Topological Methods in Nonlinear Analysis

A variable exponent diffusion problem of concave-convex nature

Jorge García-Melián, Julio D. Rossi, and José C. Sabina de Lis

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Abstract

We deal with the problem $$ \begin{cases} -\Delta u = \lambda u^{q(x)} & \text{if } x\in \Omega,\\ u = 0 & \text{if } x\in \p\Omega, \end{cases} $$ where $\Omega\subset \mathbb R^N$ is a bounded smooth domain, $\lambda> 0$ is a parameter and the exponent $q(x)$ is a continuous positive function that takes values both greater than and less than one in $\overline{\Omega}$. It is therefore a kind of concave-convex problem where the presence of the interphase $q=1$ in $\overline{\Omega}$ poses some new difficulties to be tackled. The results proved in this work are the existence of $\lambda^* > 0$ such that no positive solutions are possible for $\lambda > \lambda^*$, the existence and structural properties of a branch of minimal solutions, $u_\lambda$, $0 < \lambda < \lambda^*$, and, finally, the existence for all $ \lambda \in (0,\lambda^*)$ of a second positive solution.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 47, Number 2 (2016), 613-639.

Dates
First available in Project Euclid: 13 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1468413755

Digital Object Identifier
doi:10.12775/TMNA.2016.019

Mathematical Reviews number (MathSciNet)
MR3559923

Zentralblatt MATH identifier
1362.35116

Citation

García-Melián, Jorge; Rossi, Julio D.; de Lis, José C. Sabina. A variable exponent diffusion problem of concave-convex nature. Topol. Methods Nonlinear Anal. 47 (2016), no. 2, 613--639. doi:10.12775/TMNA.2016.019. https://projecteuclid.org/euclid.tmna/1468413755


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References

  • H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), no. 4, 620–709.
  • H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Differential. Equations 146 (1998), 336–374.
  • A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994), 519–543.
  • A. Ambrosetti, J. García-Azorero and I. Peral, Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal. 137 (1996), 219–242.
  • L. Boccardo, M. Escobedo and I. Peral, A Dirichlet problem involving critical exponents, Nonlinear Anal. 24 (1995), no. 11, 1639–1648.
  • H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.
  • H. Brezis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl. (9) 58 (1979), no. 2, 137–151.
  • H. Brezis and L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal. 10 (1986), no. 1, 55–64.
  • F. Charro, E. Colorado, I. Peral, Multiplicity of solutions to uniformly elliptic Fully Nonlinear equations with concave-convex right hand side, J. Differential Equations 246 (2009), 4221–4248.
  • C. Cortázar, M. Elgueta and P. Felmer, On a semilinear elliptic problem in $\R^N$ with a non-Lipschitzian nonlinearity, Adv. Differential Equations 1 (1996), no. 2, 199–218.
  • M. G. Crandall and P.H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal. 52 (1973), 161–180.
  • M. Delgado, J. López-Gómez and A. Suárez, Combining linear and non-linear diffusion, Adv. Nonlinear Stud. 4 (2004), 273–287.
  • P.C. Fife, Mathematical aspects of reacting and diffusing systems, Lecture Notes in Biomathematics no. 28, Springer, Berlin, 1979.
  • J.P. García Azorero, J.J. Manfredi and I. Peral Alonso, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Comm. Contemp. Maths. 2 (2000), 385–404.
  • J. García–Azorero and I. Peral Alonso, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term, Trans. Amer. Math. Soc. 323 (1991), 877–895.
  • J. García–Azorero, I. Peral Alonso, Some results about the existence of a second positive solution in a quasilinear critical problem, Indiana Univ. Math. J. 43 (1994), no. 3, 941–957.
  • J. García-Azorero, I. Peral and J.D. Rossi, A convex-concave problem with a nonlinear boundary condition, J. Differential Equations 198 (1) (2004), 91–128.
  • J. García-Melián, J.D. Rossi and J. Sabina de Lis, Large solutions for the Laplacian with a power nonlinearity given by a variable exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), 889–902.
  • ––––, Existence, asymptotic behavior and uniqueness for large solutions to $\De u = e^{q(x)u}$, Adv. Nonlinear Stud. 9 (2) (2009), 395–424.
  • ––––, A convex-concave elliptic problem with a parameter on the boundary condition, Discrete Contin. Dyn. Syst. 32 (2012), no. 4, 1095–1124.
  • B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations. Comm. Pure Appl. Math. 34 (1981), 525–598.
  • ––––, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 (1981), no. 8, 883–901.
  • D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, Springer–Verlag, 1983.
  • R. Gómez-Reñasco and J. López-Gómez, The effect of varying coefficients on the dynamics of a class of superlinear indefinite reaction-diffusion equations, J. Differential Equations 167 (2000), no. 1, 36–72.
  • H. Kielhöfer, Bifurcation theory. An introduction with applications to PDEs. Springer-Verlag, New York, 2004.
  • A. V. Lair, A. Mohammed, Entire large solutions to elliptic equations of power non-linearities with variable exponents. Adv. Nonlinear Stud. 13 (2013), no. 3, 699–719.
  • P. Lindqvist, On the equation $\hbox{div}(|\nabla u|^{p-2}\nabla u)+\la |u|^{p-2} u=0$, Proc. Amer. Math. Soc. 109 (1990), 157–164.
  • P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev. 24 (1982), 441–467.
  • J. López-Gómez, Varying stoichometric exponents I: Classical steady states and metasolutions, Adv. Nonl. Stud. 3 (2003), 327–354.
  • J. López-Gómez and A. Suárez, Combining fast, linear and slow diffusion, Topol. Methods Nonl. Anal. 23 (2004), 275–300.
  • F. Mignot and J.P. Puel, Sur une classe de problèmes nonlinéaires avec nonlinéarité positive, croissante, convexe. Comm. Partial Differential Equations 5 (1980), 791–836.
  • W. M. Ni, The mathematics of diffusion, CBMS-NSF Regional Conference Series in Applied Mathematics, 82. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2011.
  • C. V. Pao, Nonlinear parabolic and elliptic equations, Plenum Press, New York, 1992.
  • P. Pucci and Q. Zhang, Existence of entire solutions for a class of variable exponent elliptic equations, J. Differential Equations 257 (2014), 1529–1566.
  • D. Ruiz, A priori estimates and existence of positive solutions for strongly nonlinear problems, J. Differential Equations 199 (2004), 96–114.
  • J. Sabina de Lis and S. Segura de León, Multiplicity of solutions to a nonlinear boundary value problem of concave–convex type, Adv. Nonlinear Studies 15 (2015), 61–90.
  • J. Serrin, Local behavior of solutions of quasilinear equations, Acta Math. 111 (1964), 247–302.
  • J. Serrin, H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math. 189 (2002), 79–142.
  • J. Smoller, Shock waves and reaction-diffusion equations, Springer–Verlag, New York, 1994.
  • M. Struwe, Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems, Springer–Verlag, Berlin, 2008.