On the first curve of Fučik Spectrum Of $p$-fractional Laplacian Operator with nonlocal normal boundary conditions

Abstract

In this article, we study the Fučik spectrum of the $p$-fractional Laplace operator with nonlocal normal derivative conditions which is defined as the set of all $(a,b)\in \mb R^2$ such that $$ \mc (F_p)\left\{ \begin{array}{lr} Λ_{n,p}(1-\al)(-Δ)_{p}^{\al} u + |u|^{p-2}u = \frac{χ_{Ω_\e}}{\e} (a (u^{+})^{p-1} - b (u^{-})^{p-1}) \;\quad \text{in}\; Ω,\quad \\ \mc{N}_{\al,p} u = 0 \; \quad \mbox{in}\; \mb R^n \setminus \overlineΩ, \end{array} \right. $$ has a non-trivial solution $u$, where $\Omega$ is a bounded domain in $\mb R^n$ with Lipschitz boundary, $p\ge 2$, $n>p \al $, $\e, \al \in(0,1)$ and $Ω{_\e}:=\{x \in Ω: d(x,\pa Ω)\leq \e \}$. We showed existence of the first non-trivial curve $\mc C$ of this spectrum which is used to obtain the variational characterization of a second eigenvalue of the problem $\mc (F_p)$. We also discuss some properties of this curve $\mc C$, e.g. Lipschitz continuous, strictly decreasing and asymptotic behaviour and nonresonance with respect to the Fučik spectrum.

arXiv admin note: text overlap with arXiv:1306.4761