Hamilton Bueno, Eduardo Huerto Caqui, Olimpio Miyagaki, Fábio Pereira
In this paper we consider a class of critical concave convex Ambrosetti-Prodi type problems for the fractional $p$-Laplacian operator. By applying the Linking Theorem and the Mountain Pass Theorem as well, the interaction of the nonlinearities with the first eigenvalue of fractional $p$-Laplacian will be used to prove existence and multiplicity of solutions.
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For Dirichlet boundary conditions on a bounded domain, what happens to the critical set of the Ambrosetti-Prodi operator if the nonlinearity is only a Lipschitz map? It turns out that many properties which hold in the smooth case are preserved, despite of the fact that the operator is not even differentiable at some points. In particular, a global Lyapunov-Schmidt decomposition of great convenience for numerical inversion is still available.
We establish Ambrosetti -Prodi type results for periodic solutions of one -dimensional nonlinear problems with drift term and drift -less whose principal operator is the fractional Laplacian of order $s\in(0,1)$. We establish conditions for the existence and nonexistence of solutions. The proofs of the existence results are based on the sub-supersolution method combined with topological degree type arguments. We also establish a priori bounds in order to get multiplicity results. We also prove that the solutions are $C^{1,α}$ under some regularity assumptions in the nonlinearities, that is, th…
In this paper we study results of existence and non-existence of solutions for the following Ambrosetti-Prodi type problem $$ \left\{ \begin{array}{lcl} -Δu=P(x)\Big( g(u)+f(x)\Big) \mbox{ in } \mathbb{R}^N,\\ u \in D^{1,2}(\R^N),\ \lim_{|x|\to +\infty}u(x)=0, \end{array} \right. \eqno{(P)} $$ where $N\geq3$, $P\in C(\R^N,\R^+)$, $f\in C(\R^N)\cap L^{\infty}(\R^N)$ and $g\in C^1(\R)$. The main tools used are the sub-supersolution method and Leray-Schauder topological degree theory.
According to the Ambrosetti-Prodi theorem, the map $F(u)= - Δu - f(u)$ between appropriate functional spaces is a global fold. Among the hypotheses, the convexity of the function $f$ is required. We show in two different ways that, under mild conditions, convexity is indeed necessary. If $f$ is not convex, there is a point with at least four preimages under $F$. More, $F$ generically admits cusps among its critical points. We present a larger class of nonlinearities $f$ for which the critical set of $F$ has cusps. The results are true for a class of boundary conditions.
The well-known Ambrosetti-Prodi theorem considers perturbations of the Dirichlet Laplacian by a nonlinear function whose derivative jumps over the principal eigenvalue of the operator. Various extensions of this landmark result were obtained for self-adjoint operators, in particular by Berger and Podolak, who gave a geometrical description of the solution set. In this text we show that similar theorems are valid for non self-adjoint operators. In particular, we prove that the semilinear operator is a global fold. As a consequence, we obtain what appears to be the first exact multiplicity resul…
We consider the following class of fractional parametric problems \begin{equation*} \left\{ \begin{array}{ll} (-Δ_{Dir})^{s} u= f(x, u)+tφ_{1}+h &\mbox{ in } Ω\\ u=0 &\mbox{ on } \partial Ω, \end{array} \right. \end{equation*} where $Ω\subset \mathbb{R}^{N}$ is a smooth bounded domain, $s\in (0, 1)$, $N> 2s$, $(-Δ_{Dir})^{s}$ is the fractional Dirichlet Laplacian, $f: \barΩ \times \mathbb{R} \rightarrow \mathbb{R}$ is a locally Lipschitz nonlinearity having linear or superlinear growth and satisfying Ambrosetti-Prodi type assumptions, $t\in \mathbb{R}$, $φ_{1}$ is the first eigenfunction of th…
In this paper we study the existence of solution for the following class of nonlocal problems \[ L_0u =f(x,u)+g(x) , \ \mbox{in} \ Ω, \] where $Ω\subset \mathbb{R}^{N}$, $N\geq 1$, is a bounded connected open, $g \in C(\overlineΩ)$, $f:\overlineΩ \times \mathbb{R} \to \mathbb{R}$ are function, and $L_0 : C(\overlineΩ) \to C(\overlineΩ)$ is a nonlocal dispersal operator. Using a sub-supersolution method and the degree theory for $γ$-Condensing maps, we have obtained a result of the Ambrosetti-Prodi type, that is, we obtain a necessary condition on $g$ for the non-existence of solutions, the exi…
We study the degenerate elliptic equation $-\mathop{\rm div}(|x|^α\nabla u) =f(u)+tϕ(x)+h(x)$ in a bounded open set $Ω$ with homogeneous Neumann boundary condition, where $α\in(0,2)$ and $f$ has a linear growth. The main result establishes the existence of real numbers $t_*$ and $t^*$ such that the problem has at least two solutions if $t\leq t_*$, there is at least one solution if $t_*<t\leq t^*$, and no solution exists for all $t>t^*$. The proof combines a priori estimates with topological degree arguments.
We establish Ambrosetti--Prodi type results for viscosity and classical solutions of nonlinear Dirichlet problems for the fractional Laplace and comparable operators. In the choice of nonlinearities we consider semi-linear and super-linear growth cases separately. We develop a new technique using a functional integration-based approach, which is more robust in the non-local context than a purely analytic treatment.
Multiplicity results for critical fractional Ambrosetti-Prodi type system with nonlinearities interacting with the spectrum
本文研究涉及分数阶Laplacian算子、非线性项达临界增长且与谱相互作用的Ambrosetti-Prodi型系统，证明了解的存在性，并进一步处理了高阶特征值处的共振情形。
Critical Ambrosetti-Prodi type problems on Carnot groups
本文研究了Carnot群上涉及子拉普拉斯算子的临界Ambrosetti-Prodi问题，探讨了参数λ<λ₁和λ>λ₁时解的存在性和多重性，以及在λ=λ₁时共振情况下的解的存在性及特征。
A Numerical Algorithm for Ambrosetti-Prodi Type Operators
针对具有有界连续导数的非线性项 f 的方程 -Δu - f(u) = g，提出一种结合有限元法和全局 Lyapunov-Schmidt 分解的数值算法。
Solutions with clustering concentration layers to the Ambrosetti-Prodi type problem
本文研究了具有聚类集中层的Ambrosetti-Prodi类型问题，通过分析临界点和相关算子的性质，证明了当参数t趋近于正无穷时，问题存在具有聚类集中层的解。
On the existence of a second positive solution to mixed local-nonlocal concave-convex critical problems
本文秉承Ambrosetti等人1994年的精神，证明了带有次线性扰动的混合局部-非局部临界半线性椭圆问题存在第二正弱解。
Since the seminal work of Ambrosetti and Prodi, the study of global folds was enriched by geometric concepts and extensions accomodating new examples. We present the advantages of considering fibers, a construction dating to Berger and Podolak's view of the original theorem. A description of folds in terms of properties of fibers gives new perspective to the usual hypotheses in the subject. The text is intended as a guide, outlining arguments and stating results which will be detailed elsewhere.
We consider a class of semilinear nonlocal problems with vanishing exterior condition and establish a Ambrosetti-Prodi type phenomenon when the nonlinear term satisfies certain conditions. Our technique makes use of the probabilistic tools and heat kernel estimates.
Critical concave-convex problems in Carnot groups
研究Carnot群中具有凹凸和临界非线性的Dirichlet问题，利用变分Perron方法和Sobolev不等式的极小函数估计，证明存在两个正解。
The classical result of A. Ambrosetti and G. Prodi [1], in the form of M.S. Berger and E. Podolak [4], gives the exact number of solutions for the problem \[ Δu+g(u)= μϕ_1(x)+e(x) \;\; \mbox{in $D$} , \;\; u=0 \;\; \mbox{on $\partial D$} \,, \] depending on the real parameter $μ$, for a class of convex $g(u)$, and $\int _D e(x) ϕ_1(x)\, dx=0$ (where $ϕ_1(x)>0$ is the principal eigenfunction of the Laplacian on $D$, and $D \subset R^n$ is a smooth domain). By considering generalized harmonics, we give a similar result for the problem \[ Δu+g(u)= μf(x) \;\; \mbox{in $D$} , \;\; u=0 \;\; \mbox{on…
Multiplicity and asymptotics of positive solutions for critical-concave Kirchhoff equation
本文研究临界Kirchhoff方程在凹扰动下的正解乘法及渐进行为，利用约束极小化方法、山峰定理和集中紧致原理，证明小λ下的正解存在性，并分析b和λ趋近于0时的渐近行为。
We study the bifurcation of solutions of semilinear elliptic boundary value problems of the form \begin{align*} \begin{aligned} -Δu &= f_λ(|x|,u,|\nabla u|) &&\text{in }Ω, u &= 0 &&\text{on }\partialΩ, \end{aligned} \end{align*} on an annulus $Ω\subset\mathbb{R}^N$, with a concave-convex nonlinearity, a special case being the nonlinearity first considered by Ambrosetti, Brezis and Cerami: $f_λ(|x|,u,|\nabla u|)=λ|u|^{q-2}u + |u|^{p-2}u$ with $1<q<2<p$. Although the trivial solution $u_0\equiv0$ is nondegenerate if $λ=0$ we prove that $(λ_0,u_0)=(0,0)$ is a bifurcation point. In fact, the bifur…
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…
Global folds between Banach spaces are obtained from a simple geometric construction: a Fredholm operator $T$ of index zero with one dimensional kernel is perturbed by a compatible nonlinear term $P$. The scheme encapsulates most of the known examples and suggests new ones. Concrete examples rely on the positivity of an eigenfunction. For the standard Nemitskii case $P(u) = f(u)$ (but $P$ might be nonlocal, non-variational), $T$ might be the Laplacian with different boundary conditions, as in the Ambrosetti-Prodi theorem, or the Schrödinger operators associated with the quantum harmonic oscill…
We consider the problem of Ambrosetti-Prodi type \begin{equation}\label{0}\quad\begin{cases} Δu + e^u = sϕ_1 + h(x) &\hbox{in} Ω, u=0 & \hbox{on} \partial Ω, \end{cases} \nonumber \end{equation} where $Ω$ is a bounded, smooth domain in $\R^2$, $ϕ_1$ is a positive first eigenfunction of the Laplacian under Dirichlet boundary conditions and $h\in\mathcal{C}^{0,α}(\barΩ)$. We prove that given $k\ge 1$ this problem has at least $k$ solutions for all sufficiently large $s>0$, which answers affirmatively a conjecture by Lazer and McKenna \cite{LM1} for this case. The solutions found exhibit multiple…
Quasilinear problems with mixed local-nonlocal operator and concave-critical nonlinearities: Multiplicity of positive solutions
研究混合局部-非局部算子驱动的凹-临界问题正解的存在性和多重性，证明了参数λ>0时存在正解的临界值，并扩展了相关经典结果到一般p情况。
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