In this paper we study quasilinear elliptic systems driven by so-called double phase operators and nonlinear right-hand sides depending on the gradients of the solutions. Based on the surjectivity result for pseudomonotone operators we prove the existence of at least one weak solution of such systems. Furthermore, under some additional conditions on the data, the uniqueness of weak solutions is shown.
In this paper we establish the uniqueness of a solution to a stationary convection-diffusion equation in divergence form with an exponentially summable generalized divergence-free drift.
In this paper we consider quasilinear elliptic equations with double phase phenomena and a reaction term depending on the gradient. Under quite general assumptions on the convection term we prove the existence of a weak solution by applying the theory of pseudomonotone operators. Imposing some linear conditions on the gradient variable the uniqueness of the solution is obtained.
In this paper, we consider the existence and uniqueness of weak solutions of a nonlinear elliptic equation with a variable exponent, a monotonic type operator and a convection term. With the topological degree theory, we prove the existence of at least one weak solution under some Leray-Lions and growth conditions. Moreover, we obtain the uniqueness of the solution of the problem under some additional assumptions. Our results generalize and improve existing results with another approach.
We consider the inverse boundary value problem for the steady state convection diffusion equation. We prove that a velocity field $V$, is uniquely determined by the Dirichlet-to-Neumann map, when $V \in C^{0,γ} (Ω)$, $2/3< γ\leq 1$, i.e. when $V$ is a Hölder continuous vector field with $2/3< γ\leq 1$.
This work contributes to an understanding of the domain size's effect on the existence and uniqueness of the linear convection--diffusion equation with integral-type boundary conditions, where boundary conditions depend non-locally on unknown solutions. Generally, the uniqueness result of this type of equation is unclear. In this preliminary study, a uniqueness result is verified when the domain is sufficiently large or small. The main approach has an advantage of transforming the integral boundary conditions into new Dirichlet boundary conditions so that we can obtain refined estimates, and t…
We show the existence of positive solutions for a class of singular elliptic systems with convection term. The approach combines pseudomonotone operator theory, sub and supersolution method and perturbation arguments involving singular terms.
The present paper concerns with the existence of blow-up solution for a class of elliptic system with convection term. Here, we prove a result involving sub and supersolution for a class of elliptic system whose nonlinearity can depend of the gradient of the solution. This result permits to study the existence of blow-up solution for a large class of systems.
We study radial symmetry of large solutions of the semi-linear elliptic problem Δu + \nabla h.\nabla u = f(|x|,u), and we provide sharp conditions under which the problem has a radial solution. The result is independent of the rate of growth of the solution at infinity.
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.
In this paper we present some very recent results regarding existence, uniqueness, and multiplicity of solutions for quasilinear elliptic equations and systems, exhibiting both singular and convective reaction terms. The importance of boundary conditions (Dirichlet, Neumann, or Robin) is also discussed. Existence is achieved via sub-supersolution and truncation techniques, fixed point theory, nonlinear regularity, and set-valued analysis, while uniqueness and multiplicity are obtained by monotonicity arguments.
We consider a scalar parabolic equation in one spatial dimension. The equation is constituted by a convective term, a reaction term with one or two equilibria, and a positive diffusivity which can however vanish. We prove the existence and several properties of traveling-wave solutions to such an equation. In particular, we provide a sharp estimate for the minimal speed of the profiles and improve previous results about the regularity of wavefronts. Moreover, we show the existence of an infinite number of semi-wavefronts with the same speed.
In this note we present $L^\infty$-results for problems of the form $A(x,u,Du)=B(x,u,Du)$ in $Ω$, $u>0$ in $Ω$, $u=0$ on $\partialΩ$, where the growth condition for the function $B\colon Ω\times \mathbb{R}\times \mathbb{R}^N\to \mathbb{R}$ contains both a singular and a convection term. We use ideas from the works of Giacomoni-Schindler-Takac (2007) and the authors (2019) to prove the boundedness of weak solutions for such general problem by applying appropriate bootstrap arguments.
We establish the existence of positive solutions for a nonlinear elliptic Dirichlet problem in dimension $N$ involving the $N$-Laplacian. The nonlinearity considered depends on the gradient of the unknown function and an exponential term. In such case, variational methods cannot be applied. Our approach is based on approximation scheme, where we consider a new class of normed spaces of finite dimension. As a particular case, we extended the result achieved by De Araujo and Montenegro [2016] for any $N>2$.
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 consider a reaction-diffusion equation with a convection term in one space variable, where the diffusion changes sign from the positive to the negative and the reaction term is bistable. We study the existence of wavefront solutions, their uniqueness and regularity. The presence of convection reveals several new features of wavefronts: according to the mutual positions of the diffusivity and reaction, profiles can occur either for a single value of the speed or for a bounded interval of such values; uniqueness (up to shifts) is lost; moreover, plateaus of arbitrary length can appear; profil…
The weak and strong comparison principles, respectively, are investigated for quasi-linear elliptic boundary value problems with the $p$-Laplacian in one space dimension. We treat the degenerate case of $2 < p < \infty$ and allow also for the nontrivial convection velocity $b\colon [-1,1]\to \mathbb{R}$ in the underlying domain $(-1,1)$. We establish the weak comparison principle under a rather general, natural sufficient condition on the convection velocity, $b(x)$, and the reaction function, $φ(x,u)$. Furthermore, we establish also the strong comparison principle under a number of various ad…
We prove the existence of unique regular solutions of steady-state buoyancy-driven flows of viscous incompressible heat-conducting fluids in 3D open channels under mixed boundary conditions. The model takes into account the phenomena of the viscous energy dissipation.
We study the existence and uniqueness for weak solutions to some classes of anisotropic elliptic Dirichlet problems with data belonging to the natural dual space.
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 prove the uniqueness of positive radial solutions for a class of quasi-linear elliptic problems containing, in particular, the quasi-linear Schrodinger equation.
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 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.
Based on a variant of frequency function, we improve the vanishing order of solutions for Schrödinger equations which describes quantitative behavior of strong uniqueness continuation property. For the first time, we investigate the quantitative uniqueness of higher order elliptic equations and show the vanishing order of solutions. Furthermore, strong unique continuation is established for higher order elliptic equations using this variant of frequency function.
In this article we are interested in addressing the question of existence and uniqueness of positive solution to a quasilinear elliptic equation involving p-laplacian in Heisenberg Group. The idea is to prove the uniqueness by using Diaz-Saa Inequality in Heisenberg Group which we obtain via a generalized version of Picone's Identity.
We give a partial uniqueness result concerning comparable renormalized solutions of the nonlinear elliptic problem $-\diw(\aop(x,Du))=μ$ in $Ω$, $u=0$ on $\partialΩ$, where $μ$ is a Radon measure with bounded variation on $Ω$.
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 quantify the uniqueness of continuation from Cauchy or interior data. Our approach consists in extending the existing results in the linear case. As by product we obtain a new stability estimate in the linear case. We also show the so-called strong uniqueness of continuation and the uniqueness of continuation from a set of positive measure. These results are derived by using a linearization procedure.
We study a class of elliptic competition-diffusion systems of long range segregation models for two and more competing species. The existence and uniqueness of the solution are shown. We prove that as the competition rate goes to infinity the solution converges, along with suitable sequences, to a spatially long range segregated state satisfying some free boundary problems.
We study boundary blow-up solutions of semilinear elliptic equations $Lu=u_+^p$ with $p>1$, or $Lu=e^{au}$ with $a>0$, where $L$ is a second order elliptic operator with measurable coefficients. Several uniqueness theorems and an existence theorem are obtained.
This paper deals with global existence and uniqueness for the lake equations with a bottom topography vanishing on the shore. Our result generalizes previous studies that assumed the depth to be nondegenerate. Elliptic estimates for degenerate equations has to be established studying the behavior of the associated Green function.
This paper is devoted to discussing the existence and uniqueness of weak solutions to time-fractional elliptic equations having time-dependent variable coefficients. To obtain the main result, our strategy is to combine the Galerkin method, a basic inequality for the fractional derivative of convex Lyapunov candidate functions, the Yoshida approximation sequence and the weak compactness argument.
In this paper we study quantitative uniqueness estimates of solutions to general second order elliptic equations with magnetic and electric potentials. We derive lower bounds of decay rate at infinity for any nontrivial solution under some general assumptions. The lower bounds depend on asymptotic behaviors of magnetic and electric potentials. The proof is carried out by the Carleman method and the bootstrapping arguments.
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.
In this paper, the existence of smooth positive solutions to a Robin boundary-value problem with non-homogeneous differential operator and reaction given by a nonlinear convection term plus a singular one is established. Proofs chiefly exploit sub-super-solution and truncation techniques, set-valued analysis, recursive methods, nonlinear regularity theory, as well as fixed point arguments. A uniqueness result is also presented.
We examine the equation \[Δ^2 u = λf(u) \qquad Ω, \] with either Navier or Dirichlet boundary conditions. We show some uniqueness results under certain constraints on the parameter $ λ$. We obtain similar results for the sytem {equation*} \{{array}{rrl} -Δu &=& λf(v) \qquad Ω, -Δv &=& γg(u) \qquad Ω, u&=& v = 0 \qquad \partial Omega. {array}. {equation*}
We study the quantitative unique continuation property of some higher order elliptic operators. In the case of $P=(-Δ)^m$, where $m$ is a positive integer, we derive lower bounds of decay at infinity for any nontrivial solutions under some general assumptions. Furthermore, in dimension 2, we can obtain essentially sharp lower bounds for some forth order elliptic operators, the sharpness is shown by constructing a Meshkov type example.
In this study we prove the existence-uniqueness of a coupled non-linear elliptic PDE system using Lax-Milgram theorem, Galerkin Method, Brouwer's fixed point theorem. Later we derive the finite element scheme for the numerical solution of the PDE system and also carry out the convergence analysis for the derived scheme. Further successfully apply the scheme to an application related to free convection phenomena.
In the present paper we prove uniqueness results for solutions to a class of Neumann boundary value problems whose prototype is --div((1 + |$\nabla$u| 2) (p--2)/2 $\nabla$u) -- div(c(x)|u| p--2 u) = f in $Ω$, (1 + |$\nabla$u| 2) (p--2)/2 $\nabla$u + c(x)|u| p--2 u $\times$ n = 0 on $\partial$$Ω$,
We study the uniqueness problem of $σ$-regular solution of the equation, $$-Δ_p u+ \abs u^{q-1}u =h \quad on\quad \RN, $$ where $q>p-1>0.$ and $N> p.$ Other coercive type equations associated to more general differential operators are also investigated. Our uniqueness results hold for equations associated to the mean curvature type operators as well as for more general quasilinear subelliptic operators.
This paper deals with an existence and uniqueness result of the weak solution for a quasilinear elliptic PDE with nonlinear Robin boundary conditions.This problem is defined on a domain whose boundary is the union of two disjoint surfaces.where a Dirichelet condition is prescribed on the exterior boundary and a nonlinear Robin condition on the interior one.To achieve that, we use the theory of monotone operators
We establish existence and uniqueness of solution for the homogeneous Dirichlet problem associated to a fairly general class of elliptic equations modeled by $$ -Δu= h(u){f} \ \ \text{in}\,\ Ω, $$ where $f$ is an irregular datum, possibly a measure, and $h$ is a continuous function that may blow up at zero. We also provide regularity results on both the solution and the lower order term depending on the regularity of the data, and we discuss their optimality.
In this paper, we study the existence and uniqueness of positive solutions for the following nonlinear fractional elliptic equation: \begin{eqnarray*} (-Δ)^αu=λa(x)u-b(x)u^p&{\rm in}\,\,\R^N, \end{eqnarray*} where $ α\in(0,1) $, $ N\ge 2 $, $λ>0$, $a$ and $b$ are positive smooth function in $\R^N$ satisfying \[ a(x)\rightarrow a^\infty>0\quad {\rm and}\quad b(x)\rightarrow b^\infty>0\quad{\rm as}\,\,|x|\rightarrow\infty. \] Our proof is based on a comparison principle and existence, uniqueness and asymptotic behaviors of various boundary blow-up solutions for a class of elliptic equations invo…
We consider the following elliptic system with fractional laplacian $$ -(-Δ)^su=uv^2,\ \ -(-Δ)^sv=vu^2,\ \ u,v>0 \ \mbox{on}\ \R^n,$$ where $s\in(0,1)$ and $(-Δ)^s$ is the $s$-Lapalcian. We first prove that all positive solutions must have polynomial bound. Then we use the Almgren monotonicity formula to perform a blown-down analysis to $s$-harmonic functions. Finally we use the method of moving planes to prove the uniqueness of the one dimensional profile, up to translation and scaling.
We use a Carleman type inequality of Koch and Tataru to obtain quantitative estimates of unique continuation for solutions of second order elliptic equations with singular lower order terms. First we prove a three sphere inequality and then describe two methods of propagation of smallness from sets of positive measure.
