Existence and nonexistence results for quasilinear elliptic equations involving the $p$-Laplacian

Abdellaoui, Boumediene ; Felli, Veronica ; Peral, Ireneo

Bollettino dell'Unione Matematica Italiana, Tome 9-A (2006), p. 445-484 / Harvested from Biblioteca Digitale Italiana di Matematica

Abstract

The paper deals with the study of a quasilinear elliptic equation involving the p-laplacian with a Hardy-type singular potential and a critical nonlinearity. Existence and nonexistence results are first proved for the equation with a concave singular term. Then we study the critical case related to Hardy inequality, providing a description of the behavior of radial solutions of the limiting problem and obtaining existence and multiplicity results for perturbed problems through variational and topological arguments.

Résumé

L’articolo riguarda lo studio di un’equazione ellittica quasi-lineare con il p-laplaciano, caratterizzata dalla presenza di un termine singolare di tipo Hardy ed una nonlinearità critica. Si dimostrano dapprima risultati di esistenza e non esistenza per l’equazione con un termine singolare concavo. Quindi si passa a studiare il caso critico legato alla disuguaglianza di Hardy, fornendo una descrizione del comportamento delle soluzioni radiali del problema limite e ottenendo risultati di esistenza e molteplicità mediante metodi variazionali e topologici.

Détail

Publié le : 2006-06-01

MR 2233146 | Zbl 1118.35010

Bibliographie

[1] Abdellaoui, B. - Felli, V. - Peral, I., Existence and multiplicity for perturbations of an equation involving Hardy inequality and critical Sobolev exponent in the whole $\mathbb{R}^N$, Adv. Diff. Equations, 9 (2004), 481-508. | Zbl 1220.35041

[2] Abdellaoui, B. - Peral, I., Existence and nonexistence results for quasilinear elliptic equations involving the p-laplacian, Ann. Mat. Pura. Applicata, 182 (2003), 247-270. | Zbl 1223.35151

[3] Allegretto, W. - Huang, Yin Xi, A Picone’s identity for the p-Laplacian and applications, Nonlinear. Anal. TMA., 32, no. 7 (1998), 819-830. | Zbl 0930.35053

[4] Ambrosetti, A., Critical points and nonlinear variational problems, Mém. Soc. Math. France (N.S.), no. 49 (1992). | Zbl 0766.49006

[5] Ambrosetti, A. - Brezis, H. - Cerami, G., Combined Effects of Concave and Convex Nonlinearities in some Elliptic Problems, Journal of Functional Anal., 122, no. 2 (1994), 519-543. | Zbl 0805.35028

[6] Ambrosetti, A. - Garcia Azorero, J. - Peral, I., Elliptic variational problems in $\mathbb{R}^N$ with critical growth, J. Diff. Equations, 168, no. 1 (2000), 10-32. | Zbl 0979.35050

[7] Brezis, H. - Cabré, X., Some simple PDE’s without solution, Boll. Unione. Mat. Ital. Sez. B, 8, no. 1 (1998), 223-262.

[8] Brothers, J. - Ziemer, W., Minimal rearrangements of Sobolev functions, Acta Univ. Carolin. Math. Phys. 28, no. 2 (1987), 13-24. | Zbl 0659.46029

[9] Cao, D. - Chabrowski, J., Multiple solutions of nonhomogeneous elliptic equation with critical nonlinearity, Differential Integral Equations, 10, no. 5 (1997), 797-814. | Zbl 0889.35033

[10] Franchi, B. - Lanconelli, E. - Serrin, J., Existence and uniqueness of nonnegative solutions of quasilinear equations in $\mathbb{R}^n$, Adv. Math., 118, no. 2 (1996), 177-243. | Zbl 0853.35035

[11] García Azorero, J. - Montefusco, E. - Peral, I., Bifurcation for the p-laplacian in $\mathbb{R}^N$, Adv. Differential Equations, 5, no. 4-6 (2000), 435-464. | Zbl 1004.35042

[12] García Azorero, J. - Peral, I., Hardy Inequalities and some critical elliptic and parabolic problems, J. Diff. Eq., 144 (1998), 441-476. | Zbl 0918.35052

[13] García Azorero, J. - Peral, I., Multiplicity of solutions for elliptic problems with critical exponent or with a non-symmetric term, Trans. Amer. Math. Soc., 323, no. 2 (1991), 877-895. | Zbl 0729.35051

[14] Ghoussoub, N. - Yuan, C., Multiple solution for Quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc., 352, no. 12 (2000), 5703-5743. | MR 1695021 | Zbl 0956.35056

[15] Lions, P. L., The concentration-compactness principle in the calculus of variations. The limit case, part 1, Rev. Matemática Iberoamericana, 1, no. 1 (1985), 541-597. 484 | MR 850686

[16] Lions, P. L., The concentration-compactness principle in the calculus of variations. The limit case, part 2, Rev. Matemática Iberoamericana, 1, no. 2 (1985), 45-121. | MR 850686 | Zbl 0704.49006

[17] Musina, R., Multiple positive solutions of a scalar field equation in $\mathbb{R}^N$, Top. Methods Nonlinear Anal., 7 (1996), 171-186. | MR 1422010 | Zbl 0909.35042

[18] Peral, I., Some results on Quasilinear Elliptic Equations: Growth versus Shape, in Proceedings of the Second School of Nonlinear Functional Analysis and Applications to Differential Equations, I.C.T.P. Trieste, Italy, A. Ambrosetti and it alter editors. World Scientific, 1998. | MR 1703531

[19] Picone, M., Sui valori eccezionali di un parametro da cui dipende una equazione differenziale lineare ordinaria del secondo ordine, Ann. Scuola. Norm. Pisa., 11 (1910), 1-144. | MR 1556637 | Zbl 41.0351.01

[20] Polya, G. - Szego, G., Isoperimetric inequalities in mathematical physics, Gosudarstv. Izdat. Fiz. Mat., Moscow 1962. | MR 253161 | Zbl 0101.41203

[21] Smets, D., Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities, Trans. AMS, to appear. | MR 2139932

[22] Simon, J., Regularité de la solution d’une equation non lineaire dans $\mathbb{R}^N$, Lectures Notes in Math, no. 665, P. Benilan editor, Springer Verlag, 1978. | MR 519432

[23] Tarantello, G., On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire.9, no. 3 (1992), 281-304. | MR 1168304 | Zbl 0785.35046

[24] Tolksdorf, P., Regularity for more general class of quasilinear elliptic equations, J. Diff. Eq., 51 (1984), 126-150. | MR 727034 | Zbl 0488.35017

[25] Terracini, S., On positive entire solutions to a class of equations with singular coefficient and critical exponent, Adv. Diff. Equ., 1, no. 2 (1996), 241-264. | MR 1364003 | Zbl 0847.35045

[26] Vázquez, J. L., A Strong Maximum Principle for Some Quasilinear Elliptic Equations, Applied Math. and Optimization., 12, no. 3 (1984), 191-202. | MR 768629

[27] Willem, M., Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, 24, Birkhäuser Boston, Inc., Boston, MA, 1996. | MR 1400007