This paper analyzes the structure of the set of positive solutions of a class of one-dimensional superlinear indefinite bvp's. It is a paradigm of how mathematical analysis aids the numerical study of a problem, whereas simultaneously its numerical study confirms and illuminates the analysis. On the analytical side, we establish the fast decay of the positive solutions as $λ\downarrow -\infty$ in the region where $a(x)<0$ (see (1.1)), as well as the decay of the solutions of the parabolic counterpart of the model (see (1.2)) as $λ\downarrow-\infty$ on any subinterval of $[0,1]$ where $u_0=0$, provided $u_0$ is a subsolution of (1.1). This result provides us with a proof of a conjecture of [28] under an additional condition of a dynamical nature. On the numerical side, this paper ascertains the global structure of the set of positive solutions on some paradigmatic prototypes whose intricate behavior is far from predictable from existing analytical results.
We define some Nehari-type constraints using an orthogonal decomposition of the Sobolev space $H^1_0$ and prove the existence of multibump nodal solutions for an indefinite superlinear elliptic problem.
In this paper we consider 6-superlinear Chern-Simons-Schrödinger systems. In contrast to most studies, we consider the case where the potential $V$ is indefinite so that the Schrödinger operator $-Δ+V$ possesses a finite-dimensional negative space. We obtain nontrivial solutions for the problem via Morse theory.
We consider a superlinear perturbation of the eigenvalue problem for the Robin Laplacian plus an indefinite and unbounded potential. Using variational tools and critical groups, we show that when $λ$ is close to a nonprincipal eigenvalue, then the problem has seven nontrivial solutions. We provide sign information for six of them.
We study the following Kirchhoff equation $$- \left(1 + b \int_{\mathbb{R}^3} |\nabla u|^2 dx \right) Δu + V(x) u = f(x,u), \ x \in \mathbb{R}^3.$$ A special feature of this paper is that the nonlinearity $f$ and the potential $V$ are indefinite, hence sign-changing. Under some appropriate assumptions on $V$ and $f$, we prove the existence of two different solutions of the equation via the Ekeland variational principle and Mountain Pass Theorem.
We provide the structure of regular/singular fast/slow decay radially symmetric solutions for a class of superlinear elliptic equations with an in- definite weight on the nonlinearity f (u, r). In particular we are interested in the case where f is positive in a ball and negative outside, or in the re- versed situation. We extend the approach to elliptic equations in presence of Hardy potentials. By the use of Fowler transformation we study the corresponding dynamical systems, presenting the construction of invariant manifolds when the global existence of solutions is not ensured.
We are concerned with the multiplicity of positive solutions for the singular superlinear and subcritical Schrödinger equation $$ \begin{array}{c} -Δu +V(x)u=λa(x)u^{-γ}+b(x)u^{p}~\mbox{in}~ \mathbb{R}^{N}, \end{array} $$ beyond the Nehari extremal value, as defined in Il'yasov [Topol. Methods Nonlinear Anal., 2017], when the potential $b \in L^{\infty}(\mathbb{R}^{N})$ may change its sign, $0<a\in L^{\frac{2}{1+γ}}(\mathbb{R}^{N})$, $V$ is a positive continuous function, $N\geq 3$ and $λ>0$ is a real parameter. The main difficulties come from the non-differentiability of the energy functional…
This paper analyzes the structure of the set of nodal solutions of a class of one-dimensional superlinear indefinite boundary values problems with an indefinite weight functions in front of the spectral parameter. Quite astonishingly, the associated high order eigenvalues might not be concave as it is the lowest one. As a consequence, in many circumstances the nodal solutions can bifurcate from three or even four bifurcation points from the trivial solution. This paper combines analytical and numerical tools. The analysis carried over on it is a paradigm of how mathematical analysis aids the n…
We consider a class of stationary Schrödinger-Poisson systems with a general nonlinearity $f(u)$ and coercive sign-changing potential $V$ so that the Schrödinger operator $-Δ+V$ is indefinite. Previous results in this framework required $f$ to be strictly $3$-superlinear, thus missing the paramount case of the Gross-Pitaevskii-Poisson system, where $f(t)=|t|^{2}t$; in this paper we fill this gap, obtaining non-trivial solutions when $f$ is not necessarily $3$-superlinear.
We consider bifurcation of solutions from a given trivial branch for a class of strongly indefinite elliptic systems via the spectral flow. Our main results establish bifurcation invariants that can be obtained from the coefficients of the systems without using explicit solutions of their linearisations at the given branch. Our constructions are based on a comparison principle for the spectral flow and a generalisation of a bifurcation theorem due to Szulkin.
We consider a parametric semilinear Robin problem driven by the Laplacian plus an indefinite potential. The reaction term involves competing nonlinearities. More precisely, it is the sum of a parametric sublinear (concave) term and a superlinear (convex) term. The superlinearity is not expressed via the Ambrosetti-Rabinowitz condition. Instead, a more general hypothesis is used. We prove a bifurcation-type theorem describing the set of positive solutions as the parameter $λ> 0$ varies. We also show the existence of a minimal positive solution $\tilde{u}_λ$ and determine the monotonicity and co…
The aim of this paper is analyzing existence, multiplicity, and regularity issues for the positive solutions of a Neumann boundary value problem of superlinear indefinite type related to the mean curvature operator with a sublinear potential at infinity.
In this paper, we use bifurcation method to investigate the existence and multiplicity of one-sign solutions of the $p$-Laplacian involving a linear/superlinear nonlinearity with zeros. To do this, we first establish a bifurcation theorem from infinity for nonlinear operator equation with homogeneous operator. To deal with the superlinear case, we establish several topological results involving superior limit.
In this paper we study an index of a critical orbit, defined in terms of the degree for invariant strongly indefinite functionals. We establish a relationship of this index with the index of a critical point of the mapping restricted to the space normal to the orbit. The second aim of the article is to use the index of a critical orbit to prove the bifurcation of nontrivial solutions of nonlinear elliptic systems with Neumann boundary conditions. We consider also the existence of unbounded sets of solutions of such systems.
We consider semilinear Robin problems driven by the negative Laplacian plus an indefinite potential and with a superlinear reaction term which need not satisfy the Ambrosetti-Rabinowitz condition. We prove existence and multiplicity theorems (producing also an infinity of smooth solutions) using variational tools, truncation and perturbation techniques and Morse theory (critical groups).
We generalise the semi-Riemannian Morse index theorem to elliptic systems of partial differential equations on star-shaped domains. Moreover, we apply our theorem to bifurcation from a branch of trivial solutions of semilinear systems, where the bifurcation parameter is introduced by shrinking the domain to a point. This extends recent results of the authors for scalar equations.
We study perturbations of the eigenvalue problem for the negative Laplacian plus an indefinite and unbounded potential and Robin boundary condition. First we consider the case of a sublinear perturbation and then of a superlinear perturbation. For the first case we show that for $λ<\widehatλ_{1}$ ($\widehatλ_{1}$ being the principal eigenvalue) there is one positive solution which is unique under additional conditions on the perturbation term. For $λ\geq\widehatλ_{1}$ there are no positive solutions. In the superlinear case, for $λ<\widehatλ_{1}$ we have at least two positive solutions and for…
We consider a nonlinear Dirichlet problem driven by a variable exponent $p$-Laplacian plus an indefinite potential term. The reaction has the competing effects of a parametric concave (sublinear) term and of a convex (superlinear) perturbation (an anisotropic concave-convex problem). We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the positive parameter $λ$ varies. Also, we prove the existence of minimal positive solutions.
We consider the semilinear elliptic equations $$ \left\{ \begin{array}{ll} &-Δu+V(x)u=\left(I_α\ast |u|^p\right)|u|^{p-2}u+λu\quad \hbox{for } x\in\mathbb R^N, \\ &u(x) \to 0 \hbox{ as } |x| \to\infty, \end{array} \right. $$ where $I_α$ is a Riesz potential, $p\in(\frac{N+α}N,\frac{N+α}{N-2})$, $N\geq3$, and $V $ is continuous periodic. We assume that $0$ lies in the spectral gap $(a,b)$ of $-Δ+ V$. We prove the existence of infinitely many geometrically distinct solutions in $H^1(\mathbb R^N)$ for each $λ\in(a, b)$, which bifurcate from $b$ if $\frac{N+α}N< p < 1 +\frac{2+α}{N}$. Moreover, $b…
We investigate positive steady states of an indefinite superlinear reaction-diffusion equation arising from population dynamics, coupled with a nonlinear boundary condition. Both the equation and the boundary condition depend upon a positive parameter $λ$, which is inversely proportional to the diffusion rate. We establish several multiplicity results when the diffusion rate is large and analyze the asymptotic profiles and the stability properties of these steady states as the diffusion rate grows to infinity. In particular, our results show that in some cases bifurcation from zero and from in…
We construct multibump nodal solutions of the elliptic equation $$ -Δu=a^+[λu+ f(\, \cdot\,, u)]-μa^- g(\, \cdot\,, u) $$ in $H^1_0(Ω)$, when $μ$ is large, under appropriate assumptions, for $f$ superlinear and subcritical and such that the eigenvalues of the associated linearized operator on $H^1_0(\{x\inΩ:\: a(x)>0\})$ at zero, $u\longmapsto u-λ(-Δ)^{-1}(a^+ u)$, are positive. The solutions are of least energy in some Nehari-type set defined by imposing suitable conditions on orthogonal components of functions in $H^1_0(Ω)$.
In this paper, based on some prior estimates, we show that the essential spectrum $λ=0$ is a bifurcation point for an superlinear elliptic equation with only local conditions, which generalizes a series of earlier results on an open problem proposed by C. A. Stuart in 1983 [Lecture Notes in Mathematics, 1017].
We consider a nonlinear Robin problem driven by the sum of $p$-Laplacian and $q$-Laplacian (i.e. the $(p,q)$-equation). In the reaction there are competing effects of a singular term and a parametric perturbation $λf(z,x)$, which is Carathéodory and $(p-1)$-superlinear at $x\in\mathbb{R},$ without satisfying the Ambrosetti-Rabinowitz condition. Using variational tools, together with truncation and comparison techniques, we prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter $λ>0$ varies.
A $p$-Laplacian elliptic problem in the presence of both strongly singular and $(p-1)$-superlinear nonlinearities is considered. We employ bifurcation theory, approximation techniques and sub-supersolution method to establish the existence of an unbounded branch of positive solutions, which is bounded in positive $λ-$direction and bifurcates from infinity at $λ=0$. As consequence of the bifurcation result, we determine intervals of existence, nonexistence and, in particular cases, global multiplicity.
We study a Dirichlet problem driven by the degenerate fractional p-Laplacian and involving a nonlinear reaction, which depends on a positive parameter. The reaction is assumed to be (p-1)-sublinear near the origin and (p-1)-superlinear at infinity (including the concave-convex case). Following a variational approach based on a combination of critical point theory and suitable truncation techniques, we prove a bifurcation-type result for the existence of positive solutions.
We consider an elliptic problem with nonlinear boundary condition involving nonlinearity with superlinear and subcritical growth at infinity and a bifurcation parameter as a factor. We use re-scaling method, degree theory and continuation theorem to prove that there exists a connected branch of positive solutions bifurcating from infinity when the parameter goes to zero. Moreover, if the nonlinearity satisfies additional conditions near zero, we establish a global bifurcation result, and discuss the number of positive solution(s) with respect to the parameter using bifurcation theory and degre…
We study the existence and multiplicity of positive solutions for a family of fractional Kirchhoff equations with critical nonlinearity of the form \begin{equation*} M\left(\int_Ω|(-Δ)^{\fracα{2}}u|^2dx\right)(-Δ)^α u= λf(x)|u|^{q-2}u+|u|^{2^*_α-2}u\;\; \text{in}\; Ω,\;u=0\;\textrm{in}\;\mathbb R^n\setminus Ω, \end{equation*} where $Ω\subset \mathbb R^n$ is a smooth bounded domain, $ M(t)=a+\varepsilon t, \; a, \; \varepsilon>0,\; 0<α<1, \; 2α<n<4α$ and $ \; 1<q<2$. Here $2^*_α={2n}/{(n-2α)}$ is the fractional critical Sobolev exponent, $λ$ is a positive parameter and the coefficient $f(x)$ is…
We prove the existence of a loop type component of non-negative solutions for an indefinite elliptic equation with homogeneous Neumann boundary conditions. This result complements our previous results obtained in [12], where the existence of another loop type component was established in a different situation. Our proof combines local and global bifurcation theory, rescaling and regularizing arguments, a priori bounds, and Whyburn's topological method. A further investigation of the loop type component established in [12] is also provided.
The purpose of this paper is to study properties of continua (closed connected sets) of nontrivial solutions of non-cooperative elliptic systems considered on geodesic balls in $S^n$. In particular, we have shown that if the geodesic ball is a hemisphere, then these continua are unbounded. It is also shown that the phenomenon of global symmetry-breaking bifurcation of such solutions occurs. Since the problem is variational and SO(n)-symmetric, we apply the techniques of equivariant bifurcation theory to prove the main results of this article. As the topological tool we use the degree theory fo…
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 is parametric. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the positive parameter $λ$ varies. This work continues our research published in arXiv:2004.12583, where $ξ\equiv 0 $ and in the reaction the parametric term is the singular one.
We consider a nonlinear elliptic equation driven by the Robin $p$-Laplacian plus an indefinite potential. In the reaction we have the competing effects of a strictly $(p-1)$-sublinear parametric term and of a $(p-1)$-linear and nonuniformly nonresonant term. We study the set of positive solutions as the parameter $λ>0$ varies. We prove a bifurcation-type result for large values of the positive parameter $λ$. Also, we show that for all admissible $λ>0$, the problem has a smallest positive solution $\overline{u}_λ$ and we study the monotonicity and continuity properties of the map $λ\mapsto\over…
In this work we obtain a Liouville theorem for positive, bounded solutions of the equation $$ (-Δ)^s u= h(x_N)f(u) \quad \hbox{in }\mathbb{R}^{N} $$ where $(-Δ)^s$ stands for the fractional Laplacian with $s\in (0,1)$, and the functions $h$ and $f$ are nondecreasing. The main feature is that the function $h$ changes sign in $\mathbb{R}$, therefore the problem is sometimes termed as indefinite. As an application we obtain a priori bounds for positive solutions of some boundary value problems, which give existence of such solutions by means of bifurcation methods.
We study the second-order boundary value problem \begin{equation*} \begin{cases} \, -u''=a_{λ,μ}(t) \, u^{2}(1-u), & t\in(0,1), \\ \, u'(0)=0, \quad u'(1)=0, \end{cases} \end{equation*} where $a_{λ,μ}$ is a step-wise indefinite weight function, precisely $a_{λ,μ}\equivλ$ in $[0,σ]\cup[1-σ,1]$ and $a_{λ,μ}\equiv-μ$ in $(σ,1-σ)$, for some $σ\in\left(0,\frac{1}{2}\right)$, with $λ$ and $μ$ positive real parameters. We investigate the topological structure of the set of positive solutions which lie in $(0,1)$ as $λ$ and $μ$ vary. Depending on $λ$ and based on a phase-plane analysis and on time-map…
We investigate the existence of a curve $q\mapsto u_{q}$, with $q\in(0,1)$, of positive solutions for the problem $(P_{a,q})$: $-Δu=a(x)u^{q}$ in $Ω$, $u=0$ on $\partialΩ$, where $Ω$ is a bounded and smooth domain of $\mathbb{R}^{N}$ and $a:Ω\rightarrow\mathbb{R}$ is a sign-changing function (in which case the strong maximum principle does not hold). In addition, we analyze the asymptotic behavior of $u_{q}$ as $q\rightarrow0^{+}$ and $q\rightarrow1^{-}$. We also show that in some cases $u_{q}$ is the ground state solution of $(P_{a,q})$. As a byproduct, we obtain existence results for a singu…
This paper infers from a generalized Picone identity the uniqueness of the stable positive solution for a class of semilinear equations of superlinear indefinite type, as well as the uniqueness and global attractivity of the coexistence state in two generalized diffusive prototypes of the symbiotic and competing species models of Lotka-Volterra. The optimality of these uniqueness theorems reveals the tremendous strength of the Picone identity.
Let $Ω\subset\mathbb{R}^{N}$ ($N\geq1$) be a smooth bounded domain, $a\in C(\barΩ)$ a sign-changing function, and $0\leq q<1$. We investigate the Robin problem \[ \begin{cases} -Δu=a(x)u^{q} & \mbox{in $Ω$},\\ u\geq0 & \mbox{in $Ω$},\\ \partial_νu=αu & \mbox{on $\partial Ω$}, \end{cases} \] where $α\in\lbrack-\infty,\infty)$ and $ν$ is the unit outward normal to $\partialΩ$. Due to the lack of strong maximum principle structure, this problem may have \textit{dead core} solutions. However, for a large class of weights $a$ we recover a \textit{positivity} property when $q$ is close to $1$, which…
We study a superlinear and subcritical Kirchhoff type equation which is variational and depends upon a real parameter $λ$. The nonlocal term forces some of the fiber maps associated with the energy functional to have two critical points. This suggest multiplicity of solutions and indeed we show the existence of a local minimum and a mountain pass type solution. We characterize the first parameter $λ_0^*$ for which the local minimum has non-negative energy when $λ\ge λ_0^*$. Moreover we characterize the extremal parameter $λ^*$ for which if $λ>λ^*$, then the only solution to the Kirchhoff equat…
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 $λ$ varies. We also show that there exists a minimal positive solution $\overline{u}_λ$ and determine the monotonicity and continuity properties of the map $λ\mapsto\overline{u}_λ$. Special attention is given to the particular case of the $p$-Laplacian.
We consider a Dirichlet problem driven by the anisotropic $(p,q)$-Laplacian and with a reaction that has the competing effects of a singular term and of a parametric superlinear perturbation. Based on variational tools along with truncation and comparison techniques, we prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter varies.
We consider a nonlinear Dirichlet problem driven by the $(p,q)$-Laplacian and with a reaction having the combined effects of a singular term and of a parametric $(p-1)$-superlinear perturbation. We prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter $λ>0$ varies. Moreover, we prove the existence of a minimal positive solution $u^*_λ$ and study the monotonicity and continuity properties of the map $λ\to u^*_λ$.
Let $Ω\subset\mathbb{R}^{N}$ ($N\geq1$) be a bounded and smooth domain and $a:Ω\rightarrow\mathbb{R}$ be a sign-changing weight satisfying $\int_Ωa<0$. We prove the existence of a positive solution $u_{q}$ for the problem $(P_{a,q})$: $-Δu=a(x)u^{q}$ in $Ω$, $\frac{\partial u}{\partialν}=0$ on $\partialΩ$, if $q_{0}<q<1$, for some $q_{0}=q_{0}(a)>0$. In doing so, we improve the existence result previously established in [16]. In addition, we provide the asymptotic behavior of $u_{q}$ as $q\rightarrow1^{-}$. When $Ω$ is a ball and $a$ is radial, we give some explicit conditions on $q$ and $a$ e…
In this paper we consider a Dirichlet problem driven by an anisotropic $(p,q)$-differential operator and a parametric reaction having the competing effects of a singular term and of a superlinear perturbation. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter moves. Moreover, we prove the existence of a minimal positive solution and determine the monotonicity and continuity properties of the minimal solution map.
We study a parametric Robin problem driven by a nonlinear nonhomogeneous differential operator and with a superlinear Carathéodory reaction term. We prove a bifurcation-type theorem for small values of the parameter. Also, we show that as the parameter $λ>0$ approaches zero we can find positive solutions with arbitrarily big and arbitrarily small Sobolev norm. Finally we show that for every admissible parameter value there is a smallest positive solution $u^*_λ$ of the problem and we investigate the properties of the map $λ\mapsto u^*_λ$.
We consider a nonlinear parametric Neumann problem driven by the anisotropic $(p,q)$-Laplacian and a reaction which exhibits the combined effects of a singular term and of a parametric superlinear perturbation. We are looking for positive solutions. Using a combination of topological and variational tools together with suitable truncation and comparison techniques, we prove a bifurcation-type result describing the set of positive solutions as the positive parameter $λ$ varies. We also show the existence of minimal positive solutions $u_λ^*$ and determine the monotonicity and continuity propert…
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 $λ$ moves on the positive semiaxis. We also show that for every $λ>0$, the problem …
We prove the existence of multiple positive radial solutions to the sign-indefinite elliptic boundary blow-up problem \[ \left\{\begin{array}{ll} Δu + \bigl(a^+(\vert x \vert) - μa^-(\vert x \vert)\bigr) g(u) = 0, & \; \vert x \vert < 1, \\ u(x) \to \infty, & \; \vert x \vert \to 1, \end{array} \right. \] where $g$ is a function superlinear at zero and at infinity, $a^+$ and $a^-$ are the positive/negative part, respectively, of a sign-changing function $a$ and $μ> 0$ is a large parameter. In particular, we show how the number of solutions is affected by the nodal behavior of the weight functi…
