Multiplicity of positive solutions for some quasilinear Dirichlet problems on bounded domains in $\Bbb R^n$
Kandilakis, Dimitrios A. ; Lyberopoulos, Athanasios N.
Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003), p. 645-658 / Harvested from Czech Digital Mathematics Library
Access to full text
Full (PDF)

We show that, under appropriate structure conditions, the quasilinear Dirichlet problem $$ \cases -\operatorname{div}(|\nabla u|^{p-2}\nabla u) =f(x,u), \quad & x\in\Omega, \ u=0, & x\in\partial\Omega, \endcases $$ where $\Omega $is a bounded domain in $\Bbb R^n$, $1

Publié le : 2003-01-01
Classification:  35J20,  35J60,  35J70,  47J30 
@article{119419,
     author = {Dimitrios A. Kandilakis and Athanasios N. Lyberopoulos},
     title = {Multiplicity of positive solutions for some quasilinear Dirichlet problems on bounded domains in $\Bbb R^n$},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {44},
     year = {2003},
     pages = {645-658},
     zbl = {1105.35311},
     mrnumber = {2062881},
     language = {en},
     url = {http://dml.mathdoc.fr/item/119419}
}

Kandilakis, Dimitrios A.; Lyberopoulos, Athanasios N. Multiplicity of positive solutions for some quasilinear Dirichlet problems on bounded domains in $\Bbb R^n$. Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) pp. 645-658. http://gdmltest.u-ga.fr/item/119419/

Ambrosetti A.; Azorero J.G.; Peral I. Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal. 137 (1996), 219-242. (1996) | MR 1383017 | Zbl 0852.35045

Ambrosetti A.; Azorero J.G.; Peral I. Existence and multiplicity results for some nonlinear elliptic equations: a survey, Rend. Matem., Ser. VII, 20 (2000), 167-198. (2000) | MR 1823096 | Zbl 1011.35049

Ambrosetti A.; Brezis H.; Cerami G. Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994), 519-543. (1994) | MR 1276168 | Zbl 0805.35028

Ambrosetti A.; Rabinowitz P. Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381. (1973) | MR 0370183 | Zbl 0273.49063

Anane A. Simplicité et isolation de la première valeur propre du p-laplacien avec poids, C.R.A.S. Paris Série I 305 (1987), 725-728. (1987) | MR 0920052 | Zbl 0633.35061

Azorero J.G.; Alonso I.P. Some results about the existence of a second positive solution in a quasilinear critical problem, Indiana Univ. Math. J. 43 3 (1994), 941-957. (1994) | MR 1305954 | Zbl 0822.35048

Azorero J.G.; Alonso I.P.; Manfredi J.J. Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Comm. Contemp. Math. 2 3 (2000), 385-404. (2000) | MR 1776988

Boccardo L.; Escobedo M.; Peral I. A Dirichlet problem involving critical exponents, Nonlinear Anal. 24 (1995), 11 1639-1648. (1995) | MR 1328589 | Zbl 0828.35042

Brezis H.; Nirenberg L. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437-477. (1983) | MR 0709644 | Zbl 0541.35029

Brezis H.; Nirenberg L. $H^{1}$ versus $C^{1}$ local minimizers, C.R.A.S. Paris Série I 317 (1993), 465-472. (1993) | MR 1239032

Drábek P.; Hernandez J. Existence and uniqueness of positive solutions for some quasilinear elliptic problems, Nonlinear Anal. 44 2 (2001), 189-204. (2001) | MR 1816658 | Zbl 0991.35035

Ghoussoub N.; Preiss D. A general mountain pass principle for locating and classifying critical points, Ann. Inst. H. Poincaré Anal. Non Linéaire 6 5 (1989), 321-330. (1989) | MR 1030853 | Zbl 0711.58008

Guedda M.; Veron L. Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal. 13 8 (1989), 879-902. (1989) | MR 1009077 | Zbl 0714.35032

Ladyzhenskaya O.; Uraltseva N. Linear and Quasilinear Elliptic Equations, Academic Press, 1968. | MR 0244627

Lieberman G. Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), 1203-1219. (1988) | MR 0969499 | Zbl 0675.35042

Moser J. A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math. 13 (1960), 457-478. (1960) | MR 0170091 | Zbl 0111.09301

Sattinger D.H. Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 21 (1972), 979-1000. (1972) | MR 0299921 | Zbl 0223.35038

Vázquez J.L. A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), 191-202. (1984) | MR 0768629