We study inverse problems for semilinear elliptic equations with fractional power type nonlinearities. Our arguments are based on the higher order linearization method, which helps us to solve inverse problems for certain nonlinear equations in cases where the solution for a corresponding linear equation is not known. By using a fractional order adaptation of this method, we show that the results of [LLLS20a, LLLS20b] remain valid for general power type nonlinearities.
In this article, we establish the existence of solutions to the fractional $p-$Kirchhoff type equations with a generalized Choquard nonlinearities without assuming the Ambrosetti-Rabinowitz condition.
In this paper we investigate the existence of positive solutions and ground state solution for a class of fractional Schrödinger-Poisson equations in $\mathbb R^3$ with general nonlinearities.
We study the well-posedness of a semilinear fractional diffusion equation and formulate an associated inverse problem. We determine fractional power type nonlinearities from the exterior partial measurements of the Dirichlet-to-Neumann map. Our arguments are based on a first order linearization as well as the parabolic Runge approximation property.
We study the existence of nontrivial solutions for a nonlinear fractional elliptic equation in presence of logarithmic and critical exponential nonlinearities. This problem extends [5] to fractional $N/s$-Laplacian equations with logarithmic nonlinearity. We overcome the lack of compactness due to the critical exponential nonlinearity by using the fractional Trudinger-Moser inequality. The existence result is established via critical point theory.
We investigate partial symmetry of solutions to semi-linear and quasi-linear elliptic problems with convex nonlinearities, in domains that are either axially symmetric or radially symmetric.
We establish some existence results for a class of critical elliptic problems with singular exponential nonlinearities. We do not assume any global sign conditions on the nonlinearity, which makes our results new even in the nonsingular case.
We introduce a method for solving Calderón type inverse problems for semilinear equations with power type nonlinearities. The method is based on higher order linearizations, and it allows one to solve inverse problems for certain nonlinear equations in cases where the solution for a corresponding linear equation is not known. Assuming the knowledge of a nonlinear Dirichlet-to-Neumann map, we determine both a potential and a conformal manifold simultaneously in dimension $2$, and a potential on transversally anisotropic manifolds in dimensions $n \geq 3$. In the Euclidean case, we show that one…
In this paper, we consider a fractional p-Laplacian system with both concave-convex nonlinearities and sign-changing weight functions in bounded domains. With the help of the Nehari\ manifold, we prove that the system has at least two nontrivial solutions when the pair of the parameters $(λ,μ)$ belongs to a certain subset of $\mathbb{R}^{n}$
We study the combined effect of concave and convex nonlinearities on the number of positive solutions for a fractional system involving critical Sobolev exponents. With the help of the Nehari manifold, we prove that the system admits at least two positive solutions when the pair of parameters $(λ,μ)$ belongs to a suitable subset of $\R^2$.
We consider the uniqueness of positive solutions to semilinear elliptic equations with double power nonlinearities. We deduce the uniqueness from the argument in the classical paper by Peletier and Serrin, thereby recovering a part of the uniqueness result of Ouyang and Shi.
In this paper we prove the existence of infinitely many nontrivial solutions for the class of $(p,\, q)$ fractional elliptic equations involving concave-critical nonlinearities in bounded domains in $\mathbb{R}^N$. Further, when the nonlinearity is of convex-critical type, we establish the multiplicity of nonnegative solutions using variational methods. In particular, we show the existence of at least $cat_Ω(Ω)$ nonnegative solutions.
In this article we find the equivalent conditions to assure the existence and uniqueness of positive solutions to semilinear elliptic equations wih double power nonlinearities. As a bonus, we give a simpler proof of our former result that the uniqueness condition comes from the existence condition.
We investigate qualitative properties of positive singular solutions of some elliptic systems in bounded and unbounded domains. We deduce symmetry and monotonicity properties via the moving plane procedure. Moreover, in the unbounded case, we study some cooperative elliptic systems involving critical nonlinearities in $\mathbb{R}^n$.
We consider the semilinear fractional equation $ (I-Δ)^s u = a(x) |u|^{p-2}u$ in $\mathbb{R}^N$, where $N \geq 3$, $0<s<1$, $2<p<2N/(N-2s)$ and $a$ is a bounded weight function. Without assuming that $a$ has an asymptotic profile at infinity, we prove the existence of a ground state solution.
We investigate the monotonicity method for fractional semilinear elliptic equations with power type nonlinearities. We prove that if-and-only-if monotonicity relations between coefficients and the derivative of the Dirichlet-to-Neumann map hold. Based on the strong monotonicity relations, we study a constructive global uniqueness for coefficients and inclusion detection for the fractional Calderón type inverse problem. Meanwhile, we can also derive the Lipschitz stability with finitely many measurements. The results hold for any $n\geq 1$.
We consider semilinear elliptic equations with double power nonlineaities. The condition to assure the existence of positive solutions is well-known. In the present paper, we remark that the additional condition to assure uniqueness proposed by Ouyang and Shi is unnecessary.
In this paper, we study the existence of multiple positive solutions for a class of fractional Schrödinger-Poisson systems involving sign-changing potential and critical nonlinearities on an unbounded domain. With the help of Nehari manifold and Ljusternik-Schnirelmann category, we investigate how the coefficient $g(x)$ of the critical nonlinearity affects the number of positive solutions. Moreover, we present a novel relationship between the number of positive solutions and the category of the global maximum set of $g(x)$.
We consider uniqueness of positive solutions to semilinear elliptic equations with double power nonlinearities. The condition to assure the existence of positive solutions to these types of equations has long been known. On the other hand for uniqueness, quite technical additional condition is proposed by Ouyang and Shi in 1998. In the present paper we remark that this additional condition is unnecessary.
We study the following nonlinear Choquard equation driven by a fractional Laplacian: $$ (-Δ)^{s}u+ u =(|x|^{-μ}\ast F(u))f(u)|{4.14mm}{in}|{1.14mm} \mathbb{R}^N, $$ with $N\geq3$, $s\in(0,1)$ and $μ\in(0,N)$. By Supposing that the nonlinearities satisfy the general Berestycki-Lions type conditions \cite{BL}, we are able to prove the existence of groundstates for this equation by variational methods.
We consider a class of singularly perturbed elliptic problems with nonautonomous asymptotically linear nonlinearities. The dependence on the spatial coordinates comes from the presence of a potential and of a function representing a saturation effect. We investigate the existence of nontrivial nonnegative solutions concentrating around local minima of both the potential and of the saturation function. Necessary conditions to locate the possible concentration points are also given.
We consider positive solutions to a singular semilinear elliptic equation in bounded smooth domains, with zero Dirichlet boundary conditions. We provide some weak and strong maximum principles for the H^1_0 part of the solution that allow to deduce symmetry and monotonicity properties of the solutions, via the Moving Plane Method.
We study the existence of least energy sign-changing solution for the fractional equation $(-Δ)^{s} u=|u|^{2_{s}^{*}-2}u+λf(x,u)$ in a smooth bounded domain $Ω$ of $\mathbb{R}^{N},$ $u=0$ in $\mathbb{R}^{N}\setminus Ω$, where $s\in (0,1)$ and $2_{s}^{*}$ is the fractional critical Sobolev exponent. The proof is based on constrained minimization in a subset of Nehari manifold, containing all the possible sign-changing solutions of the equation.
We deal with the existence of infinitely many solutions for a class of elliptic problems with non-symmetric nonlinearities. Our result, which is motivated by a well known conjecture formulated by A. Bahri and P.L. Lions, suggests a new approach to tackle these problems. The proof is based on a method which does not require to use techniques of deformation from the symmetry and may be applied to more general non-symmetric problems.
The theory of elliptic equations involving singular nonlinearities is well studied topic but the interaction of singular type nonlinearity with nonlocal nonlinearity in elliptic problems has not been investigated so far. In this article, we study the very singular and doubly nonlocal singular problem $(P_λ)$(See below). Firstly, we establish a very weak comparison principle and the optimal Sobolev regularity. Next using the critical point theory of non-smooth analysis and the geometry of the energy functional, we establish the global multiplicity of positive weak solutions.
We prove an abstract critical point theorem based on a cohomological index theory that produces pairs of nontrivial critical points with nontrivial higher critical groups. This theorem yields pairs of nontrivial solutions that are neither local minimizers nor of mountain pass type for problems with combined nonlinearities. Applications are given to subcritical and critical $p$-Laplacian problems, Kirchhoff type nonlocal problems, and critical fractional $p$-Laplacian problems.
We prove the existence of a positive solution to a semipositone $N$-Laplacian problem with a critical Trudinger-Moser nonlinearity. The proof is based on obtaining uniform $C^{1,α}$ a priori estimates via a compactness argument. Our result is new even in the semilinear case $N = 2$, and our arguments can easily be adapted to obtain positive solutions of more general semipositone problems with critical Trudinger-Moser nonlinearities.
In this article we study the existence of sign changing solution of the following p-fractional problem with concave-critical nonlinearities: \begin{eqnarray*} (-Δ)^s_pu &=& μ|u|^{q-1}u + |u|^{p^*_s-2}u \quad\mbox{in}\quad Ω, u&=&0\quad\mbox{in}\quad\mathbb{R}^N\setminusΩ, \end{eqnarray*} where $s\in(0,1)$ and $p\geq 2$ are fixed parameters, $0<q<p-1$, $μ\in\mathbb{R}^+$ and $p_s^*=\frac{Np}{N-ps}$. $Ω$ is an open, bounded domain in $\mathbb{R}^N$ with smooth boundary with $N>ps$ .
We show that the linear span of the set of scalar products of gradients of harmonic functions on a bounded smooth domain $Ω\subset \mathbb{R}^n$ which vanish on a closed proper subset of the boundary is dense in $L^1(Ω)$. We apply this density result to solve some partial data inverse boundary problems for a class of semilinear elliptic PDE with quadratic gradient terms.
In this paper, we consider the fractional elliptic equation \begin{align*} \left\{\begin{aligned} &(-Δ)^s u-μ\frac{u}{|x|^{2s}} = \frac{|u|^{2_s^\ast (α)-2}u}{|x|^α} + f(x,u), && \mbox{in} \ Ω,\\ &u=0, && \mbox{in} \ \mathbb{R}^{n}\backslash \ Ω, \end{aligned}\right. \end{align*} where $Ω\subset R^n$ is a smooth bounded domain, $0\inΩ$, $0<s<1$, $0<α<2s<n$, $2_{s}^{\ast}(α)=\frac{2(n-α)}{n-2s}$. Under some assumptions on $μ$ and $f$, we obtain the existence of nonnegative solutions.
This paper deals with the following fractional Schr$ \ddot{\textrm{o}}$dinger equations with Choquard-type nonlinearities \begin{equation*} \left\{\begin{array}{r@{\ \ }c@{\ \ }ll} (-Δ)^{\fracα{2}}u + u - C_{n,-β} \,(|x|^{β-n}\ast u^{p})\, u^{p-1}& = &0 & \mbox{in}\ \ \mathbb{R}^{n}\,, \\[0.05cm] u & > & 0 & \mbox{on}\ \ \mathbb{R}^{n}, \end{array}\right. \end{equation*} where $ 0< α,β< 2, 1\leq p <\infty \,\,and\,\, n\geq 2. $ First we construct a decay result at infinity and a narrow region principle for related equations. Then we establish the radial symmetry of positive solutions for the a…
In this paper, we study the existence and multiplicity results of nontrivial positive solutions to a quasilinear elliptic equation in $\RN$, when $N\geq2$, as \begin{equation} \Lp u+u^{p-1}=λ\hspace{0.2mm}k(x)u^{r-1}-h(x)u^{q-1}.\nonumber \end{equation} Here, $h(x),k(x)>0$ are Lebesgue measurable functions, $1<p<q<\infty$, $p<r<\min\{p^*,q\}$ if $p<N$ while $p<r<q$ if $p\geq N$, and $λ>0$ is a parameter.
We study critical growth elliptic problems with jumping nonlinearities. Standard linking arguments based on decompositions of $H^1_0(Ω)$ into eigenspaces of $- Δ$ cannot be used to obtain nontrivial solutions to such problems. We show that the associated variational functional admits certain linking structures based on splittings of $H^1_0(Ω)$ into nonlinear submanifolds. In order to capture these linking geometries, we prove several generalizations of the classical linking theorem of Rabinowitz that are not based on linear subspaces. We then use these new linking theorems to obtain nontrivial…
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…
This paper is devoted to the following class of nonlinear fractional Schrödinger equations: \begin{equation*} (-Δ)^{s} u + V(x)u = f(x,u) + λg(x,u), \quad \text{in}\: \mathbb{R}^N, \end{equation*} where $s\in (0,1)$, $N>2s$, $(-Δ)^{s}$ stands for the fractional Laplacian, $λ\in \mathbb{R}$ is a parameter, $V\in C(\mathbb{R}^N,R)$, $f(x,u)$ is superlinear and $g(x,u)$ is sublinear with respect to $u$, respectively. We prove the existence of infinitely many high energy solutions of the aforementioned equation by means of the Fountain theorem. Some recent results are extended and sharply improved…
In this paper, we study a class of fractional Schrödinger equations involving logarithmic and critical nonlinearities on an unbounded domain, and show that such an equation with positive or sign-changing weight potentials admits at least one positive ground state solution and the associated energy is positive (or negative). By applying the Nehari manifold method and Ljusternik-Schnirelmann category, we deeply investigate how the weight potential affects the multiplicity of positive solutions, and obtain the relationship between the number of positive solutions and the category of some sets rel…
In this paper, we develop a sliding method for the fractional Laplacian. We first obtain the key ingredients needed in the sliding method either in a bounded domain or in the whole space, such as narrow region principles and maximum principles in unbounded domains. Then using semi-linear equations involving the fractional Laplacian in both bounded domains and in the whole space, we illustrate how this new sliding method can be employed to obtain monotonicity of solutions. Some new ideas are introduced. Among which, one is to use Poisson integral representation of $s$-subharmonic functions in d…
We give pointwise gradient bounds for solutions of (possibly non-uniformly) elliptic partial differential equations in the entire Euclidean space. The operator taken into account is very general and comprises also the singular and degenerate nonlinear case with non-standard growth conditions. The sourcing term is also allowed to have a very general form, depending on the space variables, on the solution itself, on its gradient, and possibly on higher order derivatives if additional structural conditions are satisfied.
It is established existence of bound and ground state solutions for quasilinear elliptic systems driven by (ϕ1, ϕ2)-Laplacian operator. The main feature here is to consider quasilinear elliptic systems involving both nonsingular nonlinearities combined with indefinite potentials and singular cases perturbed by superlinear and subcritical couple terms. These prevent us to use arguments based on Ambrosetti-Rabinowitz condition and variational methods for differentiable functionals. By exploring the Nehari method and doing a fine analysis on the fibering map associated, we get estimates that allo…
We show that the elliptic equation with a non-Lipschitz right-hand side, $-Δu = λ|u|^{β-1}u - |u|^{α-1}u$ with $λ>0$ and $0<α<β<1$, considered on a smooth star-shaped domain $Ω$ subject to zero Dirichlet boundary conditions, might possess a nonnegative ground state solution which violates Hopf's maximum principle only on a nonempty subset $Γ$ of the boundary $\partialΩ$ such that $Γ\neq \partialΩ$.
In this paper, we consider the singularly perturbed fractional Schrödinger equation \begin{equation*} ε^{2α}(-Δ)^αu+V(x)u=f(u),\quad x\in \mathbb{R}^N, \end{equation*} where $ε>0$ is a small parameter, $α\in(0,1)$, $(-Δ)^α$ is the fractional Laplacian operator of order $α$, $V$ possesses global minimum points, and $f$ is asymptotically linear at infinity. We investigate the relationship between the number of positive solutions and the topology of the set where the potential $V$ attains its global minimum. We also construct multiple concentrating solutions if $V$ has several strict global minim…
This paper is devoted to the analysis of blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities \[ i\partial_t u-(-Δ)^su+λ_1|u|^{2p_1}u+λ_2|u|^{2p_2}u=0, \] where $0<p_1<p_2<\frac{2s}{N-2s}$. Firstly, we obtain some sufficient conditions about existence of blow-up solutions, and then derive some sharp thresholds of blow-up and global existence by constructing some new estimates. Moreover, we find the sharp threshold mass of blow-up and global existence in the case $0<p_1<\frac{2s}{N}$ and $p_2=\frac{2s}{N}$. Finally, we investigate the dynam…
In this article, we study the Brezis-Nirenberg type problem of nonlinear Choquard equation involving a fractional Laplacian \[ (-\De)^s u = \left( \int_{\Om}\frac{|u|^{2^*_{μ,s}}}{|x-y|^μ}\mathrm{d}y \right)|u|^{2^*_{μ,s}-2}u +\la u \; \text{in } \Om,\] where $\Om $ is a bounded domain in $\mathbb R^n$ with Lipschitz boundary, $\la $ is a real parameter, $s \in (0,1)$, $n >2s$ and $2^*_{μ,s}= (2n-μ)/(n-2s)$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. We obtain some existence, multiplicity, regularity and nonexistence results for solution of the above equation …
In this article, we study the following fractional Laplacian equation with critical growth and singular nonlinearity $$\quad (-Δ)^s u = λa(x) u^{-q} + u^{2^*_s-1}, \quad u>0 \; \text{in}\; Ω,\quad u = 0 \; \mbox{in}\; \mathbb{R}^n \setminusΩ,$$ where $Ω$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary $\partial Ω$, $n > 2s,\; s \in (0,1),\; λ>0,\; 0 < q \leq 1 $, $θ\leq a(x) \in L^\infty(Ω)$, for some $θ>0$ and $2^*_s=\frac{2n}{n-2s}$. We use variational methods to show the existence and multiplicity of positive solutions of the above problem with respect to the parameter $λ$.
In the paper we consider elliptic equations of the form $-Au=u^{-γ}\cdotμ$, where $A$ is the operator associated with a regular symmetric Dirichlet form, $μ$ is a positive nontrivial measure and $γ>0$. We prove the existence and uniqueness of solutions of such equations as well as some regularity results. We also study stability of solutions with respect to the convergence of measures on the right-hand side of the equation. For this purpose, we introduce some type of functional convergence of smooth measures, which in fact is equivalent to the quasi-uniform convergence of associated potentials…
