## Tuesday Seminar of Analysis

Seminar information archive ～09/13｜Next seminar｜Future seminars 09/14～

Date, time & place | Tuesday 16:00 - 17:30 156Room #156 (Graduate School of Math. Sci. Bldg.) |
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Organizer(s) | ISHIGE Kazuhiro, SAKAI Hidetaka, ITO Kenichi |

### 2023/07/11

16:00-17:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Nodal solutions for a class of degenerate BVP’s (English)

https://forms.gle/S3VgMSWg9wUP69cY6

**Julian López-Gómez**(Complutense University of Madrid)Nodal solutions for a class of degenerate BVP’s (English)

[ Abstract ]

In this talk we characterize the existence of nodal solutions for a generalized class of one-dimensional diffusive logistic type equations, including

\[−u''=\lambda u−a(x)u^3,\quad x∈[0,L],\]

under the boundary conditions $u(0)=u(L)=0$, where $\lambda$ is regarded as a bifurcation parameter, and the non-negative weight function $a(x)$ vanishes on some subinterval

\[ [\alpha,\beta]\subset (0,L)\]

with $\alpha<\beta$.

At a later stage, the general case when $a(x)$ vanishes on finitely many subintervals with the same length is analyzed. Finally, we construct some examples with classical non-degenerate weights, with $a(x)>0$ for all $x∈[0,L]$, where the BVP has an arbitrarily large number of solutions with one node in $(0,L)$. These are the first examples of this nature constructed in the literature.

References:

P. Cubillos, J. López-Gómez and A. Tellini, Multiplicity of nodal solutions in classical non-degenerate logistic equations, El. Res. Archive 30 (2022), 898—928.

J. López-Gómez, M. Molina-Meyer and P. H. Rabinowitz, Global bifurcation diagrams of one-node solutions on a class of degenerate boundary value problems, Disc. Cont. Dyn. Syst. B 22 (2017), 923—946.

J. López-Gómez and P. H. Rabinowitz, Nodal solutions for a class of degenerate one dimensional BVP’s, Top. Meth. Nonl. Anal. 49 (2017), 359—376.

J. López-Gómez and P. H. Rabinowitz, The estructure of the set of 1-node solutions for a class of degenerate BVP’s, J. Differential Equations 268 (2020), 4691—4732.

P. H. Rabinowitz, A note on a anonlinear eigenvalue problem for a class of differential equations, J. Differential Equations 9 (1971), 536—548.

[ Reference URL ]In this talk we characterize the existence of nodal solutions for a generalized class of one-dimensional diffusive logistic type equations, including

\[−u''=\lambda u−a(x)u^3,\quad x∈[0,L],\]

under the boundary conditions $u(0)=u(L)=0$, where $\lambda$ is regarded as a bifurcation parameter, and the non-negative weight function $a(x)$ vanishes on some subinterval

\[ [\alpha,\beta]\subset (0,L)\]

with $\alpha<\beta$.

At a later stage, the general case when $a(x)$ vanishes on finitely many subintervals with the same length is analyzed. Finally, we construct some examples with classical non-degenerate weights, with $a(x)>0$ for all $x∈[0,L]$, where the BVP has an arbitrarily large number of solutions with one node in $(0,L)$. These are the first examples of this nature constructed in the literature.

References:

P. Cubillos, J. López-Gómez and A. Tellini, Multiplicity of nodal solutions in classical non-degenerate logistic equations, El. Res. Archive 30 (2022), 898—928.

J. López-Gómez, M. Molina-Meyer and P. H. Rabinowitz, Global bifurcation diagrams of one-node solutions on a class of degenerate boundary value problems, Disc. Cont. Dyn. Syst. B 22 (2017), 923—946.

J. López-Gómez and P. H. Rabinowitz, Nodal solutions for a class of degenerate one dimensional BVP’s, Top. Meth. Nonl. Anal. 49 (2017), 359—376.

J. López-Gómez and P. H. Rabinowitz, The estructure of the set of 1-node solutions for a class of degenerate BVP’s, J. Differential Equations 268 (2020), 4691—4732.

P. H. Rabinowitz, A note on a anonlinear eigenvalue problem for a class of differential equations, J. Differential Equations 9 (1971), 536—548.

https://forms.gle/S3VgMSWg9wUP69cY6