Browsing by Subject "ground state solution"

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Access status: Open Access ,
Fractional p&q-Laplacian problems with potentials vanishing at infinity
(Wydawnictwa AGH, 2020) Isernia, Teresa
In this paper we prove the existence of a positive and a negative ground state weak solution for the following class of fractional $p\&q$-Laplacian problems $\begin{aligned} (-\Delta)_{p}^{s} u + (-\Delta)_{q}^{s} u + V(x) (|u|^{p-2}u + |u|^{q-2}u)= K(x) f(u) \quad \text{ in } \mathbb{R}^{N},\end{aligned}$ where $s \in (0,1)$, $1\lt p\lt q \lt\frac{N}{s}$, $V: \mathbb{R}^{N}\to \mathbb{R}$ and $K: \mathbb{R}^{N}\to \mathbb{R}$ are continuous, positive functions, allowed for vanishing behavior at infinity, $f$ is a continuous function with quasicritical growth and the leading operator $(-\Delta)^{s}_{t}$, with $t\in \{p,q\}$, is the fractional $t$-Laplacian operator.
Access status: Open Access ,
Properties of the least action level and the existence of ground state solution to fractional elliptic equation with harmonic potential
(Wydawnictwa AGH, 2024) Torres Ledesma, César E.; Gutierrez, Hernán C.; Rodríguez, Jesús A.; Bonilla, Manuel M.
In this article we consider the following fractional semilinear elliptic equation $(-\Delta)^su+|x|^2u =\omega u+|u|^{2\sigma}u \quad \text{ in } \mathbb{R}^N,$ where $s\in (0,1)$, $N\gt 2s$, $\sigma\in (0,\frac{2s}{N-2s})$ and $\omega\in (0, \lambda_1)$. By using variational methods we show the existence of a symmetric decreasing ground state solution of this equation. Moreover, we study some continuity and differentiability properties of the ground state level. Finally, we consider a bifurcation type result