Positive Weak Solutions Research Articles

Existence and uniqueness of weak positive solution for essential singular elliptic problem involving the square root of the Laplacian

Existence and uniqueness of weak positive solution for essential singular elliptic problem involving the square root of the Laplacian

Decay estimates for Wolff potentials in [formula omitted] and gradient-dependent quasilinear elliptic equations

We prove decay estimates of positive weak solutions u∈D1,p(RN)∩C1(RN) and their gradients |∇u| of the p-Laplacian problem (1<p<N):u∈D1,p(RN):−Δpu=a(x), where the measurable function a satisfies the decay estimate (α>0):0<a(x)≲11+|x|N+α,∀x∈RN. The positive Borel measure on RN generated by the right-hand side function a and its associated Wolff potential is known to be the main tool for pointwise estimates of u(x)>0 and |∇u(x)|. Therefore, one of the main goal of this paper is to prove decay estimates for corresponding Wolff potentials in RN. The obtained decay estimates in conjunction with a sub-supersolution method, which is developed here for quasilinear elliptic equations, are then applied to prove the existence of positive solutions for classes of gradient-dependent quasilinear equations in the formu∈D1,p(RN):−Δpu=a(x)f(u,∇u).

Existence of positive weak solutions for a class of Kirrchoff elliptic systems with multiple parameters

In this paper, by using subsuper solutions method, we study the existence of weak positive solution for a class of Kirrchoff elliptic systems in bounded domains with multiple parameters.

Corrigendum to Multiplicity of positive weak solutions to subcritical singular elliptic Dirichlet problems

Corrigendum to Multiplicity of positive weak solutions to subcritical singular elliptic Dirichlet problems

Multiple finite-energy positive weak solutions to singular elliptic problems with a parameter

Consider the problem $-\Delta u = a\left( x\right) u.{-\alpha}+f\left( \lambda, x, u\right) $ in $\Omega, $ $u = 0$ on $\partial\Omega, $ $u \gt 0$ in $\Omega, $ where $\Omega$ is a bounded domain in $\mathbb{R}.{n}$ with $C.{2}$ boundary, $0\leq a\in L.{\infty}\left( \Omega\right), $ $0 \lt \alpha \lt 3, $ and $f\left( \lambda, x, .\right) $ is nonnegative, and superlinear with subcritical growth at $\infty.$ We prove that, if $f$ satisfies some additional conditions, then, for some $\Lambda \gt 0, $ there are at least two weak solutions in $H_{0}.{1}\left( \Omega\right) \cap C\left( \overline{\Omega}\right) $ if $\lambda\in\left( 0, \Lambda\right)$, and there is no weak solution in $H_{0}.{1}\left( \Omega\right) \cap L.{\infty}\left( \Omega\right) $ if $\lambda \gt \Lambda.$ We also prove that, for each $\lambda\in\left[0, \Lambda\right] $, there exists a unique minimal weak solution $u_{\lambda }$ in $H_{0}.{1}\left( \Omega\right) \cap L.{\infty}\left( \Omega\right) $, which is strictly increasing in $\lambda.$

Nonzero positive solutions of a multi-parameter elliptic system with functional BCs

We prove, by topological methods, new results on the existence of nonzero positive weak solutions for a class of multi-parameter second order elliptic systems subject to functional boundary conditions. The setting is fairly general and covers the case of multi-point, integral and nonlinear boundary conditions. We also present a non-existence result. We provide some examples to illustrate the applicability our theoretical results.

Singular solutions of elliptic equations involving nonlinear gradient terms on perturbations of the ball

In this article we obtain positive singular solutions of(1)−Δu=|∇u|p in Ω,u=0 on ∂Ω, where Ω is a small C2 perturbation of the unit ball in RN. For NN−1<p<2 we prove that if Ω is a sufficiently small C2 perturbation of the unit ball there exists a singular positive weak solution u of (1). In the case of p>2 we prove a similar result but now the positive weak solution u is contained in C0,p−2p−1(Ω‾) and yet is not in C0,p−2p−1+ε(Ω‾) for any ε>0.

Qualitative properties of singular solutions to nonlocal problems

We consider positive weak solutions to $$(-\Delta )^s u=f(x,u)$$ in $$\Omega {\setminus } \Gamma $$ under zero Dirichlet boundary condition. The domain $$\Omega $$ is bounded or is the whole space, and the solution has a singularity on the singular set $$\Gamma $$ . Under suitable assumptions on f we prove symmetry and monotonicity properties of the solutions when the singular set $$\Gamma $$ has zero s-capacity.

A non-variational approach to the existence of nonzero positive solutions for elliptic systems

We provide new results on the existence of nonzero positive weak solutions for a class of second order elliptic systems. Our approach relies on a combined use of iterative techniques and classical fixed point index. Some examples are presented to illustrate the theoretical results.

A Variational Inequality Theory with Applications to P $P$ -Laplacian Elliptic Inequalities

The main purpose of this paper is to establish variational inequality theory in connection with demicontinuous $\psi_{p}$ -dissipative maps in reflexive smooth Banach spaces by considering the convergence of approximants. As an application of this variational inequality theory, existence, uniqueness and convergence of approximants of positive weak solution for $p$ -Laplacian elliptic inequalities are obtained under suitable conditions.

Positive solution for nonhomogeneous sublinear fractional equations in

Using a minimization argument on the Nehari manifold, we prove that the following sublinear fractional problem possesses a unique positive weak solution provided that and . Moreover, this solution converges to zero in when tends to zero.

σ-Finiteness of elliptic measures for quasilinear elliptic PDE in space

In this paper we study the Hausdorff dimension of a elliptic measure μf in space associated to a positive weak solution to a certain quasilinear elliptic PDE in an open subset and vanishing on a portion of the boundary of that open set. We show that this measure is concentrated on a set of σ-finite n−1 dimensional Hausdorff measure for p>n and the same result holds for p=n with an assumption on the boundary.We also construct an example of a domain in space for which the corresponding measure has Hausdorff dimension ≤n−1−δ for p≥n for some δ which depends on various constants including p.The first result generalizes the authors previous work in [3] when the PDE is the p-Laplacian and the second result generalizes the well known theorem of Wolff in [24] when p=2 and n=2.

Complete study of the existence and uniqueness of solutions for semilinear elliptic equations involving measures concentrated on boundary

The purpose of this paper is to study the weak solutions of the fractional elliptic problem (Section.Display) where , or , with is the fractional Laplacian defined in the principle value sense, is a bounded open set in with , is a bounded Radon measure supported in and is defined in the distribution sense, i.e. here denotes the unit inward normal vector at . In this paper, we prove that (0.1) with admits a unique weak solution when g is a continuous nondecreasing function satisfying Our interest then is to analyse the properties of weak solution when with , including the asymptotic behaviour near and the limit of weak solutions as . Furthermore, we show the optimality of the critical value in a certain sense, by proving the non-existence of weak solutions when . The final part of this article is devoted to the study of existence for positive weak solutions to (0.1) when and is a bounded nonnegative Radon measure supported in . We employ the Schauder’s fixed point theorem to obtain positive solution under the hypothesis that g is a continuous function satisfying -pagination

Positive solutions for nonlinear Choquard equation with singular nonlinearity

In this article, we study the following non-linear Choquard equation with singular non-linearitywhere is a bounded domain in with smooth boundary , and . Using variational approach and structure of associated Nehari manifold, we show the existence and multiplicity of positive weak solutions of the above problem, if is less than some positive constant. We also study the regularity of these weak solutions.

Existence of positive solutions for a class of $$\left( p\left( x\right) , q\left( x\right) \right) $$-Laplacian systems

Using sub-super solutions method, we study the existence of weak positive solutions for a new class of the system of differential equations with respect to the symmetry conditions. Our results are natural extensions from the previous ones in Fan (J Math Anal Appl 330:665–682 2007), Zhang (Nonlinear Anal 70:305–316 2009).

A cross-diffusion system derived from a Fokker\u2013Planck equation with partial averaging

A cross-diffusion system for two components with a Laplacian structure is analyzed on the multi-dimensional torus. This system, which was recently suggested by P.-L. Lions, is formally derived from a Fokker–Planck equation for the probability density associated with a multi-dimensional Itō process, assuming that the diffusion coefficients depend on partial averages of the probability density with exponential weights. A main feature is that the diffusion matrix of the limiting cross-diffusion system is generally neither symmetric nor positive definite, but its structure allows for the use of entropy methods. The global-in-time existence of positive weak solutions is proved and, under a simplifying assumption, the large-time asymptotics is investigated.

Existence of weak positive solution for a singular elliptic problem with supercritical nonlinearity

In this paper we consider the existence of weak positive solutions for an elliptic problems with the nonlinearity containing both singular and supercritical terms. By means of a priori estimate and sub-and supersolutions method, a positive weak solution is obtained.

Multiplicity of positive weak solutions to subcritical singular elliptic Dirichlet problems

Multiplicity of positive weak solutions to subcritical singular elliptic Dirichlet problems

Existence of multiple positive weak solutions and estimates for extremal values for a class of concave-convex elliptic problems with an inverse-square potential

In this paper, variational methods are used to establish some existence and multiplicity results and provide uniform estimates of extremal values for a class of elliptic equations of the form: \begin{document}$-Δ u - {{λ}\over{|x|^2}}u = h(x) u^q + μ W(x) u^p, x∈Ω\backslash\{0\}$ \end{document} with Dirichlet boundary conditions, where \begin{document}$0∈ Ω\subset\mathbb{R}^N $\end{document} ( \begin{document}$N≥q 3 $\end{document} ) be a bounded domain with smooth boundary \begin{document}$\partial Ω $\end{document} , \begin{document}$μ>0 $\end{document} is a parameter, \begin{document}$0 , \begin{document}$h(x)>0 $\end{document} and \begin{document}$W(x) $\end{document} is a given function with the set \begin{document}$\{x∈ Ω: W(x)>0\} $\end{document} of positive measure.

On a Degenerate Nonlocal Parabolic Problem Describing Infinite Dimensional Replicator Dynamics

We establish the existence of locally positive weak solutions to the homogeneous Dirichlet problem for $u_t = u \Delta u + u \int_\Omega |\nabla u|^2$ in bounded domains $\Omega\subset\mathbb{R}^n$ which arises in game theory. We prove that solutions converge to 0 if the initial mass is small, whereas they undergo blow-up in finite time if the initial mass is large. In particular, it is shown that in this case the blow-up set coincides with $\overline{\Omega}$; i.e., the finite-time blow-up is global.