Superlinear fractional elliptic problems via the nonlinear Rayleigh quotient with two parameters

Edcarlos D. Silva, M. L. M. Carvalho, C. Goulart and M. L. Silva

Mathematische Nachrichten, 2024, vol. 297, issue 3, 1062-1091

Abstract: It is established existence of weak solutions for nonlocal elliptic problems driven by the fractional Laplacian where the nonlinearity is indefinite in sign. More specifically, we shall consider the following nonlocal elliptic problem: (−Δ)su+V(x)u=μa(x)|u|q−2u−λ|u|p−2uinRN,u∈Hs(RN),$$\begin{equation*} {\left\lbrace \def\eqcellsep{&}\begin{array}{rcl}(-\Delta )^s u +V(x)u & = & \mu a(x)|u|^{q-2}u-\lambda |u|^{p-2}u \mbox{ in }\, \mathbf {R}^N, \\[3pt] u\in H^s(\mathbf {R}^N),&&{} \end{array} \right.} \end{equation*}$$where s∈(0,1),s 0$\mu , \lambda >0$. The potentials V,a:RN→R$V, a : \mathbf {R}^N \rightarrow \mathbf {R}$ satisfy some extra assumptions. The main feature is to find sharp parameters λ>0$\lambda > 0$ and μ>0$\mu > 0$ where the Nehari method can be applied. In order to do that, we employ the nonlinear Rayleigh quotient together a fine analysis on the fibering maps associated to the energy functional. It is important to mention also that for each parameters λ>0$\lambda > 0$ and μ>0$\mu > 0$, there exist degenerate points in the Nehari set that give serious difficulties. Furthermore, we consider nonlinearities that are superlinear at the origin and superlinear at infinity. In order to overcome these difficulties, we apply some estimates together with a carefully analysis on the fibering maps. Here, we also consider the asymptotic behavior of the weak solutions for our main problem when λ→0$\lambda \rightarrow 0$ or μ→∞$\mu \rightarrow \infty$. Furthermore, we consider a nonexistence result for our main problem under some appropriate conditions on the parameters λ>0$\lambda >0$ and μ>0$\mu > 0$.

Date: 2024
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