Positive solutions for some generalized p –Laplacian type problems

Previous Article
Existence theorems for generalized nonlinear quadratic integral equations via a new fixed point result
DCDS-S Home
This Issue
Next Article
Positive radial solutions for the Minkowski-curvature equation with Neumann boundary conditions

July 2020, 13(7): 1935-1945. doi: 10.3934/dcdss.2020151

Positive solutions for some generalized $ p $–Laplacian type problems

Anna Maria Candela ^, and Addolorata Salvatore

Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, Via E. Orabona 4, 70125 Bari, Italy

^* Corresponding author: Anna Maria Candela

Dedicated to Patrizia Pucci on the occasion of her 65th birthday

Received July 2018 Revised December 2018 Published November 2019

Fund Project: Partially supported by Fondi di Ricerca di Ateneo 2015/16 and Research Funds INdAM – GNAMPA Project 2018 "Problemi ellittici semilineari: alcune idee variazionali"

In this paper, we prove the existence of nontrivial weak bounded solutions of the nonlinear elliptic problem

$ \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. $

where

$ \Omega \subset \mathbb {R}^N $

is an open bounded domain,

$ N\ge 3 $

, and

$ A(x, t, \xi) $

,

$ f(x, t) $

are given functions, with

$ A_t = \frac{\partial A}{\partial t} $

,

$ a = \nabla_\xi A $

.

To this aim, we use variational arguments which are adapted to our setting and exploit a weak version of the Cerami–Palais–Smale condition.

Furthermore, if

$ A(x, t, \xi) $

grows fast enough with respect to

$ t $

, then the nonlinear term related to

$ f(x, t) $

may have also a supercritical growth.

Keywords: Quasilinear elliptic equation, positive solution, weak Cerami–Palais–Smale condition, Ambrosetti–Rabinowitz condition, supercritical growth.

Mathematics Subject Classification: Primary: 35J20, 35J92; Secondary: 35J25, 58E05.

Citation: Anna Maria Candela, Addolorata Salvatore. Positive solutions for some generalized $ p $–Laplacian type problems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 1935-1945. doi: 10.3934/dcdss.2020151

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.

[1]	Antonio Azzollini. On a functional satisfying a weak Palais-Smale condition. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1829-1840. doi: 10.3934/dcds.2014.34.1829
[2]	A. Azzollini. Erratum to: "On a functional satisfying a weak Palais-Smale condition". Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4987-4987. doi: 10.3934/dcds.2014.34.4987
[3]	Scott Nollet, Frederico Xavier. Global inversion via the Palais-Smale condition. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 17-28. doi: 10.3934/dcds.2002.8.17
[4]	Jun Wang, Junxiang Xu, Fubao Zhang. Homoclinic orbits for superlinear Hamiltonian systems without Ambrosetti-Rabinowitz growth condition. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 1241-1257. doi: 10.3934/dcds.2010.27.1241
[5]	Alain Hertzog, Antoine Mondoloni. Existence of a weak solution for a quasilinear wave equation with boundary condition. Communications on Pure and Applied Analysis, 2002, 1 (2) : 191-219. doi: 10.3934/cpaa.2002.1.191
[6]	Lauren M. M. Bonaldo, Elard J. Hurtado, Olímpio H. Miyagaki. Multiplicity results for elliptic problems involving nonlocal integrodifferential operators without Ambrosetti-Rabinowitz condition. Discrete and Continuous Dynamical Systems, 2022 doi: 10.3934/dcds.2022017
[7]	Vincenzo Ambrosio. Periodic solutions for a superlinear fractional problem without the Ambrosetti-Rabinowitz condition. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2265-2284. doi: 10.3934/dcds.2017099
[8]	Alessio Fiscella. Schrödinger–Kirchhoff–Hardy $ p $–fractional equations without the Ambrosetti–Rabinowitz condition. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 1993-2007. doi: 10.3934/dcdss.2020154
[9]	Eun Bee Choi, Yun-Ho Kim. Existence of nontrivial solutions for equations of $p(x)$-Laplace type without Ambrosetti and Rabinowitz condition. Conference Publications, 2015, 2015 (special) : 276-286. doi: 10.3934/proc.2015.0276
[10]	Sami Aouaoui, Rahma Jlel. On some elliptic equation in the whole euclidean space $ \mathbb{R}^2 $ with nonlinearities having new exponential growth condition. Communications on Pure and Applied Analysis, 2020, 19 (10) : 4771-4796. doi: 10.3934/cpaa.2020211
[11]	Tsung-Fang Wu. Multiplicity of positive solutions for a semilinear elliptic equation in $R_+^N$ with nonlinear boundary condition. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1675-1696. doi: 10.3934/cpaa.2010.9.1675
[12]	Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825
[13]	Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71
[14]	Claudianor Oliveira Alves, Paulo Cesar Carrião, Olímpio Hiroshi Miyagaki. Signed solution for a class of quasilinear elliptic problem with critical growth. Communications on Pure and Applied Analysis, 2002, 1 (4) : 531-545. doi: 10.3934/cpaa.2002.1.531
[15]	Cong He, Hongjun Yu. Large time behavior of the solution to the Landau Equation with specular reflective boundary condition. Kinetic and Related Models, 2013, 6 (3) : 601-623. doi: 10.3934/krm.2013.6.601
[16]	Elhoussine Azroul, Abdelmoujib Benkirane, and Mohammed Shimi. On a nonlocal problem involving the fractional $ p(x,.) $-Laplacian satisfying Cerami condition. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3479-3495. doi: 10.3934/dcdss.2020425
[17]	Khadijah Sharaf. A perturbation result for a critical elliptic equation with zero Dirichlet boundary condition. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1691-1706. doi: 10.3934/dcds.2017070
[18]	Xiang-Dong Fang. A positive solution for an asymptotically cubic quasilinear Schrödinger equation. Communications on Pure and Applied Analysis, 2019, 18 (1) : 51-64. doi: 10.3934/cpaa.2019004
[19]	Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5217-5226. doi: 10.3934/dcdsb.2020340
[20]	Claudianor O. Alves, César T. Ledesma. Multiplicity of solutions for a class of fractional elliptic problems with critical exponential growth and nonlocal Neumann condition. Communications on Pure and Applied Analysis, 2021, 20 (5) : 2065-2100. doi: 10.3934/cpaa.2021058

2020 Impact Factor: 2.425

Tools

Metrics

Other articles
by authors

on AIMS
Anna Maria Candela Addolorata Salvatore
on Google Scholar
Anna Maria Candela Addolorata Salvatore

2020 Impact Factor: 2.425

Tools

Article outline

2020 Impact Factor: 2.425

Tools

Metrics

Other articles
by authors

on AIMS
Anna Maria Candela Addolorata Salvatore
on Google Scholar
Anna Maria Candela Addolorata Salvatore

[Back to Top]