Existence Of Positive Radial Solutions Research Articles

On the asymptotically cubic generalized quasilinear Schrödinger equations with a Kirchhoff-type perturbation

In this paper, we consider the non-existence and existence of solutions for a generalized quasilinear Schrödinger equation with a Kirchhoff-type perturbation. When the non-linearityh(u) shows critical or supercritical growth at infinity, the non-existence result for a quasilinear Schrödinger equation is proved via the Pohožaev identity. Ifh(u) shows asymptotically cubic growth at infinity, the existence of positive radial solutions for the quasilinear Schrödinger equation is obtained whenbis large or equal to 0 andbis equal to 0 by the variational methods. Moreover, some properties are established as the parameterbtends to be 0.

Existence of positive radial solutions for nonlinear elliptic equations with gradient terms in an annulus

Existence of positive radial solutions for nonlinear elliptic equations with gradient terms in an annulus

Existence of positive radial solutions to a p-Laplacian Kirchhoff type problem on the exterior of a ball

In this paper the authors study the existence of positive radial solutions to the Kirchhoff type problem involving the \(p\)-Laplacian \[-\Big(a+b\int_{\Omega_e}|\nabla u|^p dx\Big)\Delta_p u=\lambda f\left(|x|,u\right),\ x\in \Omega_e,\quad u=0\ \text{on} \ \partial\Omega_e,\] where \(\lambda \gt 0\) is a parameter, \(\Omega_e = \lbrace x\in\mathbb{R}^N : |x|\gt r_0\rbrace\), \(r_0\gt 0\), \(N \gt p \gt 1\), \(\Delta_p\) is the \(p\)-Laplacian operator, and \(f\in C(\left[ r_0, +\infty\right)\times\left[0,+\infty\right),\mathbb{R})\) is a non-decreasing function with respect to its second variable. By using the Mountain Pass Theorem, they prove the existence of positive radial solutions for small values of \(\lambda\).

Positive radial solutions for a boundary value problem associated to a system of elliptic equations with semipositone nonlinearities

<abstract><p>In this paper we use the fixed point index theory to study the existence of positive radial solutions for a system of boundary value problems with semipositone second order elliptic equations. Some appropriate concave and convex functions are utilized to characterize coupling behaviors of our nonlinearities.</p></abstract>

Existence of positive radial solutions for a problem involving the weighted Heisenberg $ p(\cdot) $-Laplacian operator

<abstract><p>A variational principle is applied to examine a Muckenhoupt weighted $ p(\cdot) $-Laplacian equation on the Heisenberg groups. The existence of at least one positive radial solution to the problem under the Dirichlet boundary condition belongs to the first order Heisenberg-Sobolev spaces is proved.</p></abstract>

A priori bounds for positive radial solutions of quasilinear equations of Lane–Emden type

We consider the quasilinear equation $\Delta _p u +K(|x|)u^q=0$, and present the proof of the local existence of positive radial solutions near $0$ under suitable conditions on $K$. Moreover, we provide a priori estimates of positive radial solutions near $\infty $ when $r^{-\ell }K(r)$ for $\ell \ge -p$ is bounded near $\infty $.

Existence of solution of some p,q${p,q}$‐Laplacian system under local superlinear conditions

AbstractWe study existence of positive radial solutions for the following class of quasi‐linear elliptic systems where the nonlinearities satisfy some local superlinear property at . Here B is the unity ball in and . Some difficulties here are that this system, in general, is non‐variational and that might occur. Thus, our strategy to obtain a positive solution is through a priori bound methods. One of the important results in this article is that we obtain these estimates, assuming superlinearity hypotheses on the variables u and v at infinity just in some subset of the ball, allowing us assuming, for example, supercritical growths on outside of that subset. We also note that we can consider nonlinearities which are not necessary asymptotic to a power at and so it is not possible to use the classic Liouville type nonexistence results.

Positive radial solutions for a class of quasilinear Schrödinger equations in R 3

This paper is concerned with the following quasilinear Schrödinger equations of the form: − Δ u − u Δ ( u 2 ) + u = | u | p − 2 u , x ∈ R 3 , where p ∈ ( 2 , 12 ) . By making use of the constrained minimization method on a special manifold, we prove that the existence of positive radial solutions of the above problem for any p ∈ ( 2 , 12 ) .

On gauged nonlinear Schrödinger equations with Chern–Simons nonlocal term in [formula omitted

In the present paper the following gauged nonlinear Schrödinger equation is studied: −Δψ+ωψ+μ∫|x|+∞a(τ)τψ2(τ)dτ+a2(|x|)|x|2ψ=h(ψ)inR2,where μ>0 is a parameter, ω>0 is a constant, a(τ)=12∫0τsψ2(s)ds and h∈C(R,R). Under weak assumptions on h, the existence of positive radial solutions for the above equation has been derived through using perturbation arguments. The asymptotic behavior of solutions with regard to the parameter μ is also considered.

A remark on the existence of positive radial solutions to a Hessian system

<abstract><p>We give new conditions for the study of existence of positive radial solutions for a system involving the Hessian operator. The solutions to be obtained are given by successive-approximation. Our interest is to improve the works that deal with such systems at the present and to give future directions of research related to this work for researchers.</p></abstract>

Radial solutions for Hénon type fully nonlinear equations in annuli and exterior domains

<abstract><p>In this note we study existence of positive radial solutions in annuli and exterior domains for a class of nonlinear equations driven by Pucci extremal operators subject to a Hénon type weight. Our approach is based on the shooting method applied to the corresponding ODE problem, energy arguments, and the associated flow of an autonomous quadratic dynamical system.</p></abstract>

Positive Radial Solutions for Elliptic Equations with Nonlinear Gradient Terms on the Unit Ball

This paper deals with the existence of positive radial solutions of the elliptic equation with nonlinear gradient term $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u = f(|x|,\,u,\,|\nabla u|)\,,\qquad x\in \Omega \,,\qquad \qquad \\ u|_{\partial \Omega }=0\,, \end{array}\right. \end{aligned}$$ where $$\Omega =\{x\in \mathbb {R}^N:\;|x|<1\}$$ , $$N\ge 2$$ , $$f:[0,\,1]\times \mathbb {R}^+\times \mathbb {R}^+ \rightarrow \mathbb {R}$$ are continuous, $$\mathbb {R}^+=[0,\,\infty )$$ . Under some inequality conditions, the existence results of positive radial solution are obtained. The proofs of the main results are based on the method of lower and upper solutions and truncating function technique.

Radial solutions of semilinear elliptic equations with prescribed asymptotic behavior

AbstractWe consider semilinear elliptic equations of the form on . Given a suitable positive root z of g, we discuss the existence of positive radial solutions u with .

Existence of positive radial solution for Neumann problem on the Heisenberg group

The existence of at least one positive radial solution of the Neumann problem −ΔHnu+R(ξ)u=a(|ξ|Hn)|u|p−2u−b(|ξ|Hn)|u|q−2u,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ -\\Delta _{\\mathbb{H}^{n}} u+R(\\xi ) u=a \\bigl( \\vert \\xi \\vert _{\\mathbb{H}^{n}} \\bigr) \\vert u \\vert ^{p-2} u - b\\bigl( \\vert \\xi \\vert _{\\mathbb{H}^{n}}\\bigr) \\vert u \\vert ^{q-2}u, $$\\end{document} is proved on the Heisenberg group mathbb{H}^{n}, via the variational principle, where a(|xi |_{mathbb{H}^{n}}), b(|xi |_{mathbb{H}^{n}}) are nonnegative radial functions and R(xi ) satisfies suitable conditions.

Nonexistence and existence of positive radial solutions to a class of quasilinear Schr\xf6dinger equations in mathbb{R}^{N}

This paper aims to investigate the class of quasilinear Schrödinger equations 0.1−Δu−[Δ(1+u2)γ2]γu2(1+u2)2−γ2=αh(|x|)|u|p−1u+βH(|x|)|u|q−1u,x∈RN,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} \\begin{aligned}[b] &-\\Delta u-\\bigl[\\Delta \\bigl(1+u^{2}\\bigr)^{\\frac{\\gamma }{2}}\\bigr] \\frac{\\gamma u}{2(1+u^{2})^{\\frac{2-\\gamma }{2}}}\\\\ &\\quad =\\alpha h\\bigl( \\vert x \\vert \\bigr) \\vert u \\vert ^{p-1}u+ \\beta H\\bigl( \\vert x \\vert \\bigr) \\vert u \\vert ^{q-1}u, \\quad x\\in \\mathbb{R}^{N}, \\end{aligned} \\end{aligned}$$ \\end{document} where N >2, 1 le gamma le 2, alpha ,beta in mathbb{R} and either 0< p<1<q or 1< p< q. Functions h(|x|), H(|x|) are continuous and positive in mathbb{R}^{N} . Relying on some special arguments and the Schauder–Tychonoff fixed point theorem, nonexistence criteria, existence of positive ground state solutions and blow-up solutions to Eq. (0.1) with 0< p<1<q or 1< p< q will be obtained.

Positive Solutions for Systems of Quasilinear Equations with Non-homogeneous Operators and Weights

Abstract In this paper we deal with positive radially symmetric solutions for a boundary value problem containing a strongly nonlinear operator. The proof of existence of positive solutions that we give uses the blow-up method as a main ingredient for the search of a-priori bounds of solutions. The blow-up argument is one by contradiction and uses a sort of scaling, reminiscent to the one used in the theory of minimal surfaces, see [B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 1981, 883–901], and therefore the homogeneity of the operators, Laplacian or p-Laplacian, and second members powers or power like functions play a fundamental role in the method. Thus, when the differential operators are no longer homogeneous, and similarly for the second members, applying the blow-up method to obtain a-priori bounds of solutions seems an almost impossible task. In spite of this fact, in [M. García-Huidobro, I. Guerra and R. Manásevich, Existence of positive radial solutions for a weakly coupled system via blow up, Abstr. Appl. Anal. 3 1998, 1–2, 105–131], we were able to overcome this difficulty and obtain a-priori bounds for a certain (simpler) type of problems. We show in this paper that the asymptotically homogeneous functions provide, in the same sense, a nonlinear rescaling, that allows us to generalize the blow-up method to our present situation. After the a-priori bounds are obtained, the existence of a solution follows from Leray–Schauder topological degree theory.

Liouville-type results in exterior domains for radial solutions of fully nonlinear equations

We give necessary and sufficient conditions for the existence of positive radial solutions for a class of fully nonlinear uniformly elliptic equations posed in the complement of a ball in RN, and equipped with homogeneous Dirichlet boundary conditions.

Positive solutions for nonlinear Schrödinger–Kirchhoff equations in [formula omitted

In this paper, we study the nonlinear Schrödinger–Kirchhoff-type equation with pure power nonlinearities in R3 by variational methods. By carrying out the constrained minimization on a special manifold which is a combination of the Nehari manifold and Pohozaev manifold, we proved the existence of positive radial solutions of this equation for the power p∈(1,5). The results of this paper extend some existing conclusions, especially for p∈(1,2].

Radial and asymptotically constant solutions for nonautonomous elliptic equations

ABSTRACT We consider semilinear elliptic equations of the form on . Given a suitable positive z satisfying , we discuss the existence of positive radial solutions u with .

Existence of positive radial solutions for a semipositone elliptic equation

In this paper, we study the existence of positive radial solutions for a semipositone elliptic equation−Δu=λf(u)in Ω,u=0on ∂Ω, where Ω is a ball or an annulus in RN with N≥2, λ>0 is a parameter and f is a continuous function satisfying the semipositone condition f(0)<0. We give a weak and general sufficient condition on f for the existence of positive radial solutions for λ>0 large. We show that the linearized operator at each solution has the nonnegative first eigenvalue. These solutions are obtained as minimizers of a Lagrangian functional.