Positive solutions to three classes of non-local fourth-order problems with derivative-dependent nonlinearities

In the article, we investigate three classes of fourth-order boundary value problems with dependence on all derivatives in nonlinearities under the boundary conditions involving Stieltjes integrals. A Gronwall-type inequality is employed to get an a priori bound on the third-order derivative term, and the theory of fixed-point index is used on suitable open sets to obtain the existence of positive solutions. The nonlinearities have quadratic growth in the third-order derivative term. Previous results in the literature are not applicable in our case, as shown by our examples

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

Sign-changing solution for an overdetermined elliptic problem on unbounded domain

We prove the existence of two 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. The domains bifurcate from the straight cylinder $B_1\times \mathbb{R}$, where $B_1$ is the unit ball in $\mathbb{R}^N$. These results can be regarded as counterexamples to the Berenstein conjecture on unbounded domain. Unlike most previous papers in this direction, a very delicate issue here is that there may be two-dimensional kernel space at some bifurcation point. Thus a Crandall-Rabinowitz type bifurcation theorem from high-dimensional kernel space is also established to achieve the goal.Comment: 32 pages, 3 figure