Nontrivial Radial Solutions Research Articles

Nontrivial solutions for multiparameter k-Hessian systems

This paper is devoted to studying the coupled system of multiparameter k-Hessian equations where B = {x ∈ ℝ n : |x| < 1}, λ1, λ2 are positive parameters, ρi (i = 1, 2) is singular near the boundary ∂B, f i(i = 1, 2) satisfies a combined superlinear condition at ∞. Employing the fixed point theorem of Krasnosel’skii type in Banach space, the existence and multiplicity of nontrivial radial solutions are established. In particular, the dependence of the solutions on the parameters λ1, λ2 is also discussed. Finally, we study the nonexistence results of nontrivial radial solutions for the case λ1, λ2 ≫ 1.

Kirchhoff‐type critical fractional Laplacian system with convolution and magnetic field

AbstractIn this paper, we consider a class of upper critical Kirchhoff‐type fractional Laplacian system with Choquard nonlinearities and magnetic fields. With the help of the limit index theory and the concentration–compactness principles for fractional Sobolev spaces, we establish the existence of infinitely many nontrivial radial solutions for the above system. A distinguished feature of this paper is that the above Kirchhoff‐type system is degenerate, that is, the Kirchhoff term is zero at zero.

A fractional Hopf Lemma for sign-changing solutions

In this paper we prove some results on the boundary behavior of solutions to fractional elliptic problems. Firstly, we establish a Hopf Lemma for solutions to some integro-differential equations. The main novelty of our result is that we do not assume any global condition on the sign of the solutions. Secondly, we show that non-trivial radial solutions cannot have infinitely many zeros accumulating at the boundary. We provide concrete examples to show that the results obtained are sharp.

Eigenvalue problems for singular p-Monge-Ampère equations

Using the eigenvalue theory of completely continuous operators, we are concerned with the determining values of μ, in which there are nontrivial radial solutions to the singular p-Monge-Ampère problem{det(D(|Du|p−2Du))=μh(|x|)f(−u)inΩ,u=0on∂Ω, where μ>0 is a parameter, Ω is the open unit ball in Rn. In addition, as an application, we derive some criteria for the existence of nontrivial radial solutions to the power-type problem of p-Monge-Ampère equations. The dependence of nontrivial radial solution uμ on the parameter μ is also considered.

Stability and instability of radial standing waves to NLKG equation with an inverse-square potential

In this paper, we consider radial standing waves to a nonlinear Klein-Gordon equation with a repulsive inverse-square potential. It is known that there exist “radial” ground states to the stationary problem of the nonlinear Klein-Gordon equation. Here, “radial” ground states are non-trivial solutions having least energy among all non-trivial radial solutions to the stationary equation. We deal with the stability and instability of “radial” ground state standing waves.

On the solutions to weakly coupled system of ki-Hessian equations

In this paper, the existence and multiplicity of nontrivial radial convex solutions to general coupled system of ki-Hessian equations in a unit ball are studied via a fixed-point theorem. In particular, we obtain the uniqueness of nontrivial radial convex solution and nonexistence of nontrivial radial k-admissible solution to a power-type system coupled by ki-Hessian equations in a unit ball. Moreover, using a generalized Krein-Rutman theorem, the existence of k-admissible solutions to an eigenvalue problem in a general strictly (k−1)-convex domain is also obtained.

NONTRIVIAL RADIAL SOLUTIONS FOR A SYSTEM OF SECOND ORDER ELLIPTIC EQUATIONS

In this paper we use the topological degree and the Krein-Rutman theorem to investigate the existence of nontrivial radial solutions for a system of second order elliptic equations. Our results are obtained under some conditions involving the eigenvalues of a relevant linear operator.

On unique continuation principles for some elliptic systems

In this paper we prove unique continuation principles for some systems of elliptic partial differential equations satisfying a suitable superlinearity condition. As an application, we obtain nonexistence of nontrivial (not necessarily positive) radial solutions for the Lane-Emden system posed in a ball, in the critical and supercritical regimes. Some of our results also apply to general fully nonlinear operators, such as Pucci's extremal operators, being new even for scalar equations.

Convex solutions of Monge-Ampère equations and systems: Existence, uniqueness and asymptotic behavior

Abstract In this paper, the equations and systems of Monge-Ampère with parameters are considered. We first show the uniqueness of of nontrivial radial convex solution of Monge-Ampère equations by using sharp estimates. Then we analyze the existence and nonexistence of nontrivial radial convex solutions to Monge-Ampère systems, which includes some new ingredients in the arguments. Furthermore, the asymptotic behavior of nontrivial radial convex solutions for Monge-Ampère systems is also considered. Finally, as an application, we obtain sufficient conditions for the existence of nontrivial radial convex solutions of the power-type system of Monge-Ampère equations.

Existence and uniqueness of nontrivial radial solutions for k-Hessian equations

We prove existence and uniqueness of nontrivial radial solutions to the k-Hessian problem{Sk(D2u)=λf(−u)inΩ,u=0on∂Ω, where Sk(D2u) is the k-Hessian operator of u, Ω is the open unit ball in Rn, f is a continuous function and may have k-sublinear growth at ∞ with possible k-superlinear growth at 0, and λ is a large parameter.

Analysis of nontrivial radial solutions for singular superlinear [formula omitted]-Hessian equations

We consider the singular superlinear k-Hessian problem Sk(D2u)=λH(|x|)f(−u)inΩ,u=0on∂Ω, where λ is a positive parameter, Sk(D2u) is the k-Hessian operator of u, Ω is the open unit ball in Rn, H is a positive weight function which is singular near the boundary ∂Ω, and f is a continuous function and may be singular at 0 with possible k-superlinear growth at ∞. We establish an existence result for λ small, multiplicity results of nontrivial radial solutions for certain ranges of λ. The asymptotic behavior of nontrivial radial solutions on the parameter λ is also considered.

A coupled system of k‐Hessian equations

In this paper, we consider the following system coupled by multiparameter k‐Hessian equations Here, λ1 and λ2 are positive parameters, Ω is the unit ball in Rn, Sk(D2u) is the k‐Hessian operator of u, . Applying the eigenvalue theory in cones, several new results are obtained for the existence and multiplicity of nontrivial radial solutions for the above k‐Hessian system. In particular, we study the dependence of the nontrivial radial solution on the parameter λ1 and λ2. This is probably the first time that a system of equations, especially with fully nonlinear equations, has been studied by applying this technique. Finally, as an application, we obtain sufficient conditions for the existence of nontrivial radial solutions of the power‐type coupled system of k‐Hessian equations, which is new even for the special case k=1 and extends a previous result for the case k=n.

A critical fractional Choquard–Kirchhoff problem with magnetic field

In this paper, we are interested in a fractional Choquard–Kirchhoff-type problem involving an external magnetic potential and a critical nonlinearity [Formula: see text] [Formula: see text] where [Formula: see text] with [Formula: see text], [Formula: see text] is the Kirchhoff function, [Formula: see text] is the magnetic potential, [Formula: see text] is the fractional magnetic operator, [Formula: see text] is a continuous function, [Formula: see text], [Formula: see text] is a parameter, [Formula: see text] and [Formula: see text] is the critical exponent of fractional Sobolev space. We first establish a fractional version of the concentration-compactness principle with magnetic field. Then, together with the mountain pass theorem, we obtain the existence of nontrivial radial solutions for the above problem in non-degenerate and degenerate cases.

New results of coupled system of [formula omitted]-Hessian equations

Applying a new fixed point theorem due to Radu Precup, this paper investigates the existence and multiplicity of nontrivial radial solutions of coupled system of k-Hessian equations.

Radial Convex Solutions of a Singular Dirichlet Problem with the Mean Curvature Operator in Minkowski Space

In this paper, we study the existence of nontrivial radial convex solutions of a singular Dirichlet problem involving the mean curvature operator in Minkowski space. The proof is based on a well-known fixed point theorem in cones. We deal with more general nonlinear term than those in the literature.

Non-Zero Radial Solutions for Elliptic Systems with Coupled Functional BCs in Exterior Domains

AbstractWe prove new results on the existence, non-existence, localization and multiplicity of non-trivial radial solutions of a system of elliptic boundary value problems on exterior domains subject to non-local, nonlinear, functional boundary conditions. Our approach relies on fixed point index theory. As a by-product of our theory we provide an answer to an open question posed by do Ó, Lorca, Sánchez and Ubilla. We include some examples with explicit nonlinearities in order to illustrate our theory.

Symmetry breaking via Morse index for equations and systems of Hénon–Schrödinger type

We consider the Dirichlet problem for the Schr\"odinger-H\'enon system $$ -\Delta u + \mu_1 u = |x|^{\alpha}\partial_u F(u,v),\quad \qquad -\Delta v + \mu_2 v = |x|^{\alpha}\partial_v F(u,v) $$ in the unit ball $\Omega \subset \mathbb{R}^N, N\geq 2$, where $\alpha>-1$ is a parameter and $F: \mathbb{R}^2 \to \mathbb{R}$ is a $p$-homogeneous $C^2$-function for some $p>2$ with $F(u,v)>0$ for $(u,v) \not = (0,0)$. We show that, as $\alpha \to \infty$, the Morse index of nontrivial radial solutions of this problem (positive or sign-changing) tends to infinity. This result is new even for the corresponding scalar H\'enon equation and extends a previous result by Moreira dos Santos and Pacella for the case $N=2$. In particular, the result implies symmetry breaking for ground state solutions, but also for other solutions obtained by an $\alpha$-independent variational minimax principle.

A variational inequality of Kirchhoff-type in mathbb{R}^{N}

In this paper, we investigate the existence of nontrivial radial solutions for a kind of variational inequalities in mathbb{R}^{N}. Our main technique is the non-smooth critical point theory, based on the Szulkin-type functionals.

Positive Solutions for Nonvariational Systems with Asymptotically Homogeneous Operators

In this work, we deal with the existence of nontrivial radial positive solutions for nonvariational elliptic systems involving operators that belong to asymptotically homogeneous class via topological degree theory.

Solutions of perturbed Hammerstein integral equations with applications

By means of topological methods, we provide new results on the existence, non-existence, localization and multiplicity of nontrivial solutions for systems of perturbed Hammerstein integral equations. In order to illustrate our theoretical results, we study some problems that occur in applied mathematics, namely models of chemical reactors, beams and thermostats. We also apply our theory in order to prove the existence of nontrivial radial solutions of systems of elliptic boundary value problems subject to nonlocal, nonlinear boundary conditions.