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Hossein Tehrani

Publications and source records attributed to Hossein Tehrani.

7 recordsLinked to original sources
arXiv2025

A note on Trudinger-Moser Functions and Reproducing Kernel Hilbert Spaces

After a brief review of the definition of the Trudinger-Moser functions in dimension $N=2$ and some basic notions in the theory of ``Reproducing Kernel Hilbert Spaces (RKHS)'', we will show that there is a close connection between those two topics. More precisely, among other things, we start by considering a properly chosen multiple of the classical Trudinger-Moser family of functions in dimension $N=2$, which we denote by $ \gamma_t (r) := \frac{1}{2\pi}\min\,\{ log \frac{1}{r}, log \frac{1}{t} \}\,, $ where $0 < t , r < 1$, and using the theory of RKHS we will show that $\gamma_t$ can be seen as a ``bounded'' (linear) evaluation functional $u \longrightarrow u(t)$ for functions $u$ in a suitable Hilbert Space ${\cal H}$. A slightly different definition for a ''Trudinger-Moser'' type function will also be considered for $N\geq 3$.

math.FA↗
arXiv2025

Nonlocal critical elliptic equations in homogeneous fractional Sobolev spaces

We prove new multiplicity results for some nonlocal critical growth elliptic equations in homogeneous fractional Sobolev spaces. The proofs are based on an abstract critical point theorem based on the ${\mathbb Z}_2$-cohomological index and on a novel regularity result for fractional $p$-Laplacian equations as well as on some compact embeddings.

math.AP↗
arXiv2025

Global $L^\infty$ and decay estimate for fractional $p$-Laplacian equations in $D^{s,p}(\R^N)$

In this paper we present a new global $L^\infty$-estimate for solutions $u\in D^{s,p}(\R^N)$ of the fractional $p$-Laplacian equation % $$ u\in D^{s,p}(\R^N): (-\Delta_p)^s u=f(x,u) \quad\mbox{in }\R^N, $$ % of the form % $$ \|u\|_{\infty}\le C \Phi(\|u\|_{\beta}) $$ % for some $\beta> p$, where $\Phi: \R^+\to \R^+$ is a data independent function with $\lim_{s\to 0^+}\Phi(s)=0$. The obtained $L^\infty$-estimate is used to prove a decay estimate based on pointwise estimates in terms of nonlinear Wolff potentials. Taking advantage of both the $L^\infty$ and decay estimate we prove a Brezis-Nirenberg type result regarding $D^{s,2}(\R^N)$ versus $C_b\left(\R^N, 1+|x|^{N-2s}\right)$ local minimizers.

math.AP↗
arXiv2023

Global $L^\infty$-estimate for general quasilinear elliptic equations in arbitrary domains of $\mathbb{R}^N$

In this paper our main goal is to present a new global $L^\infty$-estimate for a general class of quasilinear elliptic equations of the form $$ -div \mathcal{A}(x,u,\nabla u)=\mathcal{B}(x,u,\nabla u) $$ under minimal structure conditions on the functions $\mathcal{A}$ and $\mathcal{B}$, and in arbitrary domains of $\mathbb{R}^N$. The main focus and the novelty of the paper is to prove $L^\infty$-estimate of the form $$ |u|_{\infty, \Omega}\le C \Phi(|u|_{\beta,\Omega}) $$ where $\Phi: \mathbb{R}^+\to \mathbb{R}^+$ is a data independent function with $\lim_{s\to 0^+}\Phi(s)=0$.

math.AP↗
arXiv2014

Positive solutions to logistic type equations with harvesting

We use comparison principles, variational arguments and a truncation method to obtain positive solutions to logistic type equations with harvesting both in $\mathbb{R}^N$ and in a bounded domain $Ω\subset\mathbb{R}^N$, with $N\geq 3$, when the carrying capacity of the environment is not constant. By relaxing the growth assumption on the coefficients of the differential equation we derive a new equation which is easily solved. The solution of this new equation is then used to produce a positive solution of our original problem.

math.AP↗
arXiv2014

On the Fučik spectrum of the wave operator and an asymptotically linear problem

We study generalized solutions of the nonlinear wave equation $$u_{tt}-u_{ss}=au^+-bu^-+p(s,t,u),$$ with periodic conditions in $t$ and homogeneous Dirichlet conditions in $s$, under the assumption that the ratio of the period to the length of the interval is two. When $p\equiv 0$ and $λ$ is a nonzero eigenvalue of the wave operator, we give a proof of the existence of two families of curves (which may coincide) in the Fučik spectrum intersecting at $(λ,λ)$. This result is known for some classes of self-adjoint operators (which does not cover the situation we consider here), but in a smaller region than ours. Our approach is based on a dual variational formulation and is also applicable to other operators, such as the Laplacian. In addition, we prove an existence result for the nonhomogeneous situation, when the pair $(a,b)$ is not `between' the Fučik curves passing through $(λ,λ)\neq(0,0)$ and $p$ is a continuous function, sublinear at infinity.

math.AP↗