Browsing by Subject "semipositone problems"

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Access status: Open Access ,
Positive stationary solutions of convection-diffusion equations for superlinear sources
(Wydawnictwa AGH, 2022) Orpel, Aleksandra
We investigate the existence and multiplicity of positive stationary solutions for acertain class of convection-diffusion equations in exterior domains. This problem leads to the following elliptic equation $\Delta u(x)+f(x,u(x))+g(x)x\cdot \nabla u(x)=0,$ for $x\in \Omega_{R}=\{ x \in \mathbb{R}^n, \|x\|\gt R \}$, $n \gt 2$. The goal of this paper is to show that our problem possesses an uncountable number of nondecreasing sequences of minimal solutions with finite energy in a neighborhood of infinity. We also prove that each of these sequences generates another solution of the problem. The case when $f(x,\cdot)$ may be negative at the origin, so-called semipositone problem, is also considered. Our results are based on a certain iteration schema in which we apply the sub and supersolution method developed by Noussair and Swanson. The approach allows us to consider superlinear problems with convection terms containing functional coefficient $g$ without radial symmetry.
Access status: Open Access ,
Study of fractional semipositone problems on RN
(Wydawnictwa AGH, 2024) Biswas, Nirjan
Let $s\in (0,1)$ and $N\gt 2s$. In this paper, we consider the following class of nonlocal semipositone problems: $(-\Delta)^s u= g(x)f_a(u)\text{ in }\mathbb{R}^N,\quad u \gt 0\text{ in }\mathbb{R}^N,$ where the weight $g \in L^1(\mathbb{R}^N) \cap L^{\infty}(\mathbb{R}^N)$ is positive, $a\gt 0$ is a parameter, and $f_a \in \mathcal{C}(\mathbb{R})$ is strictly negative on $(-\infty,0]$. For $f_a$ having subcritical growth and weaker Ambrosetti-Rabinowitz type nonlinearity, we prove that the above problem admits a mountain pass solution $u_a$, provided a is near zero. To obtain the positivity of $u_a$, we establish a Brezis-Kato type uniform estimate of $(u_a)$ in $L^r(\mathbb{R}^N)$ for every $r \in [\frac{2N}{N-2s}, \infty]$.