Nonexistence Of Positive Radial Solutions Research Articles

Existence and nonexistence of solutions for the mean curvature equation with weights

In this paper we study existence and nonexistence of positive radial solutions of a Dirichlet problem for the prescribed mean curvature operator with weights in a ball with a suitable radius. Because of the presence of different weights, possibly singular or degenerate, the problem under consideration appears rather delicate, it requires an accurate qualitative analysis of the solutions, as well as the use of Liouville type results based on an appropriate Pohozaev type identity. In addition, sufficient conditions for global solutions to be oscillatory are given.

Existence and nonexistence of positive radial solutions of a quasilinear Dirichlet problem with diffusion

In this paper existence and nonexistence results of positive radial solutions of a Dirichlet m-Laplacian problem with different weights and a diffusion term inside the divergence of the form (a(|x|)+g(u))−γ, with γ>0 and a, g positive functions satisfying natural growth conditions, are proved. Precisely, we obtain a new critical exponent mα,β,γ⁎, which extends the one relative to case with no diffusion and it divides existence from nonexistence of positive radial solutions. The results are obtained via several tools such as a suitable modification of the celebrated blow up technique, Liouville type theorems, a fixed point theorem and a Pohožaev-Pucci-Serrin type identity.

Positive radial solutions of critical Hénon equations on the unit ball in ℝN$$ {\mathbb{R}}^N $$

In this paper, we study the positive radial solutions for the Hénon equations with weighted critical exponents on the unit ball in with . We first confirm that with is the critical exponent for the embedding from into and name as the Hénon‐Sobolev critical exponent. Then, following the great ideas of Brezis and Nirenberg (Comm Pure Appl Math. 1983;36:437‐477), we establish the existence and nonexistence of positive radial solutions of the problems with single Hénon‐Sobolev critical exponent and linear or nonlinear but subcritical perturbations. We further study the problems with multiple critical exponents, which may be Hénon‐Sobolev critical exponents, Hardy‐Sobolev critical exponents, or Sobolev critical exponents. The methods and arguments involved with are the mountain pass theorem and the strong maximum principle and the Pohozaev identity.

Positive radial solutions to classes of [formula omitted]-Laplacian systems on the exterior of a ball with nonlinear boundary conditions

An n×n degenerate elliptic system associated with the p-Laplacian −Δpiui=λKi(|x|)fi(ui+1) in Ω,κi∂ui∂n+gi(λ,u1,u2,…,un)ui=0 on ∂Ω,ui→0 as |x|→∞,i=1,2,…,nis studied in this paper in an exterior domain Ω≔{x∈RN∣|x|>r0>0}, where 1<pi<N, λ>0, κi≥0, and the functions Ki>0, gi>0, and fi are continuous. This type of nonlinear problems with a nonlinear boundary condition arise in applications such as chemical kinetics and population distribution. Through a fixed-point theorem of the Krasnoselskii type, we prove, in different situations, the existence, multiplicity, or nonexistence of positive radial solutions according to the distinct behaviors of fi near 0 and near ∞. The new traits of the underlying system include relaxation of the restriction n=2 to any general n≥2, different p-Laplacian operators for component solutions, the mixed boundary condition, and possible singularities at 0 possessed by and negativity near 0 of the functions fi.

Positive radial solutions of a quasilinear problem in an exterior domain with vanishing boundary conditions

In this work, we study the existence and nonexistence of positive radial solutions for the quasilinear equation $\mathrm{div}(A(|\nabla u|)\nabla u)+\lambda k(|x|)f(u)=0$ in the exterior of a ball with vanishing boundary conditions using an approach based on a fixed point theorem for operators on Banach Space.

Positive radial solutions for elliptic equations with sign-changing nonlinear terms in an annulus

This paper deals with the existence and nonexistence of positive radial solutions of the elliptic equation where and , whose sign may change. We present new eigenvalue criteria for the existence and nonexistence to this problem. The discussion is based on the fixed point index theory in cones.

Existence and nonexistence of positive radial solutions for a class of $p$-Laplacian superlinear problems with nonlinear boundary conditions

We prove the existence of positive radial solutions to the problem \begin{document}$ \begin{cases} -\Delta _{p}u = \lambda K(|x|)f(u) \text{in } |x|>r_{0}, \dfrac{\partial u}{\partial n}+\tilde{c}(u)u = 0 \text{on }|x| = r_{0}, u(x)\rightarrow 0\text{ as }|x|\rightarrow \infty ,\end{cases} $\end{document} where \begin{document}$ \Delta _{p}u = div(|\nabla u|^{p-2}\nabla u), N>p>1, \Omega = \{x\in \mathbb{R}^{N}:|x|>r_{0}>0\}, $\end{document} \begin{document}$ f:(0,\infty )\rightarrow \mathbb{R} $\end{document} is \begin{document}$ p $\end{document} -superlinear at \begin{document}$ \infty $\end{document} with possible singularity at \begin{document}$ 0, $\end{document} and \begin{document}$ \lambda $\end{document} is a small positive parameter. A nonexistence result is also established when \begin{document}$ f $\end{document} has semipositone structure at \begin{document}$ 0. $\end{document}

Nonexistence of positive solutions for a class ofp-Laplacian boundary value problems

Abstract We prove the nonexistence of positive radial solutions for the problem { − Δ p u = λ f ( u ) in Ω , u = 0 on ∂ Ω , where Δ p denotes the p -Laplacian, p > 1 , Ω is a ball or an annulus in R N , N > 1 , f : [ 0 , ∞ ) → R is at least p -linear, f ( 0 ) 0 , and is not required to be increasing or to have exactly one zero. Our results extend previous nonexistence results in the literature.

Nonexistence of positive radial solutions for a problem with singular potential

Abstract. This article completes the picture in the study of positive radial solutions in the function space 𝒟 1 , 2 ( ℝ N ) ∩ L 2 ( ℝ N , | x | - α d x ) ∩ L p ( ℝ N ) ${{\mathcal {D}^{1,2}({\mathbb {R}^N}) \cap L^2({{\mathbb {R}^N}, | x |^{-\alpha } dx})\cap L^p({\mathbb {R}^N})}}$ for the equation - Δ u + A | x | α u = u p - 1 in ℝ N ∖ { 0 } with N ≥ 3 , A &gt; 0 , α &gt; 0 , p &gt; 2 . $- \Delta u + \frac{A}{| x |^\alpha } u = u^{p-1} \quad \mbox{in } {\mathbb {R}^N}\setminus \lbrace 0\rbrace \mbox{ with } N\ge 3, A&amp;gt; 0, \alpha &amp;gt; 0, p&amp;gt;2. $ An energy balance identity is employed to prove nonexistence of such solutions in the last remaining open region in the ( α , p ) ${{(\alpha , p)}}$ plane.

Positive radial solutions for a class of quasilinear boundary value problems in a ball

We prove the existence and nonexistence of positive radial solutions for the boundary value problems { − Δ p u = h ( u ) + λ f ( u ) in Ω , u = 0 on ∂ Ω , where Δ p u = div ( | ∇ u | p − 2 ∇ u ) , p > 1 , Ω is the open unit ball in R n , h , f : ( 0 , ∞ ) → R are allowed to be singular at 0 , f is asymptotically p -linear, and λ is a positive parameter.

On the number of positive solutions of elliptic systems

The paper deals with the existence, multiplicity and nonexistence of positive radial solutions for the elliptic system div(|∇|p –2∇) + λki (|x |) fi (u1, …,un) = 0, p > 1, R1 < |x | < R2, ui (x) = 0, on |x | = R1 and R2, i = 1, …, n, x ∈ ℝN , where ki and fi, i = 1, …, n, are continuous and nonnegative functions. Let u = (u1, …, un), φ (t) = |t |p –2t, fi0 = lim‖u‖→0((fi (u))/(φ (‖u‖))), fi∞= lim‖u‖→∞((fi (u))/(φ (‖u‖))), i = 1, …, n, f = (f1, …, fn), f0 = ∑n i =1 fi 0 and f∞ = ∑n i =1 fi ∞. We prove that either f0 = 0 and f∞ = ∞ (superlinear), or f0 = ∞and f∞ = 0 (sublinear), guarantee existence for all λ > 0. In addition, if fi (u) > 0 for ‖u‖ > 0, i = 1, …, n, then either f0 = f∞ = 0, or f0 = f∞ = ∞, guarantee multiplicity for sufficiently large, or small λ, respectively. On the other hand, either f0 and f∞ > 0, or f0 and f∞ < ∞ imply nonexistence for sufficiently large, or small λ, respectively. Furthermore, all the results are valid for Dirichlet/Neumann boundary conditions. We shall use fixed point theorems in a cone. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

On the structure of positive radial solutions for quasilinear equations in annular domains

We study the existence, multiplicity and nonexistence of positive radial solutions to boundary value problems for the quasilinear equation $\text{ div} \left ( A(| \nabla u|)\nabla u \right ) + \lambda h(|x|)f(u) =0$ in annular domains under general assumptions on the function $A(u)$. Various possible behaviors of the quotient $\frac{f(u)}{A(u)u}$ at zero and infinity are considered. We shall use fixed point theorems for operators on a Banach space.

Existence and nonexistence of positive radial solutions of superlinear elliptic systems

Existence and nonexistence of positive radial solutions of superlinear elliptic systems

Boundary Value Problems for theq — Laplacian on N

Nonlinear boundary value problems for the q–Laplacian in spaces of constant positive curvature are considered. The nonlinearity is of the form of a power. Existence and nonexistence of positive radial solutions in balls is established. It turns out that the situation differs considerably from the corresponding problems in the Euclidean space. Special attention is given to the critical case which has some consequences in the calculus of variation.

Non-existence of positive radial solutions for a class of quasilinear elliptic system

In this paper, our main purpose is to establish some nonexistence results of positive radial solutions to the quasilinear ordinary differential equation system. The main results of the present paper are new and extend the previously known results.

Existence and nonexistence of positive radial solutions for a class of semilinear elliptic system

Existence and nonexistence of positive radial solutions for a class of semilinear elliptic system

Existence and nonexistence of positive radial solutions of an elliptic Dirichlet problem in an exterior domain

Existence and nonexistence of positive radial solutions of an elliptic Dirichlet problem in an exterior domain

Existence and nonexistence of positive radial solutions of neumann problems with critical Sobolev exponents

Existence and nonexistence of positive radial solutions of neumann problems with critical Sobolev exponents

Existence and nonexistence of positive radial solutions for some semilinear elliptic problems in annular domains

Au + f(]x], U) = 0 in S&u = 0 on XJ, (*) when n is some annulus and the above conditions in [4,29] are not necessarily satisfied. In fact, in our main result (theorem 2.4), f is not necessarily C’, lim f(t, u)/u may not exist (or may be U-+rn finite) and f(t, u) could be zero for u in some finite interval [0, p]. We also allow lim (f(t, u)/u) > 0 (corollary 2.6) and lim (f(t, u)/u) may not exist. (In example 2.3 below, U+O u-0 neither lim f(u)/u, nor lim f(u)/u exists.) On the other hand, we show that u-0 u-m