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 & Continuous Dynamical Systems - A, 2014, 34 (7) : 2847-2860. doi: 10.3934/dcds.2014.34.2847
References:
[1]

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M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability,, Arch. Rational Mech. Anal., 52 (1973), 161.   Google Scholar

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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.  doi: 10.1016/0362-546X(79)90091-9.  Google Scholar

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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.  doi: 10.1090/S0002-9947-1989-0951886-3.  Google Scholar

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P. Korman, A global solution curve for a class of periodic problems, including the pendulum equation,, Z. Angew. Math. Phys., 58 (2007), 749.  doi: 10.1007/s00033-006-6014-6.  Google Scholar

[10]

P. Korman, Curves of equiharmonic solutions, and ranges of nonlinear equations,, Adv. Differential Equations, 14 (2009), 963.   Google Scholar

[11]

P. Korman, Global solution curves for boundary value problems, with linear part at resonance,, Nonlinear Anal., 71 (2009), 2456.  doi: 10.1016/j.na.2009.01.128.  Google Scholar

[12]

E. M. Landesman and A. C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance,, J. Math. Mech., 19 (): 609.   Google Scholar

[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.  doi: 10.1016/0022-247X(81)90166-9.  Google Scholar

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L. Nirenberg, Topics in Nonlinear Functional Analysis,, Lecture Notes, (1974), 1973.   Google Scholar

show all references

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.  doi: 10.1017/S0308210500017017.  Google Scholar

[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.  doi: 10.1007/BF02412022.  Google Scholar

[3]

J. A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis,, Cambridge Studies in Advanced Mathematics, (1993).   Google Scholar

[4]

M. S. Berger and E. Podolak, On the solutions of a nonlinear Dirichlet problem,, Indiana Univ. Math. J., 24 (): 837.   Google Scholar

[5]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability,, Arch. Rational Mech. Anal., 52 (1973), 161.   Google Scholar

[6]

L. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998).   Google Scholar

[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.  doi: 10.1016/0362-546X(79)90091-9.  Google Scholar

[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.  doi: 10.1090/S0002-9947-1989-0951886-3.  Google Scholar

[9]

P. Korman, A global solution curve for a class of periodic problems, including the pendulum equation,, Z. Angew. Math. Phys., 58 (2007), 749.  doi: 10.1007/s00033-006-6014-6.  Google Scholar

[10]

P. Korman, Curves of equiharmonic solutions, and ranges of nonlinear equations,, Adv. Differential Equations, 14 (2009), 963.   Google Scholar

[11]

P. Korman, Global solution curves for boundary value problems, with linear part at resonance,, Nonlinear Anal., 71 (2009), 2456.  doi: 10.1016/j.na.2009.01.128.  Google Scholar

[12]

E. M. Landesman and A. C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance,, J. Math. Mech., 19 (): 609.   Google Scholar

[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.  doi: 10.1016/0022-247X(81)90166-9.  Google Scholar

[14]

L. Nirenberg, Topics in Nonlinear Functional Analysis,, Lecture Notes, (1974), 1973.   Google Scholar

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