Fu\v{c}ik spectrum for the operator with rapidly increasing weight and applications

TL;DR

This paper investigates the Fučík spectrum for an operator with rapidly increasing weight on \\mathbb{R}^N, establishing properties of the first nontrivial curve and applying results to solutions of nonlinear equations with asymptotically linear behavior.

Contribution

It introduces the first nontrivial curve of the Fučík spectrum for this operator on unbounded space and analyzes its properties, extending prior bounded domain results.

Findings

Existence of a first nontrivial curve \\mathcal{C} with Lipschitz continuity

The curve \\mathcal{C} is strictly decreasing and has specific asymptotic behavior

Application to multiplicity of solutions for nonlinear problems with asymptotically linear nonlinearity

Abstract

In this paper, we study the Fu\v{c}ik spectrum for the operator with rapidly increasing weight, which is defined as a set $\Sigma$ comprising those $(\alpha, \beta) \in \mathbb{R}^2$ such that \begin{equation*} \left\{\begin{array}{l} L u:=-\Delta u-\frac{1}{2}(x \cdot \nabla u)=\alpha u^{+}-\beta u^{-}, \text{in}\ \mathbb{R}^N,\\ u\in X, \end{array}\right. \end{equation*} has a non-trivial solution $u$, where, $N\geq1$, $u^{ \pm}=\max ( \pm u, 0)$, $u=u^{+}-u^{-}$. The existence of a first nontrivial curve $\mathcal{C}$ of this spectrum, along with some of its properties (e.g., Lipschitz continuity, strict decrease and asymptotic behavior) is investigated in this paper. Our difficulty is that the problem is defined on the whole space $\mathbb{R}^N$, and therefore certain estimates do not carry over from the Fu\v{c}ik problem on bounded domains. As an application, we establish the multiplicity of solutions to the following problem \begin{equation*} \left\{\begin{array}{l} -\Delta u-\frac{1}{2}(x \cdot \nabla u)=f(x,u), \text{in}\ \mathbb{R}^N,\\ u\in X, \end{array}\right. \end{equation*} where, $N\geq1$ and the nonlinearity $f$ is asymptotically linear at zero and at infinity.