A variable exponent diffusion problem of concave-convex nature

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

DOI: http://dx.doi.org/10.12775/TMNA.2016.019

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 $\Omega\subset \R^N$ is a bounded smooth domain, $\lambda\gt 0$ 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 $\lambda^* \gt 0$ such that no positive solutions are possible for $\lambda \gt \lambda^*$, the existence and structural properties of a branch of minimal solutions, $u_\lambda$, $0 \lt \lambda \lt \lambda^*$, and, finally, the existence for all $ \lambda \in (0,\lambda^*)$ of a second positive solution.

Keywords


Variable exponent; concave-convex; minimal solution; a priori bounds; Leray-Schauder degree

Full Text:

PREVIEW FULL TEXT

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. Garcı́a-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 RN 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. Garcı́a 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. Garcı́a–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. Garcı́a–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. Garcı́a-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. Garcı́a-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.

J. Garcı́a-Melián, J.D. Rossi and J. Sabina de Lis, Existence, asymptotic behavior and uniqueness for large solutions to ∆u = eq(x)u , Adv. Nonlinear Stud. 9 (2) (2009), 395–424.

J. Garcı́a-Melián, J.D. Rossi and J. Sabina de Lis, 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.

B. Gidas and J. Spruck, 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. SpringerVerlag, New York, 2004.

A. V. Lair, A. Mohammed, Entire large solutions to elliptic equations of power nonlinearities with variable exponents. Adv. Nonlinear Stud. 13 (2013), no. 3, 699–719.

P. Lindqvist, On the equation div(|∇u|p−2 ∇u) + λ|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.


Refbacks

  • There are currently no refbacks.

Partnerzy platformy czasopism