80 No. 5 2012, 727–737.

N. Benouhiba, Z. Belyacine, On the solutions of (p, q)-Laplacian problem at resonance, Nonlinear Anal. 77 (2013), 74–81.

S. Cingolani and Degiovanni, Nontrivial Solutions for p-Laplace Equations with Right-Hand Side Having p-Linear Growth at Infinity, Comm. Partial Differential Equations, 30 (2005), 1191–1203.

M. Cuesta, Minimax Theorems on C1 manifolds via Ekeland Variational Principle, Abstract and Applied Analysis 2003:13 (2003) 757–768.

P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics, 28, Springer Verlag, Berlin-New York, 1979.

M. Guedda, L. Veron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal. 13 (1989) 879–902.

O. Kavian, Inegalité de Hardy-Sobolev, C. R. Acad. Sci., Paris Sér. I Math. 286 (1978) 779–781.

M. Otani, A remark on certain nonlinear elliptic equations, Proc. Fac. Sci. Tokai Univ. 19 (1984) 23–28.

S. A. Marano, N. S. Papageogiou, Constant-sign and nodal solutions of coercive (p, q)-Laplacian problems, Nonlinear Anal. TMA, 77(2013), 118–129.

M. Mihãilescu, An eigenvalue problem possessing a continuous family of eigenvalues plus an isolated eigenvalue, Comm. Pure Appl. Anl. 10 (2011), 701–708.

M. Montenegro, S. Lorca, The spectrum of the p-Laplacian with singular weight, Nonlinear Analysis 75 (2012) 3746–3753.

D. Motreanu, M. Tanaka, On a positive solution for (p, q)-Laplace equation with indefinite weight. Minimax Theory and its Applications, Vol. 1 (2014), In press.

N. E. Sidiropoulos, Existence of solutions to indefinite quasilinear elliptic problems of (p, q)-Laplacian type, Elect. J. Differential Equations, 2010 no. 162 (2010), 1–23.

A. Szulkin, M. Willem, Eigenvalue problems with indefnite weight, Studia Math. 135(1999), 191–201.

M. Struwe, Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin, Heidelberg, New York, 1996.

M. Tanaka, Generalized eigenvalue problems for (p, q)-Laplacian with indefinite weight, J. Math. Anal. Appl. 11/2014: 419(2), pp. 1181–1192.

N. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations. Comm. Pure Appl. Math. 20 (1967) 721–747.

H. Yin, Z. Yang, A class of p-q-Laplacian type eqation with concave-convex nonlinearities in bounded domain, J. Math. Anal. Appl., 382 (2011), 843–855.

G. Li, G. Zhang, Multiple solutions for the p-q-Laplacian problem with critical exponent, Acta Mathematica Scientia, 29B, No.4 (2009), 903–918.