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Positive radial solutions for the Minkowski-curvature equation with Neumann boundary conditions
Positive solutions for some generalized $ p $–Laplacian type problems
Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, Via E. Orabona 4, 70125 Bari, Italy |
$ \left\{ \begin{array}{ll} - {\rm div} (a(x,u,\nabla u)) + A_t(x,u,\nabla u) = f(x,u) &\hbox{in $\Omega$,}\\ u \ge 0 &\hbox{in $\Omega$,}\\ u\ = \ 0 & \hbox{on $\partial\Omega$,} \end{array} \right. $ |
$ \Omega \subset \mathbb {R}^N $ |
$ N\ge 3 $ |
$ A(x, t, \xi) $ |
$ f(x, t) $ |
$ A_t = \frac{\partial A}{\partial t} $ |
$ a = \nabla_\xi A $ |
$ A(x, t, \xi) $ |
$ t $ |
$ f(x, t) $ |
References:
[1] |
A. Ambrosetti and P. H. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
doi: 10.1016/0022-1236(73)90051-7. |
[2] |
D. Arcoya and L. Boccardo,
Critical points for multiple integrals of the calculus of variations, Arch. Rational Mech. Anal., 134 (1996), 249-274.
doi: 10.1007/BF00379536. |
[3] |
D. Arcoya and L. Boccardo,
Some remarks on critical point theory for nondifferentiable functionals, NoDEA Nonlinear Differential Equations Appl., 6 (1999), 79-100.
doi: 10.1007/s000300050066. |
[4] |
G. Autuori and P. Pucci,
Existence of entire solutions for a class of quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 977-1009.
doi: 10.1007/s00030-012-0193-y. |
[5] |
G. Autuori and P. Pucci,
Elliptic problems involving the fractional Laplacian in $ \mathbb {R}^N$, J. Differential Equations, 255 (2013), 2340-2362.
doi: 10.1016/j.jde.2013.06.016. |
[6] |
A. M. Candela and G. Palmieri,
Multiple solutions of some nonlinear variational problems, Adv. Nonlinear Stud., 6 (2006), 269-286.
doi: 10.1515/ans-2006-0209. |
[7] |
A. M. Candela and G. Palmieri,
Infinitely many solutions of some nonlinear variational equations, Calc. Var. Partial Differential Equations, 34 (2009), 495-530.
doi: 10.1007/s00526-008-0193-2. |
[8] |
A. M. Candela and G. Palmieri, Some abstract critical point theorems and applications, Discrete Contin. Dyn. Syst., Dynamical Systems, Differential Equations and Applications, 7th AIMS Conference, Suppl., (2009), 133–142. |
[9] |
A. M. Candela and G. Palmieri,
Multiplicity results for some quasilinear equations in lack of symmetry, Adv. Nonlinear Anal., 1 (2012), 121-157.
|
[10] |
A. M. Candela and G. Palmieri, An abstract three critical points theorem and applications, in Proceedings of Dynamic Systems and Applications, Dynamic Publishers Inc., Atlanta, 6
(2012), 70–77. |
[11] |
A. M. Candela and G. Palmieri, Multiplicity results for some nonlinear elliptic problems with asymptotically $p$-linear terms, Calc. Var. Partial Differential Equations, 56 (2017), Art. 72, 39 pp.
doi: 10.1007/s00526-017-1170-4. |
[12] |
A. M. Candela, G. Palmieri and K. Perera,
Multiple solutions for $p$-Laplacian type problems with asymptotically $p$-linear terms via a cohomological index theory, J. Differential Equations, 259 (2015), 235-263.
doi: 10.1016/j.jde.2015.02.007. |
[13] |
A. M. Candela, G. Palmieri and A. Salvatore, Multiple solutions for some symmetric supercritical problems, Commun. Contemp. Math., (to appear).
doi: 10.1142/S0219199719500755. |
[14] |
A. M. Candela, G. Palmieri and A. Salvatore,
Infinitely many solutions for quasilinear elliptic equations with lack of symmetry, Nonlinear Anal., 172 (2018), 141-162.
doi: 10.1016/j.na.2018.02.011. |
[15] |
A. M. Candela and A. Salvatore,
Infinitely many solutions for some nonlinear supercritical problems with break of symmetry, Opuscula Math., 39 (2019), 175-194.
doi: 10.7494/OpMath.2019.39.2.175. |
[16] |
A. Canino,
Multiplicity of solutions for quasilinear elliptic equations, Topol. Methods Nonlinear Anal., 6 (1995), 357-370.
doi: 10.12775/TMNA.1995.050. |
[17] |
P. Lindqvist,
On the equation div $ (|\nabla u|^{p-2}\nabla u) + \lambda |u|^{p-2}u =0$, Proc. Amer. Math. Soc., 109 (1990), 157-164.
doi: 10.1090/S0002-9939-1990-1007505-7. |
[18] |
B. Pellacci and M. Squassina,
Unbounded critical points for a class of lower semicontinuous functionals, J. Differential Equations, 201 (2004), 25-62.
doi: 10.1016/j.jde.2004.03.002. |
[19] |
P. Pucci and V. Rădulescu,
Combined effects in quasilinear elliptic problems with lack of compactness, Rend. Lincei Mat. Appl., 22 (2011), 189-205.
doi: 10.4171/RLM/595. |
[20] |
N. S. Trudinger,
On Harnack type inequalities and their application to quasilinear elliptic equations, Commun. Pure Appl. Math., 20 (1967), 721-747.
doi: 10.1002/cpa.3160200406. |
show all references
References:
[1] |
A. Ambrosetti and P. H. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
doi: 10.1016/0022-1236(73)90051-7. |
[2] |
D. Arcoya and L. Boccardo,
Critical points for multiple integrals of the calculus of variations, Arch. Rational Mech. Anal., 134 (1996), 249-274.
doi: 10.1007/BF00379536. |
[3] |
D. Arcoya and L. Boccardo,
Some remarks on critical point theory for nondifferentiable functionals, NoDEA Nonlinear Differential Equations Appl., 6 (1999), 79-100.
doi: 10.1007/s000300050066. |
[4] |
G. Autuori and P. Pucci,
Existence of entire solutions for a class of quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 977-1009.
doi: 10.1007/s00030-012-0193-y. |
[5] |
G. Autuori and P. Pucci,
Elliptic problems involving the fractional Laplacian in $ \mathbb {R}^N$, J. Differential Equations, 255 (2013), 2340-2362.
doi: 10.1016/j.jde.2013.06.016. |
[6] |
A. M. Candela and G. Palmieri,
Multiple solutions of some nonlinear variational problems, Adv. Nonlinear Stud., 6 (2006), 269-286.
doi: 10.1515/ans-2006-0209. |
[7] |
A. M. Candela and G. Palmieri,
Infinitely many solutions of some nonlinear variational equations, Calc. Var. Partial Differential Equations, 34 (2009), 495-530.
doi: 10.1007/s00526-008-0193-2. |
[8] |
A. M. Candela and G. Palmieri, Some abstract critical point theorems and applications, Discrete Contin. Dyn. Syst., Dynamical Systems, Differential Equations and Applications, 7th AIMS Conference, Suppl., (2009), 133–142. |
[9] |
A. M. Candela and G. Palmieri,
Multiplicity results for some quasilinear equations in lack of symmetry, Adv. Nonlinear Anal., 1 (2012), 121-157.
|
[10] |
A. M. Candela and G. Palmieri, An abstract three critical points theorem and applications, in Proceedings of Dynamic Systems and Applications, Dynamic Publishers Inc., Atlanta, 6
(2012), 70–77. |
[11] |
A. M. Candela and G. Palmieri, Multiplicity results for some nonlinear elliptic problems with asymptotically $p$-linear terms, Calc. Var. Partial Differential Equations, 56 (2017), Art. 72, 39 pp.
doi: 10.1007/s00526-017-1170-4. |
[12] |
A. M. Candela, G. Palmieri and K. Perera,
Multiple solutions for $p$-Laplacian type problems with asymptotically $p$-linear terms via a cohomological index theory, J. Differential Equations, 259 (2015), 235-263.
doi: 10.1016/j.jde.2015.02.007. |
[13] |
A. M. Candela, G. Palmieri and A. Salvatore, Multiple solutions for some symmetric supercritical problems, Commun. Contemp. Math., (to appear).
doi: 10.1142/S0219199719500755. |
[14] |
A. M. Candela, G. Palmieri and A. Salvatore,
Infinitely many solutions for quasilinear elliptic equations with lack of symmetry, Nonlinear Anal., 172 (2018), 141-162.
doi: 10.1016/j.na.2018.02.011. |
[15] |
A. M. Candela and A. Salvatore,
Infinitely many solutions for some nonlinear supercritical problems with break of symmetry, Opuscula Math., 39 (2019), 175-194.
doi: 10.7494/OpMath.2019.39.2.175. |
[16] |
A. Canino,
Multiplicity of solutions for quasilinear elliptic equations, Topol. Methods Nonlinear Anal., 6 (1995), 357-370.
doi: 10.12775/TMNA.1995.050. |
[17] |
P. Lindqvist,
On the equation div $ (|\nabla u|^{p-2}\nabla u) + \lambda |u|^{p-2}u =0$, Proc. Amer. Math. Soc., 109 (1990), 157-164.
doi: 10.1090/S0002-9939-1990-1007505-7. |
[18] |
B. Pellacci and M. Squassina,
Unbounded critical points for a class of lower semicontinuous functionals, J. Differential Equations, 201 (2004), 25-62.
doi: 10.1016/j.jde.2004.03.002. |
[19] |
P. Pucci and V. Rădulescu,
Combined effects in quasilinear elliptic problems with lack of compactness, Rend. Lincei Mat. Appl., 22 (2011), 189-205.
doi: 10.4171/RLM/595. |
[20] |
N. S. Trudinger,
On Harnack type inequalities and their application to quasilinear elliptic equations, Commun. Pure Appl. Math., 20 (1967), 721-747.
doi: 10.1002/cpa.3160200406. |
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