Set Of Positive Solutions Research Articles

Parameter dependence for the positive solutions of nonlinear, nonhomogeneous Robin problems

We consider a parametric nonlinear Robin problem driven by a nonlinear nonhomogeneous differential operator plus an indefinite potential. The reaction term is $$(p-1)$$-superlinear but need not satisfy the usual Ambrosetti–Rabinowitz condition. We look for positive solutions and prove a bifurcation-type result for the set of positive solutions as the parameter $$\lambda >0$$ varies. Also we prove the existence of a minimal positive solution $$u_\lambda ^*$$ and determine the monotonicity and continuity properties of the map $$\lambda \rightarrow u_\lambda ^*$$.

Reversed <i>S</i> -shaped connected component for second-order periodic boundary value problem with sign-changing weight

In this paper, we consider the existence of an reversed S -shaped connected component in the set of positive solutions for second order periodic boundary value problem with a sign-changing weight function. By bifurcation technique, we identify the interval of bifurcation parameter in which the periodic boundary value problem has one or two or three positive solutions according to the asymptotic behavior of f at 0 and ∞.

Positive solutions for nonlinear parametric singular Dirichlet problems

We consider a nonlinear parametric Dirichlet problem driven by the [Formula: see text]-Laplace differential operator and a reaction which has the competing effects of a parametric singular term and of a Carathéodory perturbation which is ([Formula: see text])-linear near [Formula: see text]. The problem is uniformly nonresonant with respect to the principal eigenvalue of [Formula: see text]. We look for positive solutions and prove a bifurcation-type theorem describing in an exact way the dependence of the set of positive solutions on the parameter [Formula: see text].

Nonlinear nonhomogeneous singular problems

We consider a nonlinear Dirichlet problem driven by a nonhomogeneous differential operator with a growth of order $(p-1)$ near $+\infty$ and with a reaction which has the competing effects of a parametric singular term and a $(p-1)$-superlinear perturbation which does not satisfy the usual Ambrosetti-Rabinowitz condition. Using variational tools, together with suitable truncation and strong comparison techniques, we prove a "bifurcation-type" theorem that describes the set of positive solutions as the parameter $\lambda$ moves on the positive semiaxis. We also show that for every $\lambda>0$, the problem has a smallest positive solution $u^*_\lambda$ and we demonstrate the monotonicity and continuity properties of the map $\lambda\mapsto u^*_\lambda$.

Solution components in a degenerate weighted BVP

This paper ascertains the structure of the set of positive solutions of a degenerate weighted logistic equation with a continuous kinetic under non-classical mixed boundary conditions. After establishing the uniqueness of the positive solution and the existence of two global components of positive solutions, it is shown that any component must be a continuous curve. Further, some general sufficient conditions are provided, in terms of the weight functions involved in the setting of the model, so that it possesses exactly two components of positive solutions. The problem of finding out the total number of components in the general case remains open.

Some superlinear problems with nonlocal diffusion coefficient

We study a superlinear elliptic problem with a non-local diffusion coefficient. We show that there exists a drastic change on the structure of the set of positive solutions when the non-local coefficient grows fast enough to infinity. We combine mainly sub-super and bifurcation methods to obtain our results.

Nonlinear Nonhomogeneous Robin Problems with Almost Critical and Partially Concave Reaction

We consider a nonlinear Robin problem driven by a nonhomogeneous differential operator, with reaction which exhibits the competition of two Carath\'eodory terms. One is parametric, $(p-1)$-sublinear with a partially concave nonlinearity near zero. The other is $(p-1)$-superlinear and has almost critical growth. Exploiting the special geometry of the problem, we prove a bifurcation-type result, describing the changes in the set of positive solutions as the parameter $\lambda>0$ varies.

Nonlinear singular problems with indefinite potential term

We consider a nonlinear Dirichlet problem driven by a nonhomogeneous differential operator plus an indefinite potential. In the reaction we have the competing effects of a singular term and of concave and convex nonlinearities. In this paper the concave term will be parametric. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the positive parameter lambda varies.

Continuity results for parametric nonlinear singular Dirichlet problems

AbstractIn this paper we study from a qualitative point of view the nonlinear singular Dirichlet problem depending on a parameterλ> 0 that was considered in [32]. Denoting bySλthe set of positive solutions of the problem corresponding to the parameterλ, we establish the following essential properties ofSλ:there exists a smallest element$\begin{array}{} u_\lambda^* \end{array}$inSλ, and the mappingλ↦$\begin{array}{} u_\lambda^* \end{array}$is (strictly) increasing and left continuous;the set-valued mappingλ↦Sλis sequentially continuous.

Positive solutions for nonlinear Robin problems with indefinite potential and competing nonlinearities

We consider a nonlinear Robin problem associated to the p-Laplacian plus an indefinite potential. In the reaction we have the competing effects of two nonlinear terms. One is parametric and strictly $$(p-1)$$-sublinear. The other is $$(p-1)$$-linear. We prove a bifurcation-type theorem describing the dependence of the set of positive solutions on the parameter $$\lambda >0$$. We also show that for every admissible parameter the problem has a smallest positive solution $${\bar{u}}_{\lambda }$$ and we study monotonicity and continuity properties of the map $$\lambda \rightarrow {\bar{u}}_{\lambda }$$.

Non-local Degenerate Diffusion Coefficients Break Down the Components of Positive Solutions

Abstract This paper deals with nonlinear elliptic problems where the diffusion coefficient is a degenerate non-local term. We show that this degeneration implies the growth of the complexity of the structure of the set of positive solutions of the equation. Specifically, when the reaction term is of logistic type, the continuum of positive solutions breaks into two disjoint pieces. Our approach uses mainly fixed point arguments.

On a Dirichlet problem with (p,q)-Laplacian and parametric concave-convex nonlinearity

A homogeneous Dirichlet problem with (p,q)-Laplace differential operator and reaction given by a parametric p-convex term plus a q-concave one is investigated. A bifurcation-type result, describing changes in the set of positive solutions as the parameter λ>0 varies, is proven. Since for every admissible λ the problem has a smallest positive solution u¯λ, both monotonicity and continuity of the map λ↦u¯λ are studied.

Predator-prey models with competition, Part Ⅲ: Classification of stationary solutions

For a stationary system representing prey and \begin{document}$ N $\end{document} groups of competing predators, we show classification results about the set of positive solutions. In particular, we show that if the number of components \begin{document}$ N $\end{document} is too large or if the competition between different groups is too small, then the system has only constant solutions, which we then completely characterize.

On a Robin (p,q)-equation with a logistic reaction

We consider a nonlinear nonhomogeneous Robin equation driven by the sum of a \(p\)-Laplacian and of a \(q\)-Laplacian (\((p,q)\)-equation) plus an indefinite potential term and a parametric reaction of logistic type (superdiffusive case). We prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter \(\lambda \gt 0\) varies. Also, we show that for every admissible parameter \(\lambda \gt 0\), the problem admits a smallest positive solution.

Solutions with sign information for nonlinear nonhomogeneous problems

AbstractWe consider a parametric elliptic equation driven by a nonlinear nonhomogeneous differential operator and with a Robin boundary condition. In the first part we prove the existence of positive solutions and state a bifurcation‐type result describing how the set of positive solutions changes as the parameter varies. In the second part we show that problem admits nodal (sign‐changing) solutions provided the parameter is sufficiently large.

Perturbations of nonlinear eigenvalue problems

We consider perturbations of nonlinear eigenvalue problems driven by a nonhomogeneous differential operator plus an indefinite potential. We consider both sublinear and superlinear perturbations and we determine how the set of positive solutions changes as the real parameter $\lambda$ varies. We also show that there exists a minimal positive solution $\overline{u}_\lambda$ and determine the monotonicity and continuity properties of the map $\lambda\mapsto\overline{u}_\lambda$. Special attention is given to the particular case of the $p$-Laplacian.

Non-degeneracy of positive solutions of Kirchhoff equations and its application

A non-degeneracy theorem for positive solutions of the standard Kirchhoff model is proved in all cases except for n ≥ 5 and b ∫ R n | ∇ Q | 2 d x = 2 ( n − 4 ) n − 4 2 / ( n − 2 ) n − 2 2 . In particular, for dimensions n ≥ 5 and b ∫ R n | ∇ Q | 2 d x < 2 ( n − 4 ) n − 4 2 / ( n − 2 ) n − 2 2 , we show that there exist two non-degenerate positive solutions of the Kirchhoff equations, which seem to be completely different from the result of the standard Schrödinger equation. The effects of the non-local term on the positive solution set are also studied. We show that the non-local term has no effect on the structure of the positive solution set for dimensions 1 ≤ n ≤ 3 and the effects eventually appear for dimensions n ≥ 4 . With the non-degeneracy property of positive solutions of the limit problem, we construct concentrated solutions of a singularly perturbed Kirchhoff problem via the well-known Lyapunov–Schmidt reduction method. For dimensions n ≥ 5 , the existence result of concentrated solutions is completely new and is not implied by the main result of G.M. Figueiredo et al. in [19] , where a generalized singularly perturbed Kirchhoff problem was considered.

Singular p-Laplacian equations with superlinear perturbation

We consider a nonlinear Dirichlet problem driven by the p-Laplace operator and with a right-hand side which has a singular term and a parametric superlinear perturbation. We are interested in positive solutions and prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter λ>0 varies. In addition, we show that for every admissible parameter λ>0 the problem has a smallest positive solution u‾λ and we establish the monotonicity and continuity properties of the map λ→u‾λ.

Structure of the set of positive solutions of a non-linear Schrödinger equation

In this paper we study the existence, uniqueness and multiplicity of positive solutions to a non-linear Schr¨odinger equation. We describe the set of positive solutions. We use mainly the sub-supersolution method, bifurcation and variational arguments to obtain the results.

Parametric nonlinear singular Dirichlet problems

We consider a nonlinear parametric Dirichlet problem driven by the p-Laplacian and a reaction which exhibits the competing effects of a singular term and of a resonant perturbation. Using variational methods together with suitable truncation and comparison techniques, we prove a bifurcation-type theorem describing the dependence on the parameter of the set of positive solutions.