Hamilton P. Bueno | Searcharxiv

arXiv2024

Positive solutions of singular elliptic problems with unbounded exponents and unbounded convection term

In this paper, we study the existence of a solution for a class of Dirichlet problems with a singularity and a convection term. Precisely, we consider the existence of a positive solution to the Dirichlet problem $$-Δ_p u = \fracλ{u^α} + f(x,u,\nabla u)$$ in a bounded, smooth domain $Ω$. The convection term has exponents with no upper limitations neither in $u$ nor in $\nabla u$. This is somewhat unexpected and rare. So, we address a wide range of problems not yet contained in the literature. The solution of the problem combines the definition of an auxiliary problem, the method of sub- and super-solution and Schauder's fixed point theorem.

math.AP↗

arXiv2024

On a fractional harmonic oscillator: existence and inexistence of solution, regularity and decay properties

Under simple hypotheses on the nonlinearity $f$, we consider the fractional harmonic operator problem \begin{equation}\label{abstr}\sqrt{-Δ+|x|^2}\,u=f(x,u)\ \ \textrm{in }\ \mathbb{R}^N\end{equation} or, since we work in the extension setting $\mathbb{R}^{N+1}_+$, $$\left\{\begin{aligned} -Δv +|x|^2v&=0, &&\mbox{in} \ \mathbb{R}^{N+1}_+,\\ -\displaystyle\frac{\partial v}{\partial x}(x,0)&=f(x,v(x,0)) &&\mbox{on} \ \mathbb{R}^{N}\cong\partial \mathbb{R}^{N+1}_+.\end{aligned}\right.$$ Defining the space $$\mathcal{H}(\mathbb{R}^{N+1}_+)=\left\{v\in H^1(\mathbb{R}^{N+1}_+): \iint_{\mathbb{R}^{N+1}_+}\left[|\nabla v|^2+|x|^2v^2\right]dx dy<\infty\right\},$$ we prove that the embedding $$\mathcal{H}(\mathbb{R}^{N+1}_+)\hookrightarrow L^{q}(\mathbb{R}^N)$$ is compact. We also obtain a Pohozaev-type identity for this problem, show that in the case $f(x,u)=|u|^{p^*-2}u$ the problem has no non-trivial solution, compare the extremal attached to this problem with the one of the space $H^1(\mathbb{R}^{N+1}_+)$, prove that the solution $u$ of our problem belongs to $L^p(\mathbb{R}^N)$ for all $p\in [2,\infty]$ and satisfy the polynomial decay $|u(x)|\leq C/|x|$ for any $|x|>M$. Finally, we prove the existence of a solution to a superlinear critical problem in the case $f(x,u)=|u|^{2^*-2}u+λ|u|^{q-1}$, $1<q<2^*-1$.

math.AP↗