Elliptic Equations In RN Research Articles

Optimal regularity and the Liouville property for stable solutions to semilinear elliptic equations in ℝn with n ≥ 10

Optimal regularity and the Liouville property for stable solutions to semilinear elliptic equations in ℝn with n ≥ 10

Existence of solutions to a class of quasilinear elliptic equations

This paper is concerned with the existence and nonexistence of nontrivial solutions for a class of quasilinear elliptic equations in RN involving the p-Laplacian. By utilizing a change of variables, the quasilinear equations are reduced to semilinear equations, whose associated functionals satisfy the geometric hypotheses of the mountain pass theorem. We establish the existence of nontrivial solutions via Mountain-Pass theorem. Furthermore, by using a Pohozaev type identity we prove the nonexistence of nontrivial solution under certain conditions.

Existence of a positive solution for some quasilinear elliptic equations in [formula omitted

We study the existence of a positive solution of the following quasilinear elliptic equations in RN:−Δpu=g(u),u∈W1,p(RN), where N≥2, 1<p≤N and Δpu=div(|∇u|p−2∇u) is the p-Laplacian operator. We assume that g(s) satisfies Berestycki-Lions type condition. We give a new concentration compactness type result for a Palais-Smale sequence with an additional property related to Pohozaev identity. Keys of our proof are Tartar's inequality and Brezis-Lieb Lemma.

C¹(Rᴺ) Versus W¹,ᵖ(Rᴺ) Local Minimizers

This paper deals with the energy functional associated with a quasilinear elliptic equation in RN which is driven by the p-Laplacian operator. It is shown for such functional that any C1(RN) local minimizer in an appropriate sense is a W1,p(RN) local minimizer. This extends to RN the celebrated property of Brezis-Nirenberg type known for bounded domains.

Existence of solutions for a nonlocal reaction-diffusion equation in biomedical applications

The paper is devoted to a nonlocal semi-linear elliptic equation in ℝn arising in various biological and biomedical applications. The Fredholm property studied for the corresponding linear elliptic operators with discontinuous coefficients allows the application of the implicit function theorem to prove the persistence of solutions under a small perturbation of the problem. Furthermore, the existence of solutions is established by the Leray—Schauder method based on the topological degree for Fredholm and proper operators and on a priori estimates of solutions in some special weighted spaces.

Global existence and decay estimates for the classical solution of fractional attraction–repulsion chemotaxis system

The purpose of this paper is devoted to investigate the generalized Keller–Segel system with a fractional parabolic equation and two classical elliptic equations in Rn with n⩾2. When the chemotaxis sensitivities ξ1 and ξ2 are constants, the global well-posedness of solution to (1.1) is derived if repulsion prevails over attraction in the sense that ξ1μ1−ξ2μ2>0. In the process of proving the above results, we introduce Riesz transform and Kato–Ponce’s commutator inequalities to deal with the challenge arising from fractional diffusion. However, there are few results in the case that ξ1 and ξ2 are functions. We develop a unified framework to deal with the existence, uniqueness and decay estimates in such case. At the same time, the global existence, uniqueness and decay estimates of classical solution to the generalized system (1.2) for small initial data are obtained by selecting an appropriate functional space.

On the existence of positive solutions for a nonlinear elliptic class of equations in R2 and R3

We study the existence of positive solutions for an elliptic equation in RN for N = 2, 3 which is related with the existence of standing (localized) waves and the existence of the ground state solutions for some physical model or systems in fluid mechanics to describe the evolution of weakly nonlinear water waves. We use a variational approach and the well-known principle of concentration-compactness due to P. Lions to obtain the existence of this type of solutions, even in the case that the nonlinear term g is a non-homogeneous function or an operator defined in H1(RN) with values in R.

Elliptic variational problems with mixed nonlinearities

In this paper, we study the existence and multiplicity results of nontrivial positive solutions to a quasilinear elliptic equation in RN, when N ≥ 2, as Here, h(x),k(x)&gt;0 are Lebesgue measurable functions, 1&lt;p&lt;q&lt;∞, if p&lt;N while p&lt;r&lt;q if p ≥ N, and λ&gt;0 is a parameter.

Critical elliptic systems involving multiple strongly–coupled Hardy–type terms

AbstractIn this paper, we study the radially–symmetric and strictly–decreasing solutions to a system of critical elliptic equations in RN, which involves multiple critical nonlinearities and strongly–coupled Hardy– type terms. By the ODEs analysis methods, the asymptotic behaviors at the origin and infinity of solutions are proved. It is found that the singularities ofuandvin the solution (u,v) are at the same level. Finally, an explicit form of least energy solutions is found under certain assumptions, which has all of the mentioned properties for the radial decreasing solutions.

A Nontrivial Solution of a Quasilinear Elliptic Equation Via Dual Approach

In this article, we are concerned with the existence of solutions of a quasilinear elliptic equation in ℝN which includes the so-called modified nonlinear Schrodinger equation as a special case. Combining the dual approach and the nonsmooth critical point theory, we obtain the existence of a nontrivial solution.

Solutions for critical quasilinear elliptic equations in RN

In this paper, we consider the following critical quasilinear problems in RN: −∑i,j=1NDj(bij(v)Div)+12∑i,j=1Nbij′(v)DivDjv+a(x)v=μ|v|qs−2v+|v|2*s−2v,inRN,v(x)→0 as|x| →∞, where N ≥ 4, bij(v) ∼ 1 + |v|2s−2, s ≥ 1, μ &amp;gt; 0 is a constant, q &amp;lt; 2*, 2*=2NN−2 is the Sobolev critical exponent for the embedding of H1(RN) into Lp(RN), and a(x) is a potential function which is finite and positive and satisfies some decay assumptions. We first use the truncation to overcome the difficulties caused by both the unbounded domain and the critical exponents. Then we perform a change of variables to overcome the difficulty caused by the unboundness of bij. Finally a regularization approach combined with a compact argument helps us to prove the existence of infinitely many solutions.

Multiple solutions of double phase variational problems with variable exponent

AbstractThis paper deals with the existence of multiple solutions for the quasilinear equation-div⁡𝐀⁢(x,∇⁡u)+|u|α⁢(x)-2⁢u=f⁢(x,u) in ℝN,{-\operatorname{div}\mathbf{A}(x,\nabla u)+|u|^{\alpha(x)-2}u=f(x,u)\quad\text% {in ${\mathbb{R}^{N}}$,}}which involves a general variable exponent elliptic operator𝐀{\mathbf{A}}in divergence form. The problem corresponds to double phase anisotropic phenomena, in the sense that the differential operator has various types of behavior like|ξ|q⁢(x)-2⁢ξ{|\xi|^{q(x)-2}\xi}for small|ξ|{|\xi|}and like|ξ|p⁢(x)-2⁢ξ{|\xi|^{p(x)-2}\xi}for large|ξ|{|\xi|}, where1&lt;α⁢(⋅)≤p⁢(⋅)&lt;q⁢(⋅)&lt;N{1&lt;\alpha(\,\cdot\,)\leq p(\,\cdot\,)&lt;q(\,\cdot\,)&lt;N}. Our aim is to approach variationally the problem by using the tools of critical points theory in generalized Orlicz–Sobolev spaces with variable exponent. Our results extend the previous works [A. Azzollini, P. d’Avenia and A. Pomponio, Quasilinear elliptic equations inℝN\mathbb{R}^{N}via variational methods and Orlicz–Sobolev embeddings, Calc. Var. Partial Differential Equations 49 2014, 1–2, 197–213] and [N. Chorfi and V. D. Rădulescu, Standing wave solutions of a quasilinear degenerate Schrödinger equation with unbounded potential, Electron. J. Qual. Theory Differ. Equ. 2016 2016, Paper No. 37] from cases where the exponentspandqare constant, to the case wherep⁢(⋅){p(\,\cdot\,)}andq⁢(⋅){q(\,\cdot\,)}are functions. We also substantially weaken some of the hypotheses in these papers and we overcome the lack of compactness by using the weighting method.

Compact embedding results of Sobolev spaces and positive solutions to an elliptic equation

Using a regular Borel measure μ ⩾ 0 we derive a proper subspace of the commonly used Sobolev space D1(ℝN) when N ⩾ 3. The space resembles the standard Sobolev space H1(Ω) when Ω is a bounded region with a compact Lipschitz boundary ∂Ω. An equivalence characterization and an example are provided that guarantee that is compactly embedded into L1(RN). In addition, as an application we prove an existence result of positive solutions to an elliptic equation in ℝN that involves the Laplace operator with the critical Sobolev nonlinearity, or with a general nonlinear term that has a subcritical and superlinear growth. We also briefly discuss the compact embedding of to Lp(ℝN) when N ⩾ 2 and 2 ⩽ p ⩽ N.

Dirichlet conditions at infinity for parabolic and elliptic equations

We investigate existence and uniqueness of bounded solutions of parabolic equations with unbounded coefficients in RN×R+. Under specific assumptions, we establish existence of solutions satisfying prescribed conditions at infinity, depending on the direction along which infinity is approached. Moreover, the large-time behavior of such solutions is addressed. We consider also elliptic equations in RN with similar conditions at infinity.

Infinitely many sign-changing solutions for a quasilinear elliptic equation in [formula omitted

In this paper, by using the methods of perturbation and invariant sets of descending flow, we obtain the existence of infinitely many sign-changing solutions of a quasilinear elliptic equation in RN.

Study on Existence of Solutions for p-Kirchhoff Elliptic Equation in ℝN with Vanishing Potential

In this paper, we study the existence of positive solutions to p?Kirchhoff elliptic problem a + μ ? ? N ( | ? u | p + V ( x ) | u | p ) dx ? ? Δ p u + V ( x ) | u | p ? 2 u = f ( x , u ) , in ? N , u ( x ) > 0 , in ? N , u ? D 1 , p ( ? N ) , $$\begin{array}{@{}rcl@{}} \left\{\begin{array}{lllllll} &\left(a+\mu\left({\int}_{\mathbb{R}^{N}}\!(|\nabla u|^{p}+V(x)|u|^{p})dx\right)^{\tau}\right)\left(-{\Delta}_{p}u+V(x)|u|^{p-2}u\right)=f(x,u), \quad \text{in}\; \mathbb{R}^{N}, \\ u\;\text{in}\;\; \mathbb{R}^{N},\;\; u\in \mathcal{D}^{1,p}(\mathbb{R}^{N}), \end{array}\right.\!\!\!\! \\ \end{array} $$ ?????(0.1) where a, μ > 0, ? > 0, and f(x, u) = h 1(x)|u| m?2 u + ? h 2(x)|u| r?2 u with the parameter ? ? ?, 1 0 is continuous in ? N and V(x)?0 as |x|?+?. The nontrivial solution forb Eq. (1.1) will be obtained by the Nehari manifold and fibering maps methods and Mountain Pass Theorem.

Elliptic problems involving the fractional Laplacian in [formula omitted

We study the existence and multiplicity of solutions for elliptic equations in RN, driven by a non-local integro-differential operator, which main prototype is the fractional Laplacian. The model under consideration, denoted by (Pλ), depends on a real parameter λ and involves two superlinear nonlinearities, one of which could be critical or even supercritical. The main theorem of the paper establishes the existence of three critical values of λ which divide the real line in different intervals, where (Pλ) admits no solutions, at least one nontrivial non-negative entire solution and two nontrivial non-negative entire solutions.

Existence and multiplicity of solutions for fourth-order elliptic equations in [formula omitted

In this paper, we study the fourth order elliptic equations Δ2u−Δu+λV(x)u=f(x,u) in RN, u∈H2(RN), where λ≥1 and the nonlinearity f is either superlinear or sublinear at infinity. We establish the existence and multiplicity results without the compactness of embedding of the working space, which unify and sharply improve the recent results of Liu, Chen and Wu [J. Liu, S. Chen, X. Wu, Existence and multiplicity of solutions for a class of fourth-order elliptic equations in RN, J. Math. Anal. Appl. 395 (2012) 608–615].

Existence of solutions for a class of degenerate quasilinear elliptic equation in "Equation missing" with vanishing potentials

Abstract We establish the existence of positive solution for the following class of degenerate quasilinear elliptic problem ( P ) { − L u a p + V ( x ) | x | − a p ∗ | u | p − 2 u = f ( u ) in R N , u &gt; 0 in R N ; u ∈ D a 1 , p ( R N ) , where − L u a p = − div ( | x | − a p | ∇ u | p − 2 ∇ u ) , 1 &lt; p &lt; N , − ∞ &lt; a &lt; N − p p , a ≤ e ≤ a + 1 , d = 1 + a − e , and p ∗ : = p ∗ ( a , e ) = N p N − d p denote the Hardy-Sobolev’s critical exponent, V is a bounded nonnegative vanishing potential and f has a subcritical growth at infinity. The technique used here is a truncation argument together with the variational approach. MSC:35B09, 35J10, 35J20, 35J70.

On asymptotically linear elliptic equations in [formula omitted

In this paper we obtain the existence of solution for asymptotically linear elliptic equations in RN. We mainly focus on the resonant case, the nonlinearity may not be sublinear.