In loving memory of my beloved miniature dachshund Maddie (16 March 2002 – 16 March 2020). We consider nonlocal differential equations with convolution coefficients of the form

\[{-}M\Big(\big(a*(g\circ |u|)\big)(1)\Big)u''(t)=\lambda f\big(t,u(t)\big),\quad t\in(0,1), \]

$g$

$M$

$M$

$f$

For $s_1,\,s_2\in (0,\,1)$ and $p,\,q \in (1,\, \infty )$, we study the following nonlinear Dirichlet eigenvalue problem with parameters $\alpha,\, \beta \in \mathbb {R}$ driven by the sum of two nonlocal operators:

\[ (-\Delta)^{s_1}_p u+(-\Delta)^{s_2}_q u=\alpha|u|^{p-2}u+\beta|u|^{q-2}u\ \text{in }\Omega, \quad u=0\ \text{in } \mathbb{R}^d \setminus \Omega, \quad \mathrm{(P)} \]

$\Omega \subset \mathbb {R}^d$

$\alpha,\,\beta$

$(\alpha,\, \beta )$

$p$

$q$

We consider the following Kirchhoff-type problem in an unbounded exterior domain $\Omega\subset\mathbb{R}^{3}$:(*)

\begin{align}\left\{\begin{array}{ll}-\left(a+b\displaystyle{\int}_{\Omega}|\nabla u|^{2}\,{\rm d}x\right)\triangle u+\lambda u=f(u), & x\in\Omega,\\\\u=0, & x\in\partial \Omega,\\\end{array}\right.\end{align}

$b\geq0$

$\partial\Omega\neq\emptyset$

$\mathbb{R}^{3}\backslash\Omega$

$u\in H_{0}^{1}(\Omega)$

$f\in C^1(\mathbb{R},\mathbb{R})$

\begin{equation*}-\Delta u+\lambda u=f(u), \qquad x\in \mathbb R^3 \end{equation*}

$H^1(\mathbb R^3)$

$\mathbb R^3\setminus \Omega$

In this paper, we study the asymptotic profiles of positive solutions for diffusive logistic equations. The aim is to study the sharp effect of linear growth and nonlinear function. Both the classical reaction-diffusion equation and nonlocal dispersal equation are investigated. Our main results reveal that the linear and nonlinear parts of reaction term play quite different roles in the study of positive solutions.

This paper is concerned with the existence of solutions for a class of elliptic equations on the unit ball with zero Dirichlet boundary condition. The nonlinearity is supercritical in the sense of Trudinger–Moser. Using a suitable approximating scheme we obtain the existence of at least one positive solution.

We consider a second-order elliptic operator L in skew product of an ordinary differential operator L1 on an interval (a, b) and an elliptic operator on a domain D2 of a Riemannian manifold such that the associated heat kernel is intrinsically ultracontractive. We give criteria for criticality and subcriticality of L in terms of a positive solution having minimal growth at η (η = a, b) to an associated ordinary differential equation. In the subcritical case, we explicitly determine the Martin compactification and Martin kernel for L on the basis of [24]; in particular, the Martin boundary over η is either one point or a compactification of D2, which depends on whether an associated integral near η diverges or converges. From this structure theorem we show a monotonicity property that the Martin boundary over η does not become smaller as the potential term of L1 becomes larger near η.

Positive solutions of a Kirchhoff-type nonlinear elliptic equation with a non-local integral term on a bounded domain in ℝN, N ⩾ 1, are studied by using bifurcation theory. The parameter regions of existence, non-existence and uniqueness of positive solutions are characterized by the eigenvalues of a linear eigenvalue problem and a nonlinear eigenvalue problem. Local and global bifurcation diagrams of positive solutions for various parameter regions are obtained.

We consider the boundary-value problem

where H: [0,+∞) → ℝ and f : [0, 1] × ℝ → ℝ are continuous and λ > 0 is a parameter. We show that if H satisfies a boundedness condition on a specified compact set, then this, together with an assumption that H is either affine or superlinear at +∞, implies existence of at least one positive solution to the problem, even in the case where we impose no growth conditions on f. Finally, since it can hold that f(t, y) < 0 for all (t, y) ∈ [0, 1]×ℝ, the semipositone problem is included as a special case of the existence result.

We consider a nonlinear parametric elliptic equation driven by a nonhomogeneous differential operator with a logistic reaction of the superdiffusive type. Using variational methods coupled with suitable truncation and comparison techniques, we prove a bifurcation type result describing the set of positive solutions as the parameter varies.

In this paper, our main purpose is to establish the existence of positive solution of the following system where are constants. F(x,u,υ) = λp(x)[g(x)a(u)+f(υ)], H(x,u,υ)=θq(x)[g1(x)b(υ)+h(u)], λ, θ>0 are parameters, p(x), q(x) are radial symmetric functions, is called p(x)-Laplacian. We give the existence results and consider the asymptotic behavior of the solutions. In particular, we do not assume any symmetric condition, and we do not assume any sign condition on F(x,0,0) and H(x,0,0) either.

In this paper we consider the existence of a positive solution to boundary-value problems with non-local nonlinear boundary conditions, the archetypical example being −y″(t) = λf(t,y(t)), t ∈ (0, 1), y(0) = H(φ(y)), y(1) = 0. Here, H is a nonlinear function, λ > 0 is a parameter and φ is a linear functional that is realized as a Lebesgue—Stieltjes integral with signed measure. By requiring φ to decompose in a certain way, we show that this problem has at least one positive solution for each λ ∈ (0, λ0), for a number λ0 > 0 that is explicitly computable. We also give a separate result that holds for all λ > 0.

We provide an existence result of radially symmetric, positive, classical solutions for a nonlinear Schrödinger equation driven by the infinitesimal generator of a rotationally invariant Lévy process.

The aim of this paper is to prove the existence of a positive solution for a quasi-linear elliptic problem involving the (p, q)-Laplacian and a convection term, which means an expression that is not in the principal part and depends on the solution and its gradient. The solution is constructed through an approximating process based on gradient bounds and regularity up to the boundary. The positivity of the solution is shown by applying a new comparison principle, which is established here.

In this paper, we consider the Neumann boundary value problem with a parameter λ∈(0,∞): By using fixed point theorems in a cone, we obtain some existence, multiplicity and nonexistence results for positive solutions in terms of different values of λ. We also prove an existence and uniqueness theorem and show the continuous dependence of solutions on the parameter λ.

In this paper, some criteria for the existence of positive solutions of a class of systems of impulsive dynamic equations on time scales are obtained by using a fixed point theorem in cones.

We investigate the global asymptotic stability for positive solutions to a class of general symmetric rational difference equations and prove that the unique positive equilibrium 1 of the general symmetric rational difference equations is globally asymptotically stable. As a special case of our result, we solve the conjecture raised by Berenhaut, Foley and Stević [‘The global attractivity of the rational difference equation yn=(yn−k+yn−m)/(1+yn−kyn−m)’, Appl. Math. Lett.20 (2007), 54–58].

This paper is concerned with a boundary-value problem on the half-line for nonlinear two-dimensional delay differential systems with positive delays. A theorem is established, which provides sufficient conditions for the existence of positive solutions. The application of this theorem to the special case of second-order nonlinear delay differential equations is given. Also, the application of the theorem to two-dimensional Emden–Fowler-type delay differential systems with constant delays is presented. Moreover, some general examples demonstrating the applicability of the theorem are included.

We establish that the elliptic equation defined in an exterior domain of ℝn, n≥3, has a positive solution which decays to 0 as under quite general assumptions upon f and g.

In this paper, existence criteria for multiple solutions of periodic boundary value problems for the first-order difference equation are established by using the Leggett–Williams multiple fixed point theorem and fixed point theorem of cone expansion and compression. Two examples are also given to illustrate the main results.