This article investigates the existence, nonexistence, and multiplicity of positive solutions to the sublinear fractional elliptic problem $(P_λ^s)$. We begin by establishing several a priori estimates that provide regularity results and describe the qualitative behavior of solutions. A critical threshold level for the parameter $λ$ is identified, which plays a crucial role in determining the existence or nonexistence of solutions. The sub and supersolution method is employed to obtain a weak solution. Furthermore, we establish a relation between the local minimizers of $\mathcal{D}^{s,2}(\mathbb{R}^N)$ versus $C(\mathbb{R}^N; 1+|x|^{N-2s})$. Combining these results with the Classical Linking Theorem, we demonstrate the existence of at least two distinct positive weak solutions to $(P_λ^s)$. This work extends the results of Yang, Abrantes, Ubilla, and Zhou (J. Differential Equations, 416:159-189, 2025) to the nonlocal setting, i.e., when $s \in (0,1)$. Several technical challenges arise in this framework, such as the lack of a standard comparison principle in $\mathbb{R}^N$ in the fractional setting.
We study the Lie point symmetries of semilinear Kohn-Laplace equations on the Heisenberg group H^1 and obtain a complete group classification of these equations.
We give necessary and sufficient conditions for the solvability of some semilinear elliptic boundary value problems involving the Laplace operator with linear and nonlinear highest order boundary conditions involving the Laplace-Beltrami operator.
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…
In this manuscript, in the $L^1$ scaling critical case, a lifespan estimate of solutions to the Cauchy problem for non-gauge invariant semilinear semirelativistic equations is considered. The lifespan estimate is given by the modified test function method with a fractional Laplace operator. The main obstacle to obtaining the lifespan estimate is the non-locality of the fractional Laplace operator. To treat the non-locality, special test functions are introduced.
We show that any Lie point symmetry of semilinear Kohn-Laplace equations on the Heisenberg group H^1 with power nonlinearity is a divergence symmetry if and only if the corresponding exponent assumes critical value.
We study a nonlinear elliptic boundary value problem defined on a smooth bounded domain involving the fractional Laplace operator, a concave-convex powers term together with mixed Dirichlet-Neumann boundary conditions.
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…
Using the complete group classification of semilinear differential equations on the three-dimensional Heisenberg group carried out in a preceding work, we establish the conservation laws for the critical Kohn-Laplace equations via the Noether's Theorem.
We show which Lie point symmetries of non-critical semilinear Kohn-Laplace equations on the Heisenberg group $H^1$ are Noether symmetries and we establish their respectives conservations laws.
We consider the semilinear diffusion equation $\partial$ t u = Au + |u| $α$ u in the half-space R N + := R N --1 x (0, +$\infty$), where A is a linear diffusion operator, which may be the classical Laplace operator, or a fractional Laplace operator, or an appropriate non regularizing nonlocal operator. The equation is supplemented with an initial data u(0, x) = u 0 (x) which is nonnegative in the half-space R N + , and the Dirichlet boundary condition u(t, x ' , 0) = 0 for x ' $\in$ R N --1. We prove that if the symbol of the operator A is of order a|$ξ$| $β$ near the origin $ξ$ = 0, for some …
In this paper, we study the existence of multiple solutions to a generalized $p(\cdot)$-Laplace equation with two parameters involving critical growth. More precisely, we give sufficient "local" conditions, which mean that growths between the main operator and nonlinear term are locally assumed for the cases $p(\cdot)$-sublinear, $p(\cdot)$-superlinear, and sandwich-type. Compared to constant exponent problems (for examples, $p$-Laplacian and $(p,q)$-Laplacian), this characterizes the study of variable exponent problems. We show this by applying variants of the Mountain Pass Theorem for $p(\cd…
We study the ground states of the following generalization of the Kirchhoff-Love functional, $$J_σ(u)=\int_Ω\dfrac{(Δu)^2}{2} - (1-σ)\int_Ωdet(\nabla^2u)-\int_ΩF(x,u),$$ where $Ω$ is a bounded convex domain in $\mathbb{R}^2$ with $C^{1,1}$ boundary and the nonlinearities involved are of sublinear type or superlinear with power growth. These critical points correspond to least-energy weak solutions to a fourth-order semilinear boundary value problem with Steklov boundary conditions depending on $σ$. Positivity of ground states is proved with different techniques according to the range of the pa…
Our purpose is to find positive solutions $u \in D^{1,2}(\rz^N)$ of the semilinear elliptic problem $-\laplace u = h(x) u^{p-1}$ for $2<p$. The function $h$ may have an indefinite sign. Key ingredients are a $h$-dependent concentration-compactness Lemma and a characterization of compact embeddings of $D^{1,2}(\rz^N)$ into weighted Lebesgue spaces.
We consider radial solutions of a general elliptic equation involving a weighted Laplace operator. We establish the uniqueness of the radial bound state solutions to $$ {div}\big(\mathsf A\,\nabla v\big)+\mathsf B\,f(v)=0\,,\quad\lim_{|x|\to+\infty}v(x)=0,\quad x\in\mathbb R^n,$$ $n>2$, where $\mathsf A$ and $\mathsf B$ are two positive, radial, smooth functions defined on $\mathbb R^n\setminus\{0\}$. We assume that the nonlinearity $f\in C(-c,c)$, $0<c\le\infty$ is an odd function satisfying some convexity and growth conditions, and has a zero at $b>0$, is non positive and not identically 0 i…
Let $Ω$ be a smooth bounded domain in $\mathbb{R}^N$ and $δ(x)=\text{dist}\,(x,\partial Ω)$. Assume $μ>0$, $ν$ is a nonnegative finite measure on $\partial Ω$ and $g \in C(Ω\times \mathbb{R}_+)$. We study positive solutions of $$ (P)\qquad -Δu - \fracμ{δ^2} u = g(x,u) \text{ in } Ω, \qquad \text{tr}^*(u)=ν. $$ Here $\text{tr}^*(u)$ denotes the normalized boundary trace of $u$ which was recently introduced by M. Marcus and P. T. Nguyen. We focus on the case $0<μ< C_H(Ω)$ (the Hardy constant for $Ω$) and provide some qualitative properties of solutions of (P). When $g(x,u)=u^q$ with $q>1$, we pr…
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…
We study and compare two concepts for weak solutions to semilinear parabolic path-dependent partial differential equations (PPDEs). The first is that of mild solutions as it appears, e.g., in the log-Laplace functionals of historical superprocesses. The aim of this paper is to show that mild solutions are also solutions in a viscosity sense. This result is motivated by the fact that mild solutions can provide value functions and optimal strategies for problems of stochastic optimal control. Since unique mild solutions exist under weak conditions, we obtain as a corollary a general existence re…
Recently, several works have been carried out in attempt to develop a theory for linear or sublinear elliptic equations involving a general class of nonlocal operators characterized by mild assumptions on the associated Green kernel. In this paper, we study the Dirichlet problem for superlinear equation (E) ${\mathbb L} u = u^p +λμ$ in a bounded domain $Ω$ with homogeneous boundary or exterior Dirichlet condition, where $p>1$ and $λ>0$. The operator ${\mathbb L}$ belongs to a class of nonlocal operators including typical types of fractional Laplacians and the datum $μ$ is taken in the optimal …
We study the existence and nonexistence of positive (super-)solutions to a singular semilinear elliptic equation $$-\nabla\cdot(|x|^A\nabla u)-B|x|^{A-2}u=C|x|^{A-σ}u^p$$ in cone--like domains of $\R^N$ ($N\ge 2$), for the full range of parameters $A,B,σ,p\in\R$ and $C>0$. We provide a complete characterization of the set of $(p,σ)\in\R^2$ such that the equation has no positive (super-)solutions, depending on the values of $A,B$ and the principle Dirichlet eigenvalue of the cross--section of the cone. The proofs are based on the explicit construction of appropriate barriers and involve the ana…
Dirichlet problem for Lane-Emden type equations with several sublinear terms
Sirindhorn International Institute of Technology, Thammasat University（诗琳通国际理工学院，朱拉隆功大学）；Department of Mathematics, Graduate School of Advanced Science and Engineering, Hiroshima University（广岛大学高级理工学研究科数学系）
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…
We study the existence, multiplicity, and certain qualitative properties of solutions to the zero Dirichlet problem for the equation $-Δ_p u = λ|u|^{p-2}u + a(x)|u|^{q-2}u$ in a bounded domain $Ω\subset \mathbb{R}^N$, where $1<q<p$, $λ\in\mathbb{R}$, and $a$ is a continuous sign-changing weight function. Our primary interest concerns ground states and nonnegative solutions which are positive in $\{x\in Ω: a(x)>0\}$, when the parameter $λ$ lies in a neighborhood of the critical value $λ^* = \inf\left\{\int_Ω|\nabla u|^p \, dx/\int_Ω|u|^p \, dx: u\in W_0^{1,p}(Ω) \setminus \{0\},\ \int_Ωa|u|^q\,…
We consider nonlinear diffusive evolution equations posed on bounded space domains, governed by fractional Laplace-type operators, and involving porous medium type nonlinearities. We establish existence and uniqueness results in a suitable class of solutions using the theory of maximal monotone operators on dual spaces. Then we describe the long-time asymptotics in terms of separate-variables solutions of the friendly giant type. As a by-product, we obtain an existence and uniqueness result for semilinear elliptic non local equations with sub-linear nonlinearities. The Appendix contains a revi…
This paper is devoted to study the semilinear elliptic system of Hénon-type \begin{eqnarray*} -Δ_{\mathbb{B}^{N}}u= K(d(x))Q_{u}(u,v) \\ -Δ_{\mathbb{B}^{N}}v= K(d(x))Q_{v}(u,v) \end{eqnarray*} in the hyperbolic space $\mathbb{B}^{N}$, $N\geq 3$, where $u, v \in H_{r}^{1}(\mathbb{B}^{N})=\{ϕ\in H^1(\mathbb{B}^N): ϕ\, \text{is radial}\}$ and $-Δ_{\mathbb{B}^{N}}$ denotes the Laplace-Beltrami operator on $\mathbb{B}^N$, $Q \in C^{1}(\mathbb{R}\times \mathbb{R},\mathbb{R})$ is a p-homogeneous function, $d(x)=d_{\mathbb{B}^N}(0,x)$ and $K\geq0 $ is a continuous function. We prove a compactness resu…
Our purpose is to find positive solutions $u \in D^{1,2}(\rz^N)$ of the semilinear elliptic problem $-\laplace u - λV(x) u = h(x) u^{p-1}$ for $2<p$. The functions $V$ and $h$ may have an indefinite sign and the linearized operator need not to have a first (principal) eigenvalue, e.g. we allow $V\equiv 1$. We give precise existence and nonexistence criteria, which depend on $λ$ and on the growth of $h^{-}$ and $h^{+}/V^+$. Existence theorems are obtained by constrained minimization. The mountain pass theorem leads to a second solution.
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 $0<α<\frac{n}{2}$,…
Let T be a homogeneous tree and L the Laplace operator on T. We consider the semilinear Schrodinger equation associated to L with a power-like nonlinearity F of degree d. We first obtain dispersive estimates and Strichartz estimates with no admissibility conditions. We next deduce global well-posedness for small L2 data with no gauge invariance assumption on the nonlinearity F. On the other hand if F is gauge invariant, L2 conservation leads to global well-posedness for arbitrary L2 data. Notice that, in contrast with the Euclidean case, these global well-posedness results hold with no restric…
In this paper we discuss the existence and non--existence of weak solutions to parametric equations involving the Laplace-Beltrami operator $Δ_g$ in a complete non-compact $d$--dimensional ($d\geq 3$) Riemannian manifold $(\mathcal{M},g)$ with asymptotically non--negative Ricci curvature and intrinsic metric $d_g$. Namely, our simple model is the following problem $$ \left\{ \begin{array}{ll} -Δ_gw+V(σ)w=λα(σ)f(w) & \mbox{ in } \mathcal{M}\\ w\geq 0 & \mbox{ in } \mathcal{M} \end{array}\right. $$ where $V$ is a positive coercive potential, $α$ is a positive bounded function, $λ$ is a real para…
In this paper we discuss the existence, uniqueness and regularity of solutions of the following system of coupled semilinear Poisson equations on a smooth bounded domain $Ω$ in $\mathbb{R}^n$: \[ \left\{{llll} \mathcal{A}^s u= v^p & {\rm in} \ \ Ω\mathcal{A}^s v = f(u) & {\rm in} \ \ Ωu= v=0 & {\rm on} \ \ \partialΩ \right. \] where $s\in (0, 1)$ and $\mathcal{A}^s$ denote spectral fractional Laplace operators. We assume that $1< p<\frac{2s}{n-2s}$, and the function $f$ is superlinear and with no growth restriction (for example $f(r)=re^r$); thus the system has a nontrivial solution. Another i…
Critical $p$-Laplace equations with monotone coefficients: Liouville classification and a Schoen-type Harnack inequality
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学研究所）
A Pohozaev-type neck proof of a conditional Harnack inequality in the critical $p$-Laplacian setting
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Institute of Mathematics, Academy of Mathematics and Systems Science, the Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学研究所）
