Geodesic
Multiplicity of positive solutions for some quasilinear Dirichlet problems on bounded domains in $\Bbb R^n$
Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) no. 4, pp. 645-658 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $\Bbb R^n$, $1$, admits two positive solutions $u_{0}$, $u_{1}$ in $W_{0}^{1,p}(\Omega)$ such that $0$ in $\Omega $, while $u_{0}$ is a local minimizer of the associated Euler-Lagrange functional.
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 $\Bbb R^n$, $1$, admits two positive solutions $u_{0}$, $u_{1}$ in $W_{0}^{1,p}(\Omega)$ such that $0$ in $\Omega $, while $u_{0}$ is a local minimizer of the associated Euler-Lagrange functional.
Classification : 35J20, 35J60, 35J70, 47J30
Keywords: $p$-Laplacian; positive solutions; sub- and supersolutions; local minimizers; Palais-Smale condition 
@article{CMUC_2003_44_4_a8,
     author = {Kandilakis, Dimitrios A. and Lyberopoulos, Athanasios N.},
     title = {Multiplicity of positive solutions for some quasilinear {Dirichlet} problems on bounded domains in $\Bbb R^n$},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {645--658},
     year = {2003},
     volume = {44},
     number = {4},
     mrnumber = {2062881},
     zbl = {1105.35311},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_2003_44_4_a8/}
}
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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) no. 4, pp. 645-658. http://geodesic.mathdoc.fr/item/CMUC_2003_44_4_a8/