A Variant of the Mountain Pass Theorem and Variational Gluing

Montecchiari, Piero; Rabinowitz, Paul H.

doi:10.1007/s00032-020-00318-3

Cited by 4 publications

(2 citation statements)

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“…Clearly, these degenerate situations are due to the lack of uniqueness for the associated Cauchy problem at the origin. These results should be compared with the gluing of variational solutions found in [28].…”

Section: The Simplest One-dimensional Prototype Modelmentioning

confidence: 99%

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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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Section: The Simplest One-dimensional Prototype Modelmentioning

confidence: 99%

“…Symbolics dynamics results have also been obtained for more general settings using gluing arguments from the calculus of variations. See, e.g., the survey paper [28] and the references therein. For such gluing arguments, the heads and tails are replaced by a collection of basic solutions of the equations.…”

Section: Final Commentsmentioning

confidence: 99%