This paper concerns the existence of a nontrivial solution for the following problem
\begin{equation}
\left\{\begin{aligned}
-Δu + V(x)u & \in \partial_u F(x,u)\;\;\mbox{a.e. in}\;\;\mathbb{R}^{N},\nonumber
u \in H^{1}(\mathbb{R}^{N}).
\end{aligned}
\right.\leqno{(P)}
\end{equation} where $F(x,t)=\int_{0}^{t}f(x,s)\,ds$, $f$ is a $\mathbb{Z}^{N}$-periodic Caratheodory function and $λ=0$ does not belong to the spectrum of $-Δ+V$. Here, $\partial_t F$ denotes the generalized gradient of $F$ with respect to variable $t$.
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 present a new approach to solve a Schrödinger Equation autonomous at infinity, by identifying the relation between the arrangement of the spectrum of the concerned operator and the behavior of the nonlinearity at zero and at infinity. In order to apply variational methods, we set up a suitable linking structure depending on the growth of the nonlinear term and making use of information about the autonomous problem at infinity. Our method allows us to circumvent the lack of compactness. The main novelty is that none monotonicity assumption is required on the nonlinearity, which may be sign-c…
Let $Ω$ be a smooth bounded domain in $\mathbb{R}^{N}$, $N\geq1$, let $K$, $M$ be two nonnegative functions and let $α,γ>0$. We study existence and nonexistence of positive solutions for singular problems of the form $-Δu=K\left( x\right) u^{-α}-λM\left( x\right) u^{-γ}$ in $Ω$, $u=0$ on $\partialΩ$, where $λ>0$ is a real parameter. We mention that as a particular case our results apply to problems of the form $-Δu=m\left( x\right) u^{-γ}$ in $Ω$, $u=0$ on $\partialΩ$, where $m$ is allowed to change sign in $Ω$.
In this note, we study Q-curvature flow on $S^4$ with indefinite nonlinearity. Our result is that the prescribed Q-curvature problem on $S^4$ has a solution provided the prescribed Q-curvature $f$ has its positive part, which possesses non-degenerate critical points such that $Δ_{S^4} f\not=0$ at the saddle points and an extra condition such as a nontrivial degree counting condition.
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…
In this work, we study a quasilinear elliptic problem involving the 1-laplacian operator, with a discontinuous, superlinear and subcritical nonlinearity involving the Heaviside function $H(\cdot - β)$. Our approach is based on an analysis of the associated p-laplacian problem, followed by a thorough analysis of the asymptotic behaviour or such solutions as $p \to 1^+$. We study also the asymptotic behaviour of the solutions, as $β\to 0^+$ and we prove that it converges to a solution of the original problem, without the discontinuity in the nonlinearity.
In this paper, we prove the existence of a weak solution for the Dirichlet boundary value problem related to the $p(x)-$Laplacian $$ -\mbox{div}(|\nabla u|^{p(x)-2}\nabla u)+u\in -[\underline{g}(x,u),\overline{g}(x,u)], $$ by using the degree theory after turning the problem into a Hammerstein equation. The right hand side $g$ is a possibly discontinuous function in the second variable satisfying some non-standard growth conditions..
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.
The existence of two nontrivial smooth solutions to a semilinear Robin problem with indefinite unbounded potential and asymmetric nonlinearity $f$ is established. Both crossing and resonance are allowed. A third nonzero solution exists provided $f$ is $C^1$. Proofs exploit variational methods, truncation techniques, and Morse theory.
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…
This paper concerns with the existence of nontrivial solution for the following problem \begin{equation} \left\{\begin{aligned} -Δu + V(x)u & = γH_{e}(|u|-a)|u|^{q-2}u+|u|^{2^{*}-2}u\;\;\mbox{ in}\;\;\mathbb{R}^{N},\nonumber u \in H^{1}(\mathbb{R}^{N}). \end{aligned} \right. \end{equation} where, $N\geq 3$, $γ\geq 0$, $H_{e}:\mathbb{R} \to \mathbb{R}$ denotes the Heaviside function, $a \geq 0$, $2<q<2^*$ and $V:\mathbb{R^{N}} \to \mathbb{R}$ is $\mathbb{Z}^{N}$-periodic with $β= 0$ does not belong to the spectrum of $-Δ+V$.
This paper concerns the existence of a nontrivial solution for the following problem \begin{equation} \left\{\begin{aligned} -Δu + V(x)u & \in \partial_u F(x,u)\;\;\mbox{a.e. in}\;\;\mathbb{R}^{N},\nonumber u \in H^{1}(\mathbb{R}^{N}), \end{aligned} \right.\leqno{(P)} \end{equation} where $F(x,t)=\int_{0}^{t}f(x,s)\,ds$, $f$ is a discontinuous function and asymptotically linear at infinity, $λ=0$ is in a spectral gap of $-Δ+V$, and $\partial_t F$ denotes the generalized gradient of $F$ with respect to variable $t$. Here, by employing Variational Methods for Locally Lipschitz Functionals, we es…
In this article, we prove the existence of solutions to a nonlinear nonlocal elliptic problem with a singualrity and a discontinuous critical nonlinearity which is given as follows. \begin{align} \begin{split}\label{main_prob} (-Δ)_p^su&=μg(x,u)+\fracλ{u^γ}+H(u-α)u^{p_s^*-1},~\text{in}~Ω u&>0,~\text{in}~Ω, u&=0,~\text{in}~\mathbb{R}^N\setminusΩ, \end{split} \end{align} where $Ω\subset\mathbb{R}^N$ is a bounded domain with Lipschitz boundary, $s\in (0,1)$, $2<p<\frac{N}{s}$, $γ\in (0,1)$, $λ,μ>0$, $α\geq 0$ is real, $H$ is the Heaviside function, i.e. $H(a)=0$ if $a\leq 0$, $H(a)=1$ if $a>0$ an…
Indian Institute of Technology Bhubaneswar（印度理工学院布巴内斯瓦尔分校）；University of Ljubljana（卢布尔雅那大学）；Institute of Mathematics, Physics and Mechanics（数学、物理与力学研究所）；Imam Abdulrahman Bin Faisal University（伊玛姆阿卜杜勒拉赫曼·本·费萨尔大学）
Considering the radial nonlinear Schrodinger equation - Δu + V(x)u = g(x,u) in R^N, N \geq 3 we aim to find a radial nontrivial solution for it, where V changes sign ensuring this problem is indefinite and g is an asymptotically linear nonlinearity. We work with variational methods associating to the problem an indefinite functional in order to apply our Abstract Linking Theorem for Cerami sequences in [8] to get a non-trivial critical point for this functional. Our goal is to make use of spectral properties of operator A:= - Δ+ V(x) restricted to H^1_{rad}(R^N), the space of radially symmetri…
The inverse problem for the Sturm- Liouville operator with complex periodic potential and discontinuous coefficients on the axis is studied. Main characteristics of the fundamental solutions are investigated, the spectrum of the operator is studied. We give formulation of the inverse problem, prove a uniqueness theorem and provide a constructive procedure for the solution of the inverse problem.
We prove an existence result for a class of Dirichlet boundary value problems with discontinuous nonlinearity and involving a Leray-Lions operator. The proof combines monotonicity methods for elliptic problems, variational inequality techniques and basic tools related to monotone operators.
We establish the existence of an entire solution for a class of stationary Schrödinger systems with subcritical discontinuous nonlinearities and lower bounded potentials that blow-up at infinity. The proof is based on the critical point theory in the sense of Clarke and we apply Chang's version of the Mountain Pass Lemma for locally Lipschitz functionals. Our result generalizes in a nonsmooth framework a result of Rabinowitz \cite{rabi} related to entire solutions of the Schrödinger equation.
We consider a semilinear elliptic equation in a bounded domain with zero boundary conditions. The nonlinearity is discontinuous and monotone, but it is not a Carathéodory's function. The existence theorem has been proved.
We establish the existence of a nontrivial weak solution to strongly indefinite asymptotically linear and superlinear Schrödinger equations. The novelty is to identify the essential relation between the spectrum of the operator and the behavior of the nonlinear term, in order to weaken the necessary assumptions to obtain a linking structure to the problem, for instance to allow zero being in the spectrum or the nonlinearity being sign-changing. Our main difficulty is to overcome the lack of monotonicity on the nonlinear term, as well, as the lack of compactness since the domain is unbounded. W…
In this paper, we study the following Schrödinger-Poisson system: $$ \left\{\aligned&-Δu+V_λ(x)u+K(x)ϕu=f(x,u)&\quad\text{in }\bbr^3,\\ &-Δϕ=K(x)u^2&\quad\text{in }\bbr^3,\\ &(u,ϕ)\in\h\times\D,\endaligned\right.\eqno{(\mathcal{SP}_λ)} $$ where $V_λ(x)=λa(x)+b(x)$ with a positive parameter $λ$, $K(x)\geq0$ and $f(x,t)$ is continuous including the power-type nonlinearity $|u|^{p-2}u$. By applying the method of penalized functions, the existence of one nontrivial solution for such system in the less-studied case $3<p\leq4$ is obtained for $λ$ sufficiently large. The concentration behavior of thi…
We study the minimizing problem $\inf\left\{\displaystyle\int_Ωp(x)|\nabla u|^{2}dx,\,u\in H^{1}_{0}(Ω),\,\|u\|_{L^{\frac{2N}{N-2}}(Ω)}=1\right\}$ where $Ω$ is a smooth bounded domain of $\R^{N}$, $N\geq 3$ and $p$ a positive discontinuous function. We prove the existence of a minimizer under some assumptions.
We prove the existence of solutions for the following critical Choquard type problem with a variable-order fractional Laplacian and a variable singular exponent \begin{align*} \begin{split} a(-Δ)^{s(\cdot)}u+b(-Δ)u&=λ|u|^{-γ(x)-1}u+\left(\int_Ω\frac{F(y,u(y))}{|x-y|^{μ(x,y)}}dy\right)f(x,u) & +ηH(u-α)|u|^{r(x)-2}u,~\text{in}~Ω, u&=0,~\text{in}~\mathbb{R}^N\setminusΩ. \end{split} \end{align*} where $a(-Δ)^{s(\cdot)}+b(-Δ)$ is a mixed operator with variable order $s(\cdot):\mathbb{R}^{2N}\rightarrow (0,1)$, $a, b\geq 0$ with $a+b>0$, $H$ is the Heaviside function (i.e., $H(t)=0$ if $t\leq0$, $H(…
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 analyse the existence and the stability of the ground states of the one-dimensional nonlinear Schrödinger equation with a focusing power nonlinearity and a defect located at the origin. In this paper a ground state is defined as a global minimizer of the action functional on the Nehari manifold and the defect considered is a Fülöp-Tsutsui $δ$ type, namely a $δ$ condition that allows discontinuities. The existence of ground states is proved by variational techniques, while the stability results from the Grillakis-Shatah-Strauss theory.
Let $Ω$ be a bounded domain in $\mathbb R^{N}$, $N\geq3$ with smooth boundary, $a>0, λ>0$ and $0<δ<3$ be real numbers. Define $2^*:=\displaystyle\frac{2N}{N-2}$ and the characteristic function of a set $A$ by $χ_A$. We consider the following critical problem with singular and discontinuous nonlinearity: \begin{eqnarray*} (P_\la^a)~~~~ \qquad \Biggl\{\begin{array}{rl} -Δu &= λ\left(u^{2^*-1}+ \displaystyle χ_{\{u<a\}}u^{-\de} \right), u > 0~~\text{in} ~~Ω, \\ u & = 0 ~\text{on}~ \partial Ω. \end{array} \end{eqnarray*} \noindent We study the existence and the global multiplicity of solutions to …
We consider a system of quasilinear elliptic equations, with indefinite super-linear nonlinearity, depending on two real parameters $λ,μ$. By using the Nehari manifold and the notion of extremal parameter, we extend some results concerning existence of positive solutions.
In this paper, we are concerned with a Kirchhoff problem in the presence of a strongly-singular term perturbed by a discontinuous nonlinearity of the Heaviside type in the setting of Orlicz-Sobolev space. The presence of both strongly-singular and non-continuous terms bring up difficulties in associating a differentiable functional to the problem with finite energy in the whole space $W_0^{1,Φ}(Ω)$. To overcome this obstacle, we established an optimal condition for the existence of $W_0^{1,Φ}(Ω)$-solutions to a strongly-singular problem, which allows us to constrain the energy functional to a …
This paper is devoted to the existence and non-existence of positive solutions for a $(p, q)$-Laplacian system with indefinite nonlinearity depending on two parameters $(λ,μ)$. By using the sub-supersolution method together with adaptations of ideas found in \cite{AAS,CRMZ}, we extend some previous result.
We study the existence of fixed points to a parameterized Hammertstain operator $\cH_β,$ $β\in (0,\infty],$ with sigmoid type of nonlinearity. The parameter $β<\infty$ indicates the steepness of the slope of a nonlinear smooth sigmoid function and the limit case $β=\infty$ corresponds to a discontinuous unit step function. We prove that spatially localized solutions to the fixed point problem for large $β$ exist and can be approximated by the fixed points of $\cH_\infty.$ These results are of a high importance in biological applications where one often approximates the smooth sigmoid by discon…
We address the question: Why may reaction-diffusion equations with hysteretic nonlinearities become ill-posed and how to amend this? To do so, we discretize the spatial variable and obtain a lattice dynamical system with a hysteretic nonlinearity. We analyze a new mechanism that leads to appearance of a spatio-temporal pattern called {\it rattling}: the solution exhibits a propagation phenomenon different from the classical traveling wave, while the hysteretic nonlinearity, loosely speaking, takes a different value at every second spatial point, independently of the grid size. Such a dynamics …
In this paper, we consider the following free boundary problem $$ (P)\left\{\begin{array}{ll} Δu = λϕ(x)\Sum_{i=1}^n H(u-μ_i )& \quad \mbox{ in }\ Ω=Ω_2\setminus \overlineΩ_1, \\[0.3cm]u =0 &\quad \mbox{ on } \partial Ω_2, \\[0.3cm]u =M &\quad \mbox{ on } \partial Ω_1. \end{array} \right. $$ The domain $Ω$ is a convex ring where $Ω_1$ and $Ω_2$ are bounded convex domain in $\mathbb{R}^N,$ $N\geq 2,$ $H$ is the Heaviside step function, $λ, M,μ_i$ $(i=1,..,n)$ are a given positive real parameters and $ϕ$ is a given function . We show that under suitable conditions, there exists a solution to pro…
We look for ground states and bound states $E:\mathbb{R}^3\to\mathbb{R}^3$ to the curl-curl problem $$\nabla\times(\nabla\times E)= f(x,E) \qquad\hbox{in } \mathbb{R}^3$$ which originates from nonlinear Maxwell equations. The energy functional associated with this problem is strongly indefinite due to the infinite dimensional kernel of $\nabla\times(\nabla\times \cdot)$. The growth of the nonlinearity $f$ is controlled by an $N$-function $Φ:\mathbb{R}\to [0,\infty)$ such that $\displaystyle\lim_{s\to 0}Φ(s)/s^6=\lim_{s\to+\infty}Φ(s)/s^6=0$. We prove the existence of a ground state, i.e. a lea…
We study semiclassical states of the nonlinear Dirac equation \[ -i\hbar\partial_tψ= ic\hbar\sum_{k=1}^3α_k\partial_kψ- mc^2βψ- M(x)ψ+ f(|ψ|)ψ,\quad t\in\mathbb{R},\ x\in\mathbb{R}^3, \] where $V$ is a bounded continuous potential function and the nonlinear term $f(|ψ|)ψ$ is superlinear, possibly of critical growth. Our main result deals with standing wave solutions that concentrate near a critical point of the potential. Standard methods applicable to nonlinear Schrödinger equations, like Lyapunov-Schmidt reduction or penalization, do not work, not even for the homogeneous nonlinearity $f(s)=…
In this paper, we study the asymptotic behavior of the solutions of nonlocal bistable reaction-diffusion equations starting from compactly supported initial conditions. Depending on the relationship between the nonlinearity, the interaction kernel and the diffusion coefficient, we show that the solutions can either: propagate, go extinct or remain pinned. We especially focus on the latter regime where solutions are pinned by thoroughly studying discontinuous ground state solutions of the problem for a specific interaction kernel serving as a case study. We also present a detailed numerical ana…
We survey recent results concerning ground states and bound states $u\colon\mathbb{R}^3\to\mathbb{R}^3$ to the curl-curl problem $$\nabla\times(\nabla\times u)+V(x)u= f(x,u) \quad\hbox{ in } \mathbb{R}^3,$$ which originates from the nonlinear Maxwell equations. The energy functional associated with this problem is strongly indefinite due to the infinite dimensional kernel of $\nabla\times(\nabla\times \cdot)$. The growth of the nonlinearity $f$ is superlinear and subcritical at infinity or purely critical and we demonstrate a variational approach to the problem involving the generalized Nehari…
We consider the nonlinear Poisson-Boltzmann equation in the context of electrostatic models for a biological macromolecule, embedded in a bounded domain containing a solution of an arbitrary number of ionic species which is not necessarily charge neutral. The resulting semilinear elliptic equation combines several difficulties: exponential growth and lack of sign preservation in the nonlinearity accounting for ion mobility, measure data arising from point charges inside the molecule, and discontinuous permittivities across the molecule boundary. Exploiting the modelling assumption that the poi…
In this paper we study some key effects of a discontinuous forcing term in a fourth order wave equation on a bounded domain, modeling the adhesion of an elastic beam with a substrate through an elastic-breakable interaction. By using a spectral decomposition method we show that the main effects induced by the nonlinearity at the transition from attached to detached states can be traced in a loss of regularity of the solution and in a migration of the total energy through the scales.
