Unilateral global bifurcation and nodal solutions for the pp-Laplacian with sign-changing weight

In this paper, we shall establish a Dancer-type unilateral global bifurcation result for a class of quasilinear elliptic problems with sign-changing weight. Under some natural hypotheses on perturbation function, we show that $(\mu_k^\nu(p),0)$ is a bifurcation point of the above problems and there are two distinct unbounded continua, $(\mathcal{C}_{k}^\nu)^+$ and $(\mathcal{C}_{k}^\nu)^-$, consisting of the bifurcation branch $\mathcal{C}_{k}^\nu$ from $(\mu_k^\nu(p), 0)$, where $\mu_k^\nu(p)$ is the $k$-th positive or negative eigenvalue of the linear problem corresponding to the above problems, $\nu\in\{+,-\}$. As the applications of the above unilateral global bifurcation result, we study the existence of nodal solutions for a class of quasilinear elliptic problems with sign-changing weight. Moreover, based on the bifurcation result of Dr\'{a}bek and Huang (1997) [\ref{DH}], we study the existence of one-sign solutions for a class of high dimensional quasilinear elliptic problems with sign-changing weight

Global bifurcation from intervals for Sturm-Liouville problems which are not linearizable

In this paper, we study unilateral global bifurcation which bifurcates from the trivial solutions axis or from infinity for nonlinear Sturm--Liouville problems of the form \begin{equation} \left\{ \begin{array}{l} -\left(pu'\right)'+qu=\lambda au+af\left(x,u,u',\lambda\right)+g\left(x,u,u',\lambda\right)\,\,\text{for}\,\, x\in(0,1),\\ b_0u(0)+c_0u'(0)=0,\\ b_1u(1)+c_1u'(1)=0, \end{array} \right.\nonumber \end{equation} where $a\in C([0, 1], [0,+\infty))$ and $a(x)\not\equiv 0$ on any subinterval of $[0, 1]$, $f,g\in C([0,1]\times\mathbb{R}^3,\mathbb{R})$. Suppose that $f$ and $g$ satisfy \begin{equation} \vert f(x,\xi,\eta,\lambda)\vert\leq M_0\vert \xi\vert+M_1\vert \eta\vert,\,\, \forall x\in [0,1]\,\,\text{and}\,\,\lambda \in\mathbb{R}, \nonumber \end{equation} \begin{equation} g(x,\xi,\eta,\lambda)=o(\vert \xi\vert+\vert \eta\vert),\,\, \text{uniformly in}\,\, x\in [0,1]\,\,\text{and}\,\,\lambda \in \Lambda,\nonumber \end{equation} as either $\vert \xi\vert+\vert \eta\vert\rightarrow 0$ or $\vert \xi\vert+\vert \eta\vert\rightarrow +\infty$, for some constants $M_0$, $M_1$, and any bounded interval $\Lambda$

Bifurcation of sign-changing solutions for an overdetermined boundary problem in bounded domains

We obtain a continuous family of nontrivial domains $\Omega_s\subset \mathbb{R}^N$ ($N=2,3$ or $4$), bifurcating from a small ball, such that the problem \begin{equation} -\Delta u=u-\left(u^+\right)^3\,\, \text{in}\,\,\Omega_s, \,\, u=0,\,\,\partial_\nu u=\text{const}\,\,\text{on}\,\,\partial\Omega_s \nonumber \end{equation} has a sign-changing bounded solution. Compared with the recent result obtained by Ruiz, here we obtain a family domains $\Omega_s$ by using Crandall-Rabinowitz bifurcation theorem instead of a sequence of domains.Comment: 17 pages, 1 figur

Bifurcation domains from any high eigenvalue for an overdetermined elliptic problem

Let $\lambda_k$ be the $k$-th eigenvalue of the zero-Dirichlet Laplacian on the unit ball for $k\in \mathbb{N^+}$. We prove the existence of $k$ smooth families of unbounded domains in $\mathbb{R}^{N+1}$ with $N\geq1$ such that \begin{equation} -\Delta u=\lambda u\,\, \text{in}\,\,\Omega, \,\, u=0,\,\,\partial_\nu u=\text{const}\,\,\text{on}\,\,\partial\Omega\nonumber \end{equation} admits a sign-changing solution with changing the sign by $k-1$ times. This nonsymmetric sign-changing solution can be seen as the perturbation of the eigenfunction corresponding to $\lambda_k$ with $k\geq2$. The main contribution of the paper is to provide some counterexamples to the Berenstein conjecture on unbounded domain.Comment: 26pp. arXiv admin note: substantial text overlap with arXiv:2304.0555