Topological Methods in Nonlinear Analysis

Positive solutions for a nonconvex elliptic Dirichlet problem with superlinear response

Abstract

The existence of bounded solutions of the Dirichlet problem for a ceratin class of elliptic partial differential equations is discussed here. We use variational methods based on the subdifferential theory and the comparison principle for difergence form operators. We present duality and variational principles for this problem. As a consequences of the duality we obtain also the variational principle for minimizing sequences of $J$ which gives a measure of a duality gap between primal and dual functional for approximate solutions.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 27, Number 1 (2006), 177-194.

Dates
First available in Project Euclid: 12 May 2016

https://projecteuclid.org/euclid.tmna/1463081852

Mathematical Reviews number (MathSciNet)
MR2236416

Zentralblatt MATH identifier
1135.35326

Citation

Nowakowski, Andrzej; Orpel, Aleksandra. Positive solutions for a nonconvex elliptic Dirichlet problem with superlinear response. Topol. Methods Nonlinear Anal. 27 (2006), no. 1, 177--194. https://projecteuclid.org/euclid.tmna/1463081852

References

• R. A. Adams, Sobolev Spaces, Academic Press, New York (1975) \ref\key 2
• J. Aubin and I. Ekeland, Applied Nonlinear Analysis, New York, John Wiley (1984) \ref\key 3
• S. Ahmad, A. C. Lazer and J. L. Paul, Elementary critical point theory and perturbations of elliptic boundary value problems at resonance , Indiana Univ. Math. J., 25(1976), 933–944 \ref\key 4
• A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications , J. Funct. Anal., 14 (1973), 349–381 \ref\key 5
• A. Benkirane and A. Elmahi, A strongly nonlinear elliptic equation having natural growth terms and $L^1$ data , Nolinear Anal. Ser. A: Theory Methods, 39(2000, no. 4), 403–411 \ref\key 6
• H. Brezis, Opérateurs Maximaux Monotones, North–Holand, Amsterdam (1973) \ref\key 7
• L. Cesari, Optimization –- Theory and Applications, Springer–Verlag, New York (1983) \ref\key 8
• K. C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations , J. Math. Anal. Appl., 80 (1981), 102–129 \ref\key 9
• M. Degiovanni and S. Zani, Multiple solutions of semilinear elliptic equations with one-sided growth conditions, Nonlinear operator theory , Math. Comput. Modelling, 32(2000, no. 11–13), 1377–1393 \ref\key 10
• C. Ebmeyer and J. Frehse, Mixed boundary value problems for nonlinear elliptic equations with $p$-structure in nonsmooth domains , Differential Integral Equations, 14 (2001, no. 7), 801–820 \ref\key 11
• I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North–Holand, Amsterdam (1976) \ref\key 12
• D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer–Verlag(1983) \ref\key 13
• N. Grenon, Existence and comparison results for quasilinear elliptic equations with critical growth in the gradient , J. Differential Equations, 171 (2001, no. 1), 1–23 \ref\key 14
• L. Huang and Y. Li, Multiple solutions of an elliptic equation , Fuijan Shifan Daxue Xuebao Ziran Kexue Ban, 16 (2000, no. 3), 15–19 \ref\key 15
• I. Kuzin and S. Pohozaev, Entire solutions of semilinear elliptic equations , Progr. Nonlinear Differential Equations Appl., 33 (1997) \ref\key 16
• J. L. Lions and E. Magenes, Problémes aux limites non homogénes et applications, Dunod, Paris (1968) \ref\key 17
• P. Magrone, On a class of semilinear elliptic equations with potential changing sign , Dynam. System Appl., 9 (2000, no. 4), 459–467 \ref\key 18
• J. Mawhin and M. Willem M., Critical Points Theory and Hamiltonian Systems, Springer–Verlag, New York (1989) \ref\key 20
• Ch. B. Morrey, Multiple Integrals in the Calculus of Variations, Springer–Verlag, Berlin (1966) \ref\key 21
• J. Milnor, Morse Theory, Princeton Univ. Press, Princeton, N.J. (1963) \ref\key 22
• A. Nowakowski A. and A. Rogowski, Periodic solutions in the calculus of variations and differential equations , Nonlinear Anal., 21 , 537–546 (1993) \ref\key 24 ––––, Dependence on parameters for the Dirichlet problem with superlinear nonlinearities , Topol. Methods Nonlinear Anal., 16 (2000), 145–160 \ref\key 25 ––––, Multiple positive solutions for a nonlinear Dirichlet problem with nonconvex vector-valued response , Proc. Roy. Soc. Edinburgh Sec. A, 135 (2005, no. 1) \ref\key 26
• Z. Peihao Z. and Z. Chengkui, On the infinitely many positive solutions of a supercritical elliptic problem , Nonlinear Anal., Ser. A: Theory Methods, 44 (2001, no. 1), 123–139 \ref\key 27
• P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations. , CBMS, AMS, University of Miami (1986) \ref\key 28
• B. Ricceri, Existence and location of solutions to the Dirichlet problem for a class of nonlinear elliptic equations , Appl. Math. Lett., 14 (2001, no. 2), 143–148 \ref\key 29
• M. Struwe, Infinitely many critical points for functionals which are not even and applications to superlinear boundary problems , Manuscripta Math., 32 (1980), 335–364 \ref\key 30
• Y. Sun and S. Wu, On a nonlinear elliptic equation with sublinear term at the origin , Acta Math. Sci. Ser. A Chin. Ed., 20 (2000, no. 4), 461–467 \ref\key 31
• Z. Wang, Nonlinear boundary value problems with concave nonlinearities near the origin , NoDEA Nonlinear Differentian Equations Appl., 8(2001, no. 1), 15–33 \ref\key 32
• M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, Basel, Boston, Berlin, Birkhäuser, 24 (1996) \ref\key 33
• X. Xu, The boundary value problem for nonlinear elliptic equations in annular domains , Acta Math. Sci. Ser. A Chin. Ed., suppl., 20(2000), 675–683