We study the nonlinear elliptic problem $-Δu=ρ(x)f(u)$ in $\RR^N$ ($N\geq 3$), $\lim\_{|x|\ri\infty}u(x)=\ell$, where $\ell\geq 0$ is a real number, $ρ(x)$ is a nonnegative potential belonging to a certain Kato class, and $f(u)$ has a sublinear growth. We distinguish the cases $\ell>0$ and $\ell=0$ and we prove existence and uniqueness results if the potential $ρ(x)$ decays fast enough at infinity. Our arguments rely on comparison techniques and on a theorem of Brezis and Oswald for sublinear elliptic equations.
In this work we prove the existence of a classical positive solution for an elliptic equation with a sublinear term. We use Galerkin approximations to show existence of such solution on bounded domains in RN.
We consider the following problem $ -Δ_{p}u= h(x,u) \mbox{ in }Ω$, $u\in W^{1,p}_{0}(Ω)$, where $Ω$ is a bounded domain in $\mathbb{R}^{N}$, $1<p<N$, with a smooth boundary. In this paper we assume that $h(x,u)=a(x)f(u)+b(x)g(u)$ such that $f$ is regularly varying of index $p-1$ and superlinear at infinity. The function $g$ is a $p$-sublinear function at zero. The coefficients $a$ and $b$ belong to $L^{k}(Ω)$ for some $k>\frac{N}{p}$ and they are without sign condition. Firstly, we show a priori bound on solutions, then by using variational arguments, we prove the existence of at least two non…
We study existence and uniqueness of bounded solutions to a fractional sublinear elliptic equation with a variable coefficient, in the whole space. Existence is investigated in connection to a certain fractional linear equation, whereas the proof of uniqueness relies on uniqueness of solutions to an associated fractional porous medium equation with variable density.
We investigate second order elliptic equations \[F(\mathcal{H}u) = 0\] where the function $F\colon S(n)\to\mathbb{R}$ on the space of symmetric $n\times n$ matrices is assumed to be sublinear. There is very little to be found in the literature devoted particularly to sublinear elliptic operators. When examples of such operators occur, they are often merely treated as members of the larger class of convex operators. That class has been thoroughly investigated and many of its aspects are well understood. There is, however, something to be said about sublinear operators that do not, in general, a…
In this article we deal with different forms of the unique continuation property for second order elliptic equations with nonlinear potentials of sublinear growth. Under suitable regularity assumptions, we prove the weak and the strong unique continuation property. Moreover, we also discuss the unique continuation property from measurable sets, which shows that nodal domains to these equations must have vanishing Lebesgue measure. Our methods rely on suitable Carleman estimates, for which we include the sublinear potential into the main part of the operator.
We present a necessary and sufficient condition on nonnegative Radon measures $μ$ and $ν$ for the existence of a positive continuous solution of the Dirichlet problem for the sublinear elliptic equation $-Δu=μu^q+ν$ with prescribed nonnegative continuous boundary data in a general domain. Moreover, two-sided pointwise estimates of Brezis-Kamin type for positive bounded solutions and the uniqueness of a positive continuous $L^q$-solution are investigated.
Let $L $ be a second order elliptic operator with smooth coefficients defined on a domain $Ω\subset \mathbb{R}^d$ (possibly unbounded), $d\geq 3$. We study nonnegative continuous solutions $u$ to the equation $L u(x) - φ(x, u(x))=0$ on $Ω$, where $φ$ is in the Kato class with respect to the first variable and it grows sublinearly with respect to the second variable. Under fairly general assumptions we prove that if there is a bounded non zero solution then there is no large solution.
We are concerned with a class of second order quasilinear elliptic equations driven by a nonhomogeneous differential operator introduced by C.A. Stuart and whose study is motivated by models in Nonlinear Optics. We establish sufficient conditions for the existence of at least one or two non-negative solutions. Our analysis considers the cases when the reaction has either a sublinear or a linear growth. In the sublinear case, we also prove a nonexistence property. The proofs combine energy estimates and variational methods. In particular, the monotonicity trick is applied in order to overcome t…
We establish a positivity property for a class of semilinear elliptic problems involving indefinite sublinear nonlinearities. Namely, we show that any nontrivial nonnegative solution is positive for a class of problems the strong maximum principle does not apply to. Our approach is based on a continuity argument combined with variational techniques, the sub and supersolutions method and some a priori bounds. Both Dirichlet and Neumann homogeneous boundary conditions are considered. As a byproduct, we deduce some existence and uniqueness results. Finally, as an application, we derive some posit…
We establish several results related to existence, nonexistence or bifurcation of positive solutions for a Dirichlet boundary value problem with in a smooth bounded domain. The main feature of this paper consists in the presence of a singular nonlinearity, combined with a convection term. Our approach takes into account both the sign of the potential and the decay rate around the origin of the singular nonlinearity. The proofs are based on various techniques related to the maximum principle for elliptic equations.
We study positive supersolutions to an elliptic equation $(*)$: $-Δu=c|x|^{-s}u^p$, $p,s\in\bf R$ in cone-like domains in $\bf R^N$ ($N\ge 2$). We prove that in the sublinear case $p<1$ there exists a critical exponent $p_*<1$ such that equation $(*)$ has a positive supersolution if and only if $-\infty<p<p_*$. The value of $p_*$ is determined explicitly by $s$ and the geometry of the cone.
In this paper we study the positive solutions of sub linear elliptic equations with a Hardy potential which is singular at the boundary. By means of ODE techniques a fairly complete picture of the class of radial solutions is given. Local solutions with a prescribed growth at the boundary are constructed by means of contraction operators. Some of those radial solutions are then used to construct ordered upper and lower solutions in general domains. By standard iteration arguments the existence of positive solutions is proved. An important tool is the Hardy constant.
Let $L$ be a second order elliptic operator $L$ with smooth coefficients defined on a domain $Ω$ in $\mathbb{R}^d $, $d\geq3$, such that $L1\leq 0$. We study existence and properties of continuous solutions to the following problem \begin{equation}\label{00} Lu=φ(\cdot,u),% & \hbox{in $Ω$; in the sens of distribution;} \\ \end{equation} in $Ω,$ where $Ω$ is a Greenian domain for $L$ {(possibly unbounded)} in $\mathbb{R}^d$ and $φ$ is a nonnegative function on $Ω\times [0,+\infty [$ increasing with respect to the second variable. By means of thinness, we obtain a characterization of $φ$ for whi…
Let $Ω$ be a smooth bounded domain in $\mathbb{R}^{N}$ and let $m$ be a possibly discontinuous and unbounded function that changes sign in $Ω$. Let $f:\left[ 0,\infty\right) \rightarrow\left[ 0,\infty\right) $ be a continuous function such that $k_{1}ξ^{p}\leq f\left(ξ\right) \leq k_{2}ξ^{p}$ for all $ξ\geq0$ and some $k_{1},k_{2}>0$ and $p\in\left(0,1\right) $. We study existence and nonexistence of strictly positive solutions for nonlinear elliptic problems of the form $-Δu=m\left(x\right) f\left(u\right) $ in $Ω$, $u=0$ on $\partialΩ$.
We give necessary and sufficient conditions for the existence of a positive solution with zero boundary values to the elliptic equation \[ \mathcal{L}u = σu^{q} + μ\quad \text{in} \;\; Ω, \] in the sublinear case $0<q<1$, with finite generalized energy: $\mathbb{E}_γ[u]:=\int_Ω |\nabla u|^{2} u^{γ-1}dx<\infty$, for $γ>0$. In this case $u \in L^{γ+q}(Ω, σ)\cap L^γ(Ω, μ)$, where $γ=1$ corresponds to finite energy solutions. Here $\mathcal{L} u:= -\,\text{div}(\mathcal{A}\nabla u)$ is a linear uniformly elliptic operator with bounded measurable coefficients, and $σ$, $μ$ are nonnegative functions…
Let $f$ be a continuous and non-decreasing function such that $f>0$ on $(0,\infty)$, $f(0)=0$, $\sup \_{s\geq 1}f(s)/s< \infty$ and let $p$ be a non-negative continuous function. We study the existence and nonexistence of explosive solutions to the equation $Δu+|\nabla u|=p(x)f(u)$ in $Ω,$ where $Ω$ is either a smooth bounded domain or $Ω=\RR^N$. If $Ω$ is bounded we prove that the above problem has never a blow-up boundary solution. Since $f$ does not satisfy the Keller-Osserman growth condition at infinity, we supply in the case $Ω=\RR^N$ a necessary and sufficient condition for the existenc…
We study quasilinear elliptic equations of the type $$-Δ_pu=σ\, u^q \quad \text{in} \, \, \, \mathbb{R}^n,$$ where $Δ_p u=\nabla \cdot(\nabla u |\nabla u|^{p-2})$ is the $p$-Laplacian (or a more general $\mathcal{A}$-Laplace operator $\text{div} \, \mathcal{A}(x, \nabla u)$), $0<q < p-1$, and $σ\ge 0$ is an arbitrary locally integrable function or measure on $\mathbb{R}^n$. We obtain necessary and sufficient conditions for the existence of positive solutions (not necessarily bounded) which satisfy global pointwise estimates of Brezis-Kamin type given in terms of Wolff potentials. Similar probl…
In this paper, we establish space like strong unique continuation property (sucp) for uniformly parabolic sublinear equations under appropriate structural assumptions. Our main result Theorem 1.1 constitutes the parabolic counterpart of the strong unique continuation result recently established by Ruland in [Ru] for analogous elliptic sublinear equations. Similar to that in [Ru], this is accomplished via a new $L^{2}-L^{2}$ type Carleman estimate for a class of sublinear parabolic operators.
We give a survey of nonlinear potential estimates and their applications obtained recently for positive solutions to sublinear problems of the type \[ u = \mathbf{G}(σu^q) + f \quad \textrm{in} \,\, Ω, \] where $0 < q < 1$, $σ\ge 0$ is a Radon measure in $Ω$, $ f \ge 0$ is a measurable function, and $\mathbf{G}$ is a linear integral operator with positive kernel $G$ on $Ω\timesΩ$. For quasi-metric (or quasi-metrically modifiable) kernels $G$, these bilateral pointwise estimates yield existence criteria and uniqueness of solutions $u \in L^q_{\rm loc} (Ω, σ)$. Applications are considered to sem…
We study the weighted norm inequality of $(1,q)$-type, \[ \Vert \mathbf{G}ν\Vert_{L^q(Ω, dσ)} \le C \Vert ν\Vert, \quad \text{ for all } ν\in \mathcal{M}^+(Ω), \] along with its weak-type analogue, for $0 < q < 1$, where $\mathbf{G}$ is an integral operator associated with the nonnegative kernel $G(x,y)$. Here $\mathcal{M}^+(Ω)$ denotes the class of positive Radon measures in $Ω$; $σ, ν\in \mathcal{M}^+(Ω)$, and $||ν||=ν(Ω)$. For both weak-type and strong-type inequalities, we provide conditions which characterize the measures $σ$ for which such an embedding holds. The strong-type $(1,q)$-ineq…
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…
We are concerned with singular elliptic equations of the form $-Δu= p(x)(g(u)+ f(u)+|\nabla u|^a)$ in $\RR^N$ ($N\geq 3$), where $p$ is a positive weight and $0< a <1$. Under the hypothesis that $f$ is a nondecreasing function with sublinear growth and $g$ is decreasing and unbounded around the origin, we establish the existence of a ground state solution vanishing at infinity. Our arguments rely essentially on the maximum principle.
We derive the unique continuation property of a class of semi-linear elliptic equations with non-Lipschitz nonlinearities. The simplest type of equations to which our results apply is given as $-Δu = |u|^{σ-1} u$ in a domain $Ω\subset \mathbb{R}^N$, with $0 \le σ<1$. Despite the sublinear character of the nonlinear term, we prove that if a solution vanishes in an open subset of $Ω$, then it vanishes necessarily in the whole $Ω$. We then extend the result to equations with variable coefficients operators and inhomogeneous right-hand side.
We are concerned with the nodal set of solutions to equations of the form \begin{equation*} -Δu = λ_+ \left(u^+\right)^{q-1} - λ_- \left(u^-\right)^{q-1} \quad \text{in $B_1$} \end{equation*} where $λ_+,λ_- > 0$, $q \in [1,2)$, $B_1=B_1(0)$ is the unit ball in $\mathbb{R}^N$, $N \ge 2$, and $u^+:= \max\{u,0\}$, $u^-:= \max\{-u,0\}$ are the positive and the negative part of $u$, respectively. This class includes, the \emph{unstable two-phase membrane problem} ($q=1$), as well as \emph{sublinear} equations for $1<q<2$. We prove the following main results: (a) the finiteness of the vanishing orde…
This article concerns the fractional elliptic equations \begin{equation*}(-Δ)^{s}u+λV(x)u=f(u), \quad u\in H^{s}(\mathbb{R}^N), \end{equation*}where $(-Δ)^{s}$ ($s\in (0\,,\,1)$) denotes the fractional Laplacian, $λ>0$ is a parameter, $V\in C(\mathbb{R}^N)$ and $V^{-1}(0)$ has nonempty interior. Under some mild assumptions, we establish the existence of nontrivial solutions. Moreover, the concentration of solutions is also explored on the set $V^{-1}(0)$ as $λ\to\infty$.
We study local regularity properties for solutions of linear, non-uniformly elliptic equations. Assuming certain integrability conditions on the coefficient field, we prove local boundedness and Harnack inequality. The assumed integrability assumptions are essentially sharp and improve upon classical results by Trudinger [ARMA 1971]. We then apply the deterministic regularity results to the corrector equation in stochastic homogenization and establish sublinearity of the corrector.
We report on some recent existence and uniqueness results for elliptic equations subject to Dirichlet boundary condition and involving a singular nonlinearity. We take into account the following types of problems: (i) singular problems with sublinear nonlinearity and two parameters; (ii) combined effects of asymptotically linear and singular nonlinearities in bifurcation problems; (iii) bifurcation for a class of singular elliptic problems with subquadratic convection term. In some concrete situations we also establish the asymptotic behaviour of the solution around the bifurcation point. Our …
We give bilateral pointwise estimates for positive solutions $u$ to the sublinear integral equation \[ u = \mathbf{G}(σu^q) + f \quad \textrm{in} \,\, Ω,\] for $0 < q < 1$, where $σ\ge 0$ is a measurable function, or a Radon measure, $f \ge 0$, and $\mathbf{G}$ is the integral operator associated with a positive kernel $G$ on $Ω\timesΩ$. Our main results, which include the existence criteria and uniqueness of solutions, hold for quasi-metric, or quasi-metrically modifiable kernels $G$. As a consequence, we obtain bilateral estimates, along with the existence and uniqueness, for positive soluti…
The paper concerns the discreteness of the eigenvalues and the solvability of the interior transmission problem for anisotropic media. Conditions for the ellipticity of the problem are written explicitly, and it is shown that they do not guarantee the discreteness of the eigenvalues. Some simple sufficient conditions for the discreteness and solvability are found. They are expressed in terms of the values of the anisotropy matrix and the refraction index at the boundary of the domain. The discreteness of the eigenvalues and the solvability of the interior transmission problem are shown if a sm…
We prove new results on the existence of positive radial solutions of the elliptic equation $-Δu= λh(|x|,u)$ in an annular domain in $\mathbb{R}^{N}, N\geq 2$. Existence of positive radial solutions are determined under the conditions that the nonlinearity function $h(t,u)$ is either superlinear or sublinear growth in $u$ or satisfies some upper and lower inequalities on $h$. Our discussion is based on a fixed point theorem due to a revised version of a fixed point theorem of Gustafson and Schmitt.
We study Sobolev spaces on the $n-$dimensional unit ball $B^n(1)$ endowed with a parameter-depending Finsler metric $F_a$, $a\in [0,1],$ which interpolates between the Klein metric $(a=0)$ and Funk metric $(a=1)$, respectively. We show that the standard Sobolev space defined on the Finsler manifold $(B^n(1),F_a)$ is a vector space if and only if $a\in [0,1).$ Furthermore, by exploiting variational arguments, we provide non-existence and existence results for sublinear elliptic problems on $(B^n(1),F_a)$ involving the Finsler-Laplace operator whenever $a\in [0,1).$
In this paper we study the existence and partial regularity of weak solutions to an elliptic-parabolic system that models the single-phase miscible displacement of one incompressible fluid by another in a porous media. The system is singular and involves quadratic nonlinearity. A partial regularity result is obtained via the Fefferman-Stein inequality.
We study an elliptic equation with measurable coefficients arising from photo-acoustic imaging in inhomogeneous media. We establish Holder continuity of weak solutions and obtain pointwise bounds for Green's functions subject to Dirichlet or Neumann condition.
We consider nonlinear second order elliptic problems of the type \[ -Δu=f(u) \text{ in } Ω, \qquad u=0 \text{ on } \partial Ω, \] where $Ω$ is an open $C^{1,1}$-domain in $\mathbb{R}^N$, $N\geq 2$, under some general assumptions on the nonlinearity that include the case of a sublinear pure power $f(s)=|s|^{p-1}s$ with $0<p<1$ and of Allen-Cahn type $f(s)=λ(s-|s|^{p-1}s)$ with $p>1$ and $λ>λ_2(Ω)$ (the second Dirichlet eigenvalue of the Laplacian). We prove the existence of a least energy nodal (i.e. sign changing) solution, and of a nodal solution of mountain-pass type. We then give explicit e…
We obtain new oscillation and gradient bounds for the viscosity solutions of fully nonlinear degenerate elliptic equations where the Hamiltonian is a sum of a sublinear and a superlinear part in the sense of Barles and Souganidis (2001). We use these bounds to study the asymptotic behavior of weakly coupled systems of fully nonlinear parabolic equations. Our results apply to some "asymmetric systems" where some equations contain a sublinear Hamiltonian whereas the others contain a superlinear one. Moreover, we can deal with some particular case of systems containing some degenerate equations u…
In this survey we report on some recent results related to various singular phenomena arising in the study of some classes of nonlinear elliptic equations. We establish qualitative results on the existence, nonexistence or the uniqueness of solutions and we focus on the following types of problems: (i) blow-up boundary solutions of logistic equations; (ii) Lane-Emden-Fowler equations with singular nonlinearities and subquadratic convection term. We study the combined effects of various terms involved in these problems: sublinear or superlinear nonlinearities, singular nonlinear terms, convecti…
Following the methodology of [Brasco and Volzone, Adv. Math. 2022], we study the long-time behavior for the signed Fractional Porous Medium Equation in open bounded sets with smooth boundary. Homogeneous exterior Dirichlet boundary conditions are considered. We prove that if the initial datum has sufficiently small energy, then the solution, once suitably rescaled, converges to a nontrivial constant sign solution of a sublinear fractional Lane-Emden equation. Furthermore, we give a nonlocal sufficient energetic criterion on the initial datum, which is important to identify the exact limit prof…
In this note we study periodic homogenization of Dirichlet problem for divergence type elliptic systems when both the coefficients and the boundary data are oscillating. One of the key difficulties here is the determination of the fixed boundary data corresponding to the limiting (homogenized) problem. This issue has been addressed in recent papers by D. Gérard-Varet and N. Masmoudi, and by C. Prange, however, not much is known about the regularity of this fixed data. The main objective of this note is to initiate a study of this problem, and to prove several regularity results in this connect…
We obtain necessary and sufficient conditions for the existence of a positive finite energy solution to the inhomogeneous quasilinear elliptic equation \[ -Δ_{p} u = σu^{q} + μ\quad \text{on} \;\; \mathbb{R}^n \] in the sub-natural growth case $0<q<p-1$, where $Δ_{p}$ ($1<p<\infty$) is the $p$-Laplacian, and $σ$, $μ$ are positive Borel measures on $\mathbb{R}^n$. Uniqueness of such a solution is established as well. Similar inhomogeneous problems in the sublinear case $0<q<1$ are treated for the fractional Laplace operator $(-Δ)^α$ in place of $-Δ_{p}$, on $\mathbb{R}^n$ for $0<α<\frac{n}{2}$,…
We consider the natural time-dependent fractional $p$-Laplacian equation posed in the whole Euclidean space, with parameter $1<p<2$ and fractional exponent $s\in (0,1)$. Rather standard theory shows that the Cauchy Problem for data in the Lebesgue $L^q$ spaces is well posed, and the solutions form a family of non-expansive semigroups with regularity and other interesting properties. The superlinear case $p>2$ has been dealt with in a recent paper. We study here the "fast" regime $1<p<2$ which is more complex. As main results, we construct the self-similar fundamental solution for every mass va…
We are concerned with singular elliptic problems of the form $-Δu\pm p(d(x))g(u)=\la f(x,u)+μ|\nabla u|^a$ in $Ω,$ where $Ω$ is a smooth bounded domain in $\RR^N$, $d(x)={\rm dist}(x,\partialΩ),$ $\la>0,$ $μ\in\RR$, $0<a\leq 2$, and $f,k$ are nonnegative and nondecreasing functions. We assume that $p(d(x))$ is a positive weight with possible singular behavior on the boundary of $Ω$ and that the nonlinearity $g$ is unbounded around the origin. Taking into account the competition between the anisotropic potential $p(d(x))$, the convection term $|\nabla u|^a$, and the singular nonlinearity $g$, w…
We solve time-harmonic Maxwell's equations in anisotropic, spatially homogeneous media in intersections of $L^p$-spaces. The material laws are time-independent. The analysis requires Fourier restriction-extension estimates for perturbations of Fresnel's wave surface. This surface can be decomposed into finitely many components of the following three types: smooth surfaces with non-vanishing Gaussian curvature, smooth surfaces with Gaussian curvature vanishing along one-dimensional submanifolds, but without flat points, and surfaces with conical singularities. Our estimates are based on new Boc…
