Curves of equiharmonic solutions, and problems at resonance

TL;DR

This paper investigates the solution structure of a semilinear Dirichlet problem at resonance, analyzing solution curves and multiplicity through a parameter continuation method, with applications to numerical computations.

Contribution

It introduces a novel approach to study solution curves at resonance by fixing certain coefficients and varying others, providing new existence and multiplicity results.

Findings

Established solution existence and multiplicity at resonance points.

Developed a numerical implementation illustrating theoretical results.

Analyzed solutions at both principal and higher eigenvalues.

Abstract

We consider the semilinear Dirichlet problem \[ \Delta u+kg(u)=\mu _1 \varphi _1+\cdots +\mu _n \varphi _n+e(x) \;\; \mbox{for $x \in \Omega$}, \;\; u=0 \;\; \mbox{on $\partial \Omega$}, \] where $\varphi _k$ is the $k$-th eigenfunction of the Laplacian on $\Omega$ and $e(x) \perp \varphi _k$, $k=1, \ldots, n$. Write the solution in the form $u(x)= \Sigma _{i=1}^n \xi _i \varphi _i+U(x)$, with $ U \perp \varphi _k$, $k=1, \ldots, n$. Starting with $k=0$, when the problem is linear, we continue the solution in $k$ by keeping $\xi =(\xi _1, \ldots,\xi _n)$ fixed, but allowing for $\mu =(\mu _1, \ldots,\mu _n)$ to vary. Studying the map $\xi \rightarrow \mu$ provides us with the existence and multiplicity results for the above problem. We apply our results to problems at resonance, at both the principal and higher eigenvalues. Our approach is suitable for numerical calculations, which we implement, illustrating our results.