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

show all references

References:
 [1] Shu-Zhi Song, Shang-Jie Chen, Chun-Lei Tang. Existence of solutions for Kirchhoff type problems with resonance at higher eigenvalues. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6453-6473. doi: 10.3934/dcds.2016078 [2] Jiafeng Liao, Peng Zhang, Jiu Liu, Chunlei Tang. Existence and multiplicity of positive solutions for a class of Kirchhoff type problems at resonance. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1959-1974. doi: 10.3934/dcdss.2016080 [3] Shujie Li, Zhitao Zhang. Multiple solutions theorems for semilinear elliptic boundary value problems with resonance at infinity. Discrete & Continuous Dynamical Systems, 1999, 5 (3) : 489-493. doi: 10.3934/dcds.1999.5.489 [4] Jijiang Sun, Chun-Lei Tang. Resonance problems for Kirchhoff type equations. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 2139-2154. doi: 10.3934/dcds.2013.33.2139 [5] Sergiu Aizicovici, Nikolaos S. Papageorgiou, Vasile Staicu. Nonlinear Dirichlet problems with double resonance. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1147-1168. doi: 10.3934/cpaa.2017056 [6] Nikolaos S. Papageorgiou, Patrick Winkert. Double resonance for Robin problems with indefinite and unbounded potential. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 323-344. doi: 10.3934/dcdss.2018018 [7] Francesco Della Pietra, Ireneo Peral. Breaking of resonance for elliptic problems with strong degeneration at infinity. Communications on Pure & Applied Analysis, 2011, 10 (2) : 593-612. doi: 10.3934/cpaa.2011.10.593 [8] D. Bonheure, C. Fabry, D. Smets. Periodic solutions of forced isochronous oscillators at resonance. Discrete & Continuous Dynamical Systems, 2002, 8 (4) : 907-930. doi: 10.3934/dcds.2002.8.907 [9] Nikolaos S. Papageorgiou, Calogero Vetro, Francesca Vetro. Multiple solutions for (p, 2)-equations at resonance. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 347-374. doi: 10.3934/dcdss.2019024 [10] Francisco Odair de Paiva, Humberto Ramos Quoirin. Resonance and nonresonance for p-Laplacian problems with weighted eigenvalues conditions. Discrete & Continuous Dynamical Systems, 2009, 25 (4) : 1219-1227. doi: 10.3934/dcds.2009.25.1219 [11] Shouchuan Hu, Nikolaos S. Papageorgiou. Double resonance for Dirichlet problems with unbounded indefinite potential and combined nonlinearities. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2005-2021. doi: 10.3934/cpaa.2012.11.2005 [12] Anna Capietto, Walter Dambrosio, Tiantian Ma, Zaihong Wang. Unbounded solutions and periodic solutions of perturbed isochronous Hamiltonian systems at resonance. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1835-1856. doi: 10.3934/dcds.2013.33.1835 [13] Martin Bauer, Martins Bruveris, Philipp Harms, Peter W. Michor. Soliton solutions for the elastic metric on spaces of curves. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1161-1185. doi: 10.3934/dcds.2018049 [14] David L. Finn. Convexity of level curves for solutions to semilinear elliptic equations. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1335-1343. doi: 10.3934/cpaa.2008.7.1335 [15] Jinlong Bai, Desheng Li, Chunqiu Li. A note on multiplicity of solutions near resonance of semilinear elliptic equations. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3351-3365. doi: 10.3934/cpaa.2019151 [16] Dmitriy Yu. Volkov. The Hopf -- Hopf bifurcation with 2:1 resonance: Periodic solutions and invariant tori. Conference Publications, 2015, 2015 (special) : 1098-1104. doi: 10.3934/proc.2015.1098 [17] Xuelei Wang, Dingbian Qian, Xiying Sun. Periodic solutions of second order equations with asymptotical non-resonance. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4715-4726. doi: 10.3934/dcds.2018207 [18] Shiwang Ma. Nontrivial periodic solutions for asymptotically linear hamiltonian systems at resonance. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2361-2380. doi: 10.3934/cpaa.2013.12.2361 [19] Laura Olian Fannio. Multiple periodic solutions of Hamiltonian systems with strong resonance at infinity. Discrete & Continuous Dynamical Systems, 1997, 3 (2) : 251-264. doi: 10.3934/dcds.1997.3.251 [20] Wenming Zou. Multiple solutions results for two-point boundary value problem with resonance. Discrete & Continuous Dynamical Systems, 1998, 4 (3) : 485-496. doi: 10.3934/dcds.1998.4.485

2019 Impact Factor: 1.338