Rich dynamics in planar systems with heterogeneous nonnegative weights

López-Gómez, Julián; Muñoz-Hernández, Eduardo; Zanolin, Fabio

doi:10.3934/cpaa.2023020

Cited by 3 publications

(2 citation statements)

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“…(See [27] for similar computations). Performing the change of variables u = cs, 0 < u < c, 0 < s < 1, it follows from (3.18) that…”

Section: The Simplest One-dimensional Prototype Modelmentioning

confidence: 89%

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Non-negative solution of a sublinear elliptic problem

Lopez-Gomez,

Rabinowitz,

Zanolin

2024

Preprint

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In this paper the existence of solutions, $(\lambda,u)$,of the problem$$\left\lbrace\begin{array}{ll} -\D u=\l u -a(x)|u|^{p-1}u & \quad \hbox{in }\O,\\ u=0 &\quad \hbox{on}\;\;\p\O, \end{array}\right.$$is explored for $0 < p < 1$. When $p>1$, it is known that thereis an unbounded component of such solutions bifurcating from$(\s_1, 0)$, where $\s_1$ is the smallest eigenvalue of $-\D$ in$\O$ under Dirichlet boundary conditions on $\p\O$. Thesesolutions have $u \in P$, the interior of the positive cone. Thecontinuation argument usedwhen $p>1$ to keep $u \in P$fails if $0 < p < 1$. Nevertheless when $0 < p < 1$,we are still able to show that there is a component ofsolutions bifurcating from $(\s_1, \infty)$, unbounded outside ofa neighborhood of $(\s_1, \infty)$, and having $u \gneq 0$. Thisnon-negativity for $u$ cannot be improved as is shown via adetailed analysis of the simplest autonomous one-dimensionalversion of the problem: its set of non-negative solutionspossesses a countable set of components, each of them consistingof positive solutions with a fixed (arbitrary) number of bumps.Finally, the structure of these components is fully described.

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“…(See [27] for similar computations). Performing the change of variables u = cs, 0 < u < c, 0 < s < 1, it follows from (3.18) that…”

Section: The Simplest One-dimensional Prototype Modelmentioning

confidence: 89%

“…Some other work on sublinear problems will be mentioned next. In [27], a very preliminary study has been carried out recently. The existence of a (unique) solution with u > 0 in (0, L) and u ′ (0) > 0 was established for every λ < π 2 when a is a negative constant.…”

Section: Introductionmentioning

confidence: 99%