# Article

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Keywords:
$p$-Laplacian; positive solutions; sub- and supersolutions; local minimizers; Palais-Smale condition
Summary:
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 $\Bbb R^n$, $1<p<+\infty$, admits two positive solutions $u_{0}$, $u_{1}$ in $W_{0}^{1,p}(\Omega)$ such that $0<u_{0}\leq u_{1}$ in $\Omega$, while $u_{0}$ is a local minimizer of the associated Euler-Lagrange functional.
References:
[1] Ambrosetti A., Azorero J.G., Peral I.: Multiplicity results for some nonlinear elliptic equations. J. Funct. Anal. 137 (1996), 219-242. MR 1383017 | Zbl 0852.35045
[2] 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. MR 1823096 | Zbl 1011.35049
[3] Ambrosetti A., Brezis H., Cerami G.: Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal. 122 (1994), 519-543. MR 1276168 | Zbl 0805.35028
[4] Ambrosetti A., Rabinowitz P.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14 (1973), 349-381. MR 0370183 | Zbl 0273.49063
[5] 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. MR 0920052 | Zbl 0633.35061
[6] 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. MR 1305954 | Zbl 0822.35048
[7] 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. MR 1776988
[8] Boccardo L., Escobedo M., Peral I.: A Dirichlet problem involving critical exponents. Nonlinear Anal. 24 (1995), 11 1639-1648. MR 1328589 | Zbl 0828.35042
[9] Brezis H., Nirenberg L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math. 36 (1983), 437-477. MR 0709644 | Zbl 0541.35029
[10] Brezis H., Nirenberg L.: $H^{1}$ versus $C^{1}$ local minimizers. C.R.A.S. Paris Série I 317 (1993), 465-472. MR 1239032
[11] Drábek P., Hernandez J.: Existence and uniqueness of positive solutions for some quasilinear elliptic problems. Nonlinear Anal. 44 2 (2001), 189-204. MR 1816658 | Zbl 0991.35035
[12] 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. MR 1030853 | Zbl 0711.58008
[13] Guedda M., Veron L.: Quasilinear elliptic equations involving critical Sobolev exponents. Nonlinear Anal. 13 8 (1989), 879-902. MR 1009077 | Zbl 0714.35032
[14] Ladyzhenskaya O., Uraltseva N.: Linear and Quasilinear Elliptic Equations. Academic Press, 1968. MR 0244627
[15] Lieberman G.: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 12 (1988), 1203-1219. MR 0969499 | Zbl 0675.35042
[16] 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. MR 0170091 | Zbl 0111.09301
[17] Sattinger D.H.: Monotone methods in nonlinear elliptic and parabolic boundary value problems. Indiana Univ. Math. J. 21 (1972), 979-1000. MR 0299921 | Zbl 0223.35038
[18] Vázquez J.L.: A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. 12 (1984), 191-202. MR 0768629

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