Positive solutions for diffusive Logistic equation with refuge

2019 ◽  
Vol 9 (1) ◽  
pp. 1092-1101 ◽  
Author(s):  
Jian-Wen Sun
Keyword(s):  
Boundary Condition ◽  
Weight Function ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Critical Value ◽  
Stationary Solutions ◽  
Logistic Equation ◽  
Diffusion Equations ◽  
Diffusion Operator ◽  
Diffusive Logistic Equation

Abstract In this paper, we study the stationary solutions of the Logistic equation $$\begin{array}{} \displaystyle u_t=\mathcal {D}[u]+\lambda u-[b(x)+\varepsilon]u^p \text{ in }{\it\Omega} \end{array}$$ with Dirichlet boundary condition, here 𝓓 is a diffusion operator and ε > 0, p > 1. The weight function b(x) is nonnegative and vanishes in a smooth subdomain Ω0 of Ω. We investigate the asymptotic profiles of positive stationary solutions with the critical value λ when ε is sufficiently small. We find that the profiles are different between nonlocal and classical diffusion equations.

scholarly journals On the extremal solutions of superlinear Helmholtz problems

2022 ◽  
Vol 40 ◽  
pp. 1-8
Author(s):  
Makkia Dammak ◽  
Majdi El Ghord ◽  
Saber Ali Kharrati
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Weak Sense ◽  
Critical Value ◽  
Extremal Solutions ◽  
Regular Solutions ◽  
Smooth Bounded Domain ◽  
Bifurcation Phenomena ◽  
Stable Solutions

Abstract: In this note, we deal with the Helmholtz equation −∆u+cu = λf(u) with Dirichlet boundary condition in a smooth bounded domain Ω of R n , n > 1. The nonlinearity is superlinear that is limt−→∞ f(t) t = ∞ and f is a positive, convexe and C 2 function defined on [0,∞). We establish existence of regular solutions for λ small enough and the bifurcation phenomena. We prove the existence of critical value λ ∗ such that the problem does not have solution for λ > λ∗ even in the weak sense. We also prove the existence of a type of stable solutions u ∗ called extremal solutions. We prove that for f(t) = e t , Ω = B1 and n ≤ 9, u ∗ is regular.

scholarly journals The sub-supersolution method for a nonhomogeneous elliptic equation involving Lebesgue generalized spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Giovany M. Figueiredo ◽  
A. Razani
Keyword(s):  
Boundary Condition ◽  
Weak Solution ◽  
Elliptic Equation ◽  
Bounded Domain ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Logistic Equation ◽  
Positive Weak Solution

AbstractIn this paper, a nonhomogeneous elliptic equation of the form $$\begin{aligned}& - \mathcal{A}\bigl(x, \vert u \vert _{L^{r(x)}}\bigr) \operatorname{div}\bigl(a\bigl( \vert \nabla u \vert ^{p(x)}\bigr) \vert \nabla u \vert ^{p(x)-2} \nabla u\bigr) \\& \quad =f(x, u) \vert \nabla u \vert ^{\alpha (x)}_{L^{q(x)}}+g(x, u) \vert \nabla u \vert ^{ \gamma (x)}_{L^{s(x)}} \end{aligned}$$ − A ( x , | u | L r ( x ) ) div ( a ( | ∇ u | p ( x ) ) | ∇ u | p ( x ) − 2 ∇ u ) = f ( x , u ) | ∇ u | L q ( x ) α ( x ) + g ( x , u ) | ∇ u | L s ( x ) γ ( x ) on a bounded domain Ω in ${\mathbb{R}}^{N}$ R N ($N >1$ N > 1 ) with $C^{2}$ C 2 boundary, with a Dirichlet boundary condition is considered. Using the sub-supersolution method, the existence of at least one positive weak solution is proved. As an application, the existence of at least one solution of a generalized version of the logistic equation and a sublinear equation are shown.

scholarly journals On some classes of generalized Schrödinger equations

2020 ◽  
Vol 10 (1) ◽  
pp. 522-533
Author(s):  
Amanda S. S. Correa Leão ◽  
Joelma Morbach ◽  
Andrelino V. Santos ◽  
João R. Santos Júnior
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Laplacian Operator ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nontrivial Solutions ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Stationary Problems

Abstract Some classes of generalized Schrödinger stationary problems are studied. Under appropriated conditions is proved the existence of at least 1 + $\begin{array}{} \sum_{i=2}^{m} \end{array}$ dim Vλi pairs of nontrivial solutions if a parameter involved in the equation is large enough, where Vλi denotes the eigenspace associated to the i-th eigenvalue λi of laplacian operator with homogeneous Dirichlet boundary condition.

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