Sublinear Perturbation Research Articles

On the antimaximum principle for the p-Laplacian and its sublinear perturbations

We investigate qualitative properties of weak solutions of the Dirichlet problem for the equation $$-\Delta _p u = \lambda \,m(x)|u|^{p-2}u+ \eta \,a(x)|u|^{q-2}u+ f(x)$$ in a bounded domain $$\Omega \subset \mathbb {R}^N$$ , where $$q<p$$ . Under certain regularity and qualitative assumptions on the weights m, a and the source function f, we identify ranges of the parameters $$\lambda $$ and $$\eta $$ for which solutions satisfy maximum and antimaximum principles in weak and strong forms. Some of our results, especially on the validity of the antimaximum principle under low regularity assumptions, are new for the unperturbed problem with $$\eta =0$$ , and among them there are results providing new information even in the linear case $$p=2$$ . In particular, we show that for any $$p>1$$ solutions of the unperturbed problem satisfy the antimaximum principle in a right neighborhood of the first eigenvalue of the p-Laplacian provided $$m,f \in L^\gamma (\Omega )$$ with $$\gamma >N$$ . For completeness, we also investigate the existence of solutions.

Resonant solutions for elliptic systems with Neumann boundary conditions

We consider a sublinear perturbation of an elliptic eigenvalue system with homogeneous Neumann boundary conditions. For oscillatory nonlinearities and using bifurcation from infinity, we prove the existence of an unbounded sequence of turning points and an unbounded sequence of resonant solutions.&#x0D; See also https://ejde.math.txstate.edu/special/02/d1/abstr.html

Existence and concentration of solutions for a Kirchhoff-type problem with sublinear perturbation and steep potential well

&lt;abstract&gt;&lt;p&gt;In this paper, we consider the following nonlinear Kirchhoff-type problem with sublinear perturbation and steep potential well&lt;/p&gt; &lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ \begin{eqnarray*} \left \{\begin{array}{ll} -\Big(a+b\int_{\mathbb{R}^3}|\nabla u|^2dx\Big)\Delta u+\lambda V(x)u = f(x,u)+g(x)|u|^{q-2}u\ \ \mbox{in}\ \mathbb{R}^3,\\ \\ u\in H^1(\mathbb{R}^3), \\ \end{array} \right. \label{1} \end{eqnarray*} $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt; &lt;p&gt;where $ a $ and $ b $ are positive constants, $ \lambda &amp;gt; 0 $ is a parameter, $ 1 &amp;lt; q &amp;lt; 2 $, the potential $ V\in C(\mathbb{R}^3, \mathbb{R}) $ and $ V^{-1}(0) $ has a nonempty interior. The functions $ f $ and $ g $ are assumed to obey a certain set of conditions. The existence of two nontrivial solutions are obtained by using variational methods. Furthermore, the concentration behavior of solutions as $ \lambda\rightarrow \infty $ is also explored.&lt;/p&gt;&lt;/abstract&gt;

Fractional mean curvature flow of Lipschitz graphs

We consider the fractional mean curvature flow of entire Lipschitz graphs. We provide regularity results, and we study the long time asymptotics of the flow. In particular we show that in a suitable rescaled framework, if the initial graph is a sublinear perturbation of a cone, the evolution asymptotically approaches an expanding self-similar solution. We also prove stability of hyperplanes and of convex cones in the unrescaled setting.

Fourth-order elliptic problems with critical nonlinearities by a sublinear perturbation

In this paper, we get the existence of two positive solutions for a fourth-order problem with Navier boundary condition. Our nonlinearity has a critical growth, and the method is a local minimum theorem obtained by Bonanno.

Anisotropic mean curvature flow of Lipschitz graphs and convergence to self-similar solutions

We consider the anisotropic mean curvature flow of entire Lipschitz graphs. We prove existence and uniqueness of expanding self-similar solutions which are asymptotic to a prescribed cone, and we characterize the long time behavior of solutions, after suitable rescaling, when the initial datum is a sublinear perturbation of a cone. In the case of regular anisotropies, we prove the stability of self-similar solutions asymptotic to strictly mean convex cones, with respect to perturbations vanishing at infinity. We also show the stability of hyperplanes, with a proof which is novel also for the isotropic mean curvature flow.

Fractional problems with critical nonlinearities by a sublinear perturbation

Abstract In this paper, the existence of two nontrivial solutions for a fractional problem with critical exponent, depending on real parameters, is established. The variational approach is used based on a local minimum theorem due to G. Bonanno. In addition, a numerical estimate on the real parameters is provided, for which the two solutions are obtained.

Multiple Solutions with Sign Information for a Class of Parametric Superlinear (p,\xa02)-Equations

We consider a parametric nonlinear, nonhomogeneous Dirichlet problem driven by the sum of a p-Laplacian (with p>2) and a Laplacian (a two phase equation). The reaction consists of a parametric (p-1)-superlinear term and a (p-1)-sublinear perturbation. We show that for all lambda >0 big, the problem has at least three nontrivial smooth solutions, all with sign information. Also we determine their asymptotic behaviour as the parameter lambda rightarrow infty . When we strengthen the regularity of the perturbation term, we produce a second nodal solution, for a total of four solutions, all with sign information.

Existence and multiplicity of solutions for the fractional Schrödinger equation with steep potential well and perturbation

In the present paper, we investigate the solutions of a fractional Schrodinger equation with sublinear perturbation and steep potential well. First, we obtain the existence of at least one solution by the minimisation result due to Mawhin and Willem. Second, by virtue of the symmetric mountain pass theorem, the existence criteria of infinitely many solutions are established. Lastly, an example is given to demonstrate the main results.

Positive Solutions for Nonlinear Dirichlet Problems with Convection

We consider a nonlinear Dirichlet problem driven by the p-Laplacian, a convection term and a $$(p-1)$$ -sublinear perturbation. First we assume that the coefficient in the convection term (drift coefficient) is sign changing. Using the theory of nonlinear operators of monotone type together with suitable truncation and comparison techniques we prove the existence of a positive smooth solution. When the drift coefficient is nonnegative, we are able to relax the conditions on the data of the problem.

Energy solutions and concentration problem of fractional Schr\xf6dinger equation

In this paper, we consider a fractional Schrödinger equation with steep potential well and sublinear perturbation. By virtue of variational methods, the existence criteria of infinitely many nontrivial high or small energy solutions are established. In addition, the phenomenon of the concentration of solutions is also explored. We also give some examples to demonstrate the main results.

Three solutions for an elliptic system near resonance with the principal eigenvalue

We consider an elliptic system of Hamiltonian type with linear part depending on two parameters and a sublinear perturbation. We obtain the existence of at least three solutions when the linear part is near resonance with the principal eigenvalue, either from above or from below. For two of these solutions, we also obtain information on the sign of its components. The system is associated to a strongly indefinite functional and the solutions are obtained trough saddle point theorem, after truncating the nonlinearity.

Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential

We study perturbations of the eigenvalue problem for the negative Laplacian plus an indefinite and unbounded potential and Robin boundary condition. First we consider the case of a sublinear perturbation and then of a superlinear perturbation. For the first case we show that for $\lambda<\widehat{\lambda}_{1}$ ($\widehat{\lambda}_{1}$ being the principal eigenvalue) there is one positive solution which is unique under additional conditions on the perturbation term. For $\lambda\geq\widehat{\lambda}_{1}$ there are no positive solutions. In the superlinear case, for $\lambda<\widehat{\lambda}_{1}$ we have at least two positive solutions and for $\lambda\geq\widehat{\lambda}_{1}$ there are no positive solutions. For both cases we establish the existence of a minimal positive solution $\bar{u}_{\lambda}$ and we investigate the properties of the map $\lambda\mapsto\bar{u}_{\lambda}$.

Landesman–Lazer conditions for difference equations involving sublinear perturbations

We study the existence and uniqueness for discrete Neumann and periodic problems. We consider both ordinary and partial difference equations involving sublinear perturbations. All the proofs are based on reformulating these discrete problems as a general singular algebraic system. Firstly, we use variational techniques (specifically, the Saddle Point Theorem) and prove the existence result based on a type of Landesman–Lazer condition. Then we show that for a certain class of bounded nonlinearities this condition is even necessary and therefore, we specify also the cases in which there does not exist any solution. Finally, the uniqueness is discussed.

Multiplicity and concentration of solutions for fractional Schrödinger equation with sublinear perturbation and steep potential well

In this paper, we investigate the following fractional Schrödinger equation with sublinear perturbation and steep potential well {(−△)su+λV(x)u=f(x,u)+α(x)|u|ν−2uinRN,u∈Hs(RN), where 0<s<1,2s<N,λ>0,1<ν<2, f∈C(RN×R) is of subcritical growth. By using variational methods, we prove that such a class of equations possess at least two nontrivial solutions. Moreover, the phenomenon of concentration of solutions is explored as well.

Positive solutions for perturbations of the eigenvalue problem for the Robin p-Laplacian

We study perturbations of the eigenvalue problem for the Robin p-Laplacian. First we consider the case of a (p − 1)-sublinear perturbation and prove existence, nonexistence and uniqueness of positive solutions. Then we deal with the case of a (p − 1)-superlinear perturbation which need not satisfy the Ambrosetti-Rabinowitz condition and prove a multiplicity result for positive solutions. Our approach uses variational methods together with suitable truncation and perturbation techniques.

Multiple solutions for an elliptic system near resonance

We consider an elliptic system with linear part depending on two parameters and a sublinear perturbation. We obtain the existence of at least two solutions when the linear part is near resonance. The system is associated with a strongly indefinite functional and the solutions are obtained through saddle point theorem and Galerkin approximation.

Existence of Periodic Solutions of Linear Hamiltonian Systems with Sublinear Perturbation

We investigate the existence of periodic solutions of linear Hamiltonian systems with a nonlinear perturbation. Under generalized Ahmad-Lazer-Paul type coercive conditions for the nonlinearity on the kernel of the linear part, existence of periodic solutions is obtained by saddle point theorems. A note on a result of Rabinowitz is also given.

Positive fixed points for nonlinear operators

This paper is concerned with the existence and uniqueness of positive fixed points for nonlinear decreasing operators with sublinear perturbation operators in ordered Banach spaces. A new fixed-point theorem, without the assumption of compactness and continuity of operators, is established, and an iterative sequence yielding the fixed point is also given. In addition, an application to nonlinear integral equations is shown.

On the bifurcation of large amplitude solutions for a system of wave and beam equations

We consider the existence of solutions for linearly coupled system of wave and beam equation with a sublinear perturbation. The main concept is the matrix spectrum which is a natural extension of the usual spectrum. Using the standard ‘change of degree’ argument we shall find necessary and sufficient conditions for the existence of asymptotic bifurcation with respect to the matrix spectrum both in one parameter and four parameter cases. Obviously the results can be modified for more general systems of partial differential equations.