# American Institute of Mathematical Sciences

July  2014, 34(7): 2847-2860. doi: 10.3934/dcds.2014.34.2847

## Curves of equiharmonic solutions, and problems at resonance

 1 Department Of Mathematical Sciences, University Of Cincinnati, Cincinnati Ohio 45221-0025

Received  June 2013 Revised  September 2013 Published  December 2013

We consider the semilinear Dirichlet problem $\Delta u+kg(u)=\mu_1 \varphi_1+\cdots +\mu _n \varphi_n+e(x) \; \; for \; x \in \Omega, \; \; u=0 \; \; on \; \partial \Omega,$ where $\varphi_k$ is the $k$-th eigenfunction of the Laplacian on $\Omega$ and $e(x) \perp \varphi_k$, $k=1, \ldots,n$. Write the solution in the form $u(x)= \Sigma _{i=1}^n \xi _i \varphi_i+U(x)$, with $U \perp \varphi_k$, $k=1, \ldots,n$. Starting with $k=0$, when the problem is linear, we continue the solution in $k$ by keeping $\xi =(\xi _1, \ldots,\xi _n)$ fixed, but allowing for $\mu =(\mu _1, \ldots,\mu _n)$ to vary. Studying the map $\xi \rightarrow \mu$ provides us with the existence and multiplicity results for the above problem. We apply our results to problems at resonance, at both the principal and higher eigenvalues. Our approach is suitable for numerical calculations, which we implement, illustrating our results.
Citation: Philip Korman. Curves of equiharmonic solutions, and problems at resonance. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2847-2860. doi: 10.3934/dcds.2014.34.2847
##### References:
 [1] H. Amann and P. Hess, A multiplicity result for a class of elliptic boundary value problems, Proc. Roy. Soc. Edinburgh Sect. A, 84 (1979), 145-151. doi: 10.1017/S0308210500017017. [2] A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. Mat. Pura Appl. (4), 93 (1972), 231-246. doi: 10.1007/BF02412022. [3] J. A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis, Cambridge Studies in Advanced Mathematics, 34, Cambridge University Press, Cambridge, 1993. [4] M. S. Berger and E. Podolak, On the solutions of a nonlinear Dirichlet problem,, Indiana Univ. Math. J., 24 (): 837. [5] M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180. [6] L. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998. [7] D. G. de Figueiredo and W. M. Ni, Perturbations of second order linear elliptic problems by nonlinearities without Landesman-Lazer condition, Nonlinear Anal., 3 (1979), 629-634. doi: 10.1016/0362-546X(79)90091-9. [8] R. Iannacci, M. N. Nkashama and J. R. Ward, Jr., Nonlinear second order elliptic partial differential equations at resonance, Trans. Amer. Math. Soc., 311 (1989), 711-726. doi: 10.1090/S0002-9947-1989-0951886-3. [9] P. Korman, A global solution curve for a class of periodic problems, including the pendulum equation, Z. Angew. Math. Phys., 58 (2007), 749-766. doi: 10.1007/s00033-006-6014-6. [10] P. Korman, Curves of equiharmonic solutions, and ranges of nonlinear equations, Adv. Differential Equations, 14 (2009), 963-984. [11] P. Korman, Global solution curves for boundary value problems, with linear part at resonance, Nonlinear Anal., 71 (2009), 2456-2467. doi: 10.1016/j.na.2009.01.128. [12] E. M. Landesman and A. C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance,, J. Math. Mech., 19 (): 609. [13] A. C. Lazer and P. J. McKenna, On the number of solutions of a nonlinear Dirichlet problem, J. Math. Anal. Appl., 84 (1981), 282-294. doi: 10.1016/0022-247X(81)90166-9. [14] L. Nirenberg, Topics in Nonlinear Functional Analysis, Lecture Notes, 1973-1974, Courant Institute of Mathematical Sciences, New York University, New York, 1974.

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##### References:
 [1] H. Amann and P. Hess, A multiplicity result for a class of elliptic boundary value problems, Proc. Roy. Soc. Edinburgh Sect. A, 84 (1979), 145-151. doi: 10.1017/S0308210500017017. [2] A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. Mat. Pura Appl. (4), 93 (1972), 231-246. doi: 10.1007/BF02412022. [3] J. A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis, Cambridge Studies in Advanced Mathematics, 34, Cambridge University Press, Cambridge, 1993. [4] M. S. Berger and E. Podolak, On the solutions of a nonlinear Dirichlet problem,, Indiana Univ. Math. J., 24 (): 837. [5] M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180. [6] L. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998. [7] D. G. de Figueiredo and W. M. Ni, Perturbations of second order linear elliptic problems by nonlinearities without Landesman-Lazer condition, Nonlinear Anal., 3 (1979), 629-634. doi: 10.1016/0362-546X(79)90091-9. [8] R. Iannacci, M. N. Nkashama and J. R. Ward, Jr., Nonlinear second order elliptic partial differential equations at resonance, Trans. Amer. Math. Soc., 311 (1989), 711-726. doi: 10.1090/S0002-9947-1989-0951886-3. [9] P. Korman, A global solution curve for a class of periodic problems, including the pendulum equation, Z. Angew. Math. Phys., 58 (2007), 749-766. doi: 10.1007/s00033-006-6014-6. [10] P. Korman, Curves of equiharmonic solutions, and ranges of nonlinear equations, Adv. Differential Equations, 14 (2009), 963-984. [11] P. Korman, Global solution curves for boundary value problems, with linear part at resonance, Nonlinear Anal., 71 (2009), 2456-2467. doi: 10.1016/j.na.2009.01.128. [12] E. M. Landesman and A. C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance,, J. Math. Mech., 19 (): 609. [13] A. C. Lazer and P. J. McKenna, On the number of solutions of a nonlinear Dirichlet problem, J. Math. Anal. Appl., 84 (1981), 282-294. doi: 10.1016/0022-247X(81)90166-9. [14] L. Nirenberg, Topics in Nonlinear Functional Analysis, Lecture Notes, 1973-1974, Courant Institute of Mathematical Sciences, New York University, New York, 1974.
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