Existence and multiplicity results for a class of semilinear elliptic equations
Vladimir Bobkov, Pavel Drabek, Jesus Hernandez
备注：
25 pages, 12 figures. Two figures and three additional references added according to referee's suggestions; several minor typos corrected. Published in Nonlinear Analysis
We study the existence and multiplicity of nonnegative solutions, as well as the behaviour of corresponding parameter-dependent branches, to the equation $-Δu = (1-u) u^m - λu^n$ in a bounded domain $Ω\subset \mathbb{R}^N$ endowed with the zero Dirichlet boundary data, where $0<m \leq 1$ and $n>0$. When $λ> 0$, the obtained solutions can be seen as steady states of the corresponding reaction-diffusion equation describing a model of isothermal autocatalytic chemical reaction with termination. In addition to the main new results, we formulate a few relevant conjectures.
This paper is concerned with existence and multiplicity results for the semilinear subelliptic equation with free perturbation term. By using the degenerate Rellich-Kondrachov compact embedding theorem, precise lower bound estimates of Dirichlet eigenvalues for the finitely degenerate elliptic operator and minimax method, we obtain the existence and multiplicity of weak solutions for the problem.
In this paper we look for solutions of a semilinear Maxwell type equation, in even dimension, greater than four. These solutions are critical points of a functional which is strongly degenerate because of the presence of the exterior derivative. We prove that, assuming a suitable convexity condition on the nonlinearity, the equation possesses infinitely many finite energy solutions.
In this paper we study multiplicity and qualitative behavior of solutions for semilinear elliptic problems with neumann boundary condition and asymptotically linear smooth nonlinearity. We provide sufficient conditions on the number of eigenvalues the derivative of the nonlinearity crosses to guarantee existence of at least five nontrivial solutions. The techniques we use are a combination of minimization, Leray-Schauder degree, Morse Theory and Reduction method a la Castro-Lazer.
It is shown that a fourth-order semilinear parabolic equation with time-dependent absorption admit a vast multiplicity of the so-called very singular self-similar solutions (VSSs), which can bifurcate from some eigenfunctions of the linearized operators and also are generated by branching from other critical point. For the second-order semilinear heat equations, such VSSs have been known since the beginning of the 1980s and were constructed in papers by Brezis, Friedman, Galaktionov, Kyrdyumov, Samarskii, Peletier, Kamin, Veron, and many others.
We prove the existence of infinitely many classical periodic solutions for a class of semilinear wave equations with periodic boundary conditions. Our argument relies on some new estimates for the linear problem with periodic boundary conditions, by combining Littlewood-Paley techniques, the Hausdorff-Young theorem, and a variational formulation due to Rabinowitz. We also develop a new approach to the regularity of the distributional solutions, by employing Gagliardo-Nirenberg estimates.
Given $k\in\mathbb N$, we define a class of continuous piecewise functions $f$ having abrupt but controlled magnitude changes so that the problem $$Δu +f(u)=0,\quad x\in \mathbb R^N, N> 2, $$ has at least $k$ radially symmetric ground state solutions.
We prove the existence of infinitely many classical periodic solutions for a class of degenerate semilinear wave equations: \[ u_{tt}-u_{xx}+|u|^{s-1}u=f(x,t), \] for all $s>1$. In particular we prove the existence of infinitely many classical solutions for the case $s=3$ posed by Brézis in \cite{BrezisBAMS}. The proof relies on a new upper a priori estimates for minimax values of, a pertubed from symmetry, strongly indefinite functional,depending on a small parameter.
We prove the exact multiplicity of flat and compact support stable solutions of an autonomous non-Lipschitz semilinear elliptic equation of eigenvalue type according to the dimension N and the two exponents, $0<α<β<1$, of the involved nonlinearites. Suitable assumptions are made on the spatial domain $Ω$ where the problem is formulated in order to avoid a possible continuum of those solutions and, on the contrary, to ensure the exact number of solutions according to the nature of the domain $Ω$. Our results also clarify some previous works in the literature. The main techniques of proof are a …
In this paper we give conditions on $f$ so that problem $$ Δu +f(u)=0,\quad x\in \mathbb{R}^N, N\ge 2, $$ has at least two radial bound state solutions with any prescribed number of zeros, and such that $u(0)$ belongs to a specific subinterval of $(0,\infty)$. This property will allow us to give conditions on $f$ so that this problem has at least any given number of radial solutions having a prescribed number of zeros.
We establish a lower bound for the number of sign changing solutions with precisely two nodal domains to the singularly perturbed nonlinear elliptic equation -ε^{2}Δ_{g}u+u=|u|^{p-2}u on an n-dimensional Riemannian manifold M, p in (2,2n/(n-2)), in terms of the cup-length of the configuration space of M. We give a precise description of the asymptotic profile of these solutions as ε goes to 0. We also provide new estimates for the cup-length of the configuration space of M.
In the present paper, we investigate the regularity and symmetry properties of weak solutions to semilinear elliptic equations which are locally stable.
In this paper we study existence and nonexistence of nonnegative distributional solutions for a class of semilinear fractional elliptic equations involving the Hardy potential.
We consider positive singular solutions to semilinear elliptic problems with possibly singular nonlinearity. We deduce symmetry and monotonicity properties of the solutions via the moving plane procedure.
n this paper, we prove existence of nodal solutions for singular semilinear elliptic systems without variational structure where its both components are of sign changing. Our approach is based on sub-supersolutions method combined with perturbation arguments involving singular terms.
We classify stable and finite Morse index solutions to general semilinear elliptic equations posed in Euclidean space of dimension at most 10, or in some unbounded domains.
We consider nonlinear elliptic equations which contains global coupling as a nonlinear term. We classify the existence of all possible positive solutions to this problem.
In this paper we study nonexistence of non-negative distributional supersolutions for a class of semilinear elliptic equations involving inverse-square potentials.
We show that the knowledge of the Dirichlet-to-Neumann map on an arbitrary open portion of the boundary of a domain in $\mathbb{R}^n$, $n\ge 2$, for a class of semilinear elliptic equations, determines the nonlinearity uniquely.
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.
We consider positive solutions to semilinear elliptic problems with singular nonlinearities, under zero Dirichlet boundary condition. We exploit a refined version of the moving plane method to prove symmetry and monotonicity properties of the solutions, under general assumptions on the nonlinearity.
We show an existence of a weak solution of a degenerate and/or singular semilinear elliptic boundary value (nonhomogeneous) problem lying between a given weak subsolution and a given weak supersolution. It has been applied for an existence result of large solution to a similar problem.
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.
We establish quantitative asymptotic behaviors for nonnegative solutions of the critical semilinear equation $-Δu=u^{\frac{n+2}{n-2}}$ with isolated boundary singularities, where $n\ge 3$ is the dimension.
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.
In this paper, we study a class of semilinear nonlocal elliptic equations posed on settings without compact Sobolev embedding. More precisely, we prove the existence of infinitely many solutions to the fractional Brezis-Nirenberg problems on bounded domain.
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.
In studies of superlinear parabolic equations \begin{equation*} u_t=Δu+u^p,\quad x\in {\mathbb R}^N,\ t>0, \end{equation*} where $p>1$, backward self-similar solutions play an important role. These are solutions of the form $ u(x,t) = (T-t)^{-1/(p-1)}w(y)$, where $y:=x/\sqrt{T-t}$, $T$ is a constant, and $w$ is a solution of the equation $Δw-y\cdot\nabla w/2 -w/(p-1)+w^p=0$. We consider (classical) positive radial solutions $w$ of this equation. Denoting by $p_S$, $p_{JL}$, $p_L$ the Sobolev, Joseph-Lundgren, and Lepin exponents, respectively, we show that for $p\in (p_S,p_{JL})$ there are onl…
Given $Ω$ a bounded open subset of $\mathbb{R}^N$, we consider nonnegative solutions to the singular semilinear elliptic equation $-Δ\,u\,=\,\frac{f}{u^β}$ in $H^1_{loc}(Ω)$, under zero Dirichlet boundary conditions. For $β>0$ and $f\in L^1(Ω)$, we prove that the solution is unique.
We study a class of fractional semilinear elliptic equations and formulate the corresponding Calderón problem. We determine the nonlinearity from the exterior partial measurements of the Dirichlet-to-Neumann map by using first order linearization and the Runge approximation property.
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 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.
In this article we will study the nondegeneracy properties of positive finite energy solutions of a semilinear elliptic equation in the N-dimensional Hyperbolic space. We will show that the degeneracy occurs only in an N dimensional subspace. We will prove that the positive solutions are nondegenerate in the case of geodesic balls.
We study nonexistence and existence of nontrivial positive solutions to the following semilinear elliptic inequality involving gradient terms \[ Δu+u^p\left|\nabla u\right|^q\leq0, \] on weighted graphs, where $(p,q)\in\mathbb{R}^2$. We give a complete classification of $(p,q)$ under which sharp volume growth assumptions are established.
We obtain an entire Liouville type theorem to the classical semilinear subcritical elliptic equation on Heisenberg group. A pointwise estimate near the isolated singularity was also proved. The soul of the proofs is an a priori integral estimate, which deduced from a generalized formula of that found by Jerison and Lee.
A monotonicity approach to the study of the asymptotic behavior near corners of solutions to semilinear elliptic equations in domains with a conical boundary point is discussed. The presence of logarithms in the first term of the asymptotic expansion is excluded for boundary profiles sufficiently close to straight conical surfaces.
In this paper we obtain, for a semilinear elliptic problem in R^N, families of solutions bifurcating from the bottom of the spectrum of $-Δ$. The problem is variational in nature and we apply a nonlinear reduction method which allows us to search for solutions as critical points of suitable functionals defined on finite-dimensional manifolds.
This paper is concerned with semilinear equations in divergence form \[ \diver(A(x)Du) = f(u) \] where $f :\R \to [0,\infty)$ is nondecreasing. We prove a sharp Harnack type inequality for nonnegative solutions which is closely connected to the classical Keller-Osserman condition for the existence of entire solutions.
We will present some rigidity results for solutions to semilinear elliptic equations of the form $\Deltau = W'(u)$, where W is a quite general potential with a local minimum and a local maximum. We are particularly interested in Liouvlle-type theorems and symmetry results, which generalise some known facts about the Cahn-Hilliard equation.
In this short note, we investigate simultaneous recovery inverse problems for semilinear elliptic equations with partial data. The main technique is based on higher order linearization and monotonicity approaches. With these methods at hand, we can determine the diffusion, cavity and coefficients simultaneously by knowing the corresponding localized Dirichlet-Neumann operators.
The purpose of this paper is to study the existence of solutions for semilinear elliptic system driven by fractional Laplacian and establish some new existence results which are obtained by virtue of the local linking theorem and the saddle point theorem. To make the nonlinear scheme feasible, rigorous analysis of the function space involved and corresponding energy functional is necessary.
Steady vortices for the three-dimensional Euler equation for inviscid incompressible flows and for the shallow water equation are constructed and showed to tend asymptotically to singular vortex filaments. The construction is based on the asymptotic study of solutions to a semilinear elliptic problem.
We construct positive weak solutions of a class of semilinear elliptic equation which vanish in suitable trace sense on the boundary of a given smooth bounded N-dimensional domain, but which are singular at prescribed isolated points of the boundary. Similar constructions are carried out for solutions which are singular on any given embedded submanifold of the boundary.
Under appropriate assumptions, we show that all bounded entire solutions to a class of semilinear elliptic systems are confined in a convex domain. Moreover, we prove a Liouville type theorem in the case where the domain is strictly convex. Our result represents an extension, under less regularity assumptions, of a recent result of Smyrnelis. We also provide several applications.
We study a class of semilinear elliptic equations with constraints in higher dimension. It is known that several mathematical structures of the problem are closed to those of the Liouville equation in dimension two. In this paper, we establish a classification of entire solutions, the $\sup + \inf$ type inequality and the quantized blowup mechanism.
In this paper, we deal with a class of semilinear elliptic equation in a bounded domain $Ω\subset\mathbb{R}^N$, $N\geq 3$, with $C\sp{1,1}$ boundary. Using a new fixed point result of the Krasnoselskii's type for the sum of two operators, an existence principle of strong solutions is proved. We give two examples where the nonlinearity can be critical.
Our aim in this article is to study semilinear elliptic equations involving a fractional Hardy operator, an absorption and a Radon source in a weighted distributional sense. We show various scenarios, produced by the combined effect of the fractional Hardy potential, the growth of the absorption term and the concentration of the measure, in which existence and uniqueness results holds.
We consider the second order semilinear elliptic system $Δu= p\left( x\right) v^α,$ $Δv= q\left(x\right) u^β,$ where $x \in \mathbf{R}^N,$ $N \geq 3,$ $α$ and $β$ are positive constants, $p$ and $q$ are nonnegative continuous functions. We prove that nontrivial nonnegative entire solutions fail to exist if the functions $p$ and $q$ are of slow decay.
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 show the existence of a weak solution of a semilinear elliptic Dirichlet problem on an arbitrary open set. We make no assumptions about the open set, very mild regularity assumptions on the semilinearity, plus a coerciveness assumption which depends on the optimal Poincare-Steklov constant. The proof is based on Schaefer's fixed point theorem applied to a sequence of truncated problems. We state a simple uniqueness result. We also generalize the results to Robin boundary conditions.
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.
This article is a survey on boundedness results for stable solutions to semilinear elliptic problems. For these solutions, we present the currently known $L^{\infty}$ estimates that hold for all nonlinearities. Such estimates are known to hold up to dimension 4. They are expected to be true also in dimensions 5 to 9, but this is still an open problem which has only been established in the radial case.
In this paper, we consider systems of semilinear elliptic equations \displaystyle -Δ_{\mathbb{H}^{N}}u=|v|^{p-1}v, \displaystyle -Δ_{\mathbb{H}^{N}}v=|u|^{q-1}u, in the whole of Hyperbolic space $\mathbb{H}^{N}$. We establish decay estimates and symmetry properties of positive solutions. Unlike the corresponding problem in Euclidean space $\mathbb{R}^N$, we prove that there exists a nonnegative nontrivial solution of problem.
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.
We prove a Liouville type result for bounded, entire solutions to a class of variational semilinear elliptic systems, based on the growth of their potential energy over balls with growing radius. Important special cases to which our result applies are the Ginzburg-Landau system and systems that arise in the study of multi-phase transitions.
