Nonlinear Kirchhoff Type Problems Research Articles

Two weak solutions for fully nonlinear Kirchhoff-type problem

In this article, the existence of solutions for fully nonlinear Kirchhoff-type problem -M (??(|?u|) + ?(|u|))dx) [div(a(|?u|)?u) + a(|u|)u] = ? ?ki=1 (tqi(x)-1- tri(x)-1)is proved via variational method. Finally, some new problems are introduced.

Multiplicity of concentrating positive solutions for nonlinear Kirchhoff type problems with critical growth

ABSTRACT In this paper, we study the following semilinear Kirchhoff type equation: where is a parameter, a, b are positive constants, V and f are continuous functions. Under certain assumptions on V and f, we investigate the relation between the number of solutions and the topology of the set where V attains its local minimum for small ε. We also describe the concentration phenomena of solutions as . The proof is based on the method of Nehari manifold, penalization techniques and Ljusternik–Schnirelmann category theory.

Berestycki-Lions conditions on ground state solutions for Kirchhoff-type problems with variable potentials

Based on the strategy introduced by Chen and Tang [Adv. Nonlinear Anal. 9, 496–515 (2020)] and some new tricks, we prove that the nonlinear problem of Kirchhoff-type −a+b∫R3|∇u|2dx△u+V(x)u=f(u), x∈R3 in H1(R3) admits two classes of ground state solutions under the general “Berestycki-Lions assumptions” on the nonlinearity f, which are almost necessary conditions, as well as some weak assumptions on the potential V. Moreover, we also give a simple minimax characterization of the ground state energy. Our results improve and extend recent theorems in several directions.

Existence of positive solutions for the nonlinear Kirchhoff type equations in [formula omitted

In this paper, we consider the following nonlinear problem of Kirchhoff type:{−(a+λ∫Rn|∇u|2dx)Δu+V(x)u=|u|p−1u,x∈Rn,u∈H1(Rn),u>0, where n≥3, a, λ are positive constants, 1<p<max⁡(3,n+2n−2) and V(x) is a positive continuous potential satisfying V0≤V(x)≤lim inf|x|→∞V(x)=V∞. Using variational methods and a cutoff technique, we prove the existence of positive solution to the above equation for all λ>0 small.

Steep potential well may help Kirchhoff type equations to generate multiple solutions

In this paper, we consider a nonlinear Kirchhoff type problem with steep potential well: −a∫R3∇u2dx+bΔu+λV(x)u=fx|u|p−2uin R3,u∈H1(R3),where a,b,λ>0, 2<p<4, V∈C(R3,R+) and f∈L∞(R3,R). Such problem cannot be studied by applying variational methods in a standard way, even by restricting its corresponding energy functional on the Nehari manifold, because Palais–Smale sequences may not be bounded. In this paper, we introduce a novel constraint method to prove the existence of one and two positive solutions under the different assumptions on V, respectively. We conclude that steep potential well V may help Kirchhoff type equations to generate multiple solutions, which has never been involved before.

On ground state solutions of Nehari-Pohozaev type for the nonlinear Kirchhoff type problems with a general critical nonlinearity

In this paper, we are concerned with the following Kirchhoff type problem with critical growth: \begin{equation*} -\bigg(a+b\int_{\mathbb R^3}|\nabla u|^2dx\bigg)\Delta u+V(x)u=f(u)+|u|^4u, \quad u\in H^1(\mathbb R^3), \end{equation*} where $a,b &gt; 0$ are constants. Under some certain assumptions on $V$ and $f$, we prove that the above problem has a ground state solution of Nehari-Pohozaev type and a least energy solution via variational methods. Furthermore, we also show that the mountain pass value gives the least energy level for the above problem. Our results improve and extend some recent ones in the literature.

Infinitely many sign-changing solutions for Kirchhoff type problems in [formula omitted

In this paper, we consider the following nonlinear Kirchhoff type problem: −a+b∫R3|∇u|2Δu+V(x)u=f(u),inR3,u∈H1(R3),where a,b>0 are constants, the nonlinearity f is superlinear at infinity with subcritical growth and V is continuous and coercive. For the case when f is odd in u we obtain infinitely many sign-changing solutions for the above problem by using a combination of invariant sets method and the Ljusternik–Schnirelman type minimax method. To the best of our knowledge, there are only few existence results for this problem. It is worth mentioning that the nonlinear term may not be 4-superlinear at infinity, in particular, it includes the power-type nonlinearity |u|p−2u with p∈(2,4].

Existence and multiplicity of normalized solutions for the nonlinear Kirchhoff type problems

In this paper, we prove the existence and multiplicity results of solutions with prescribed L2-norm for a class of Kirchhoff type problems −a+b∫R3|∇u|2dxΔu−λu=f(u)inR3,where a,b>0 are constants, λ∈R andf∈C(R,R). To obtain such solutions, we look into critical points of the energy functional Eb(u)=a2∫R3|∇u|2+b4∫R3|∇u|22−∫R3F(u)constrained on the L2-spheres S(c)=u∈H1(R3):||u||22=c. Here, c>0 and F(s)≔∫0sf(t)dt. Under some mild assumptions on f, we show that critical points of Eb unbounded from below on S(c) exist for c>0. In addition, we establish the existence of infinitely many radial critical points {unb} of Eb on S(c) provided that f is odd. Finally, the asymptotic behavior of unb as b↘0 is analyzed. These conclusions extend some known ones in previous papers.

Signed and sign-changing solutions of Kirchhoff type problems

We consider the following nonlinear Kirchhoff type problem of the form $$\begin{aligned} \left\{ \begin{array}{ll} -(a+b\int _{\Omega }|\nabla u|^{2})\triangle u = \mu g(x,u)+f(x,u), &{}\quad \hbox {in}\ \ \Omega ,\\ u=0, &{}\quad \hbox {on}\ \ \partial \Omega ,\end{array}\right. \end{aligned}$$ where $$\Omega \subset \mathbb {R}^{3}$$ is a bounded domain with smooth boundary $$\partial \Omega $$ and $$a>0$$ , $$b\ge 0 $$ . The nonlinearity $$\mu g(x,u)+f(x,u)$$ may involve a combination of concave and convex terms. Under some suitable conditions on $$f,g\in C(\overline{\Omega }\times \mathbb {R},\mathbb {R})$$ and $$\mu \in \mathbb {R}$$ , we prove the existence of infinitely many high-energy solutions using Fountain theorem. In particular, using the method of invariant sets of descending flow, we prove the existence of at least one sign-changing solutions.

Semiclassical ground states for a class of nonlinear Kirchhoff-type problems

In this paper, we study the following semilinear Kirchhoff-type equationwhere is a parameter, a, b are positive constants, V and Q are positive bounded functions. For the equation with three times growth, we develop an argument to establish the existence of ground states for small. We also describe the concentration phenomena of ground states as .

Existence and asymptotic behavior of high energy normalized solutions for the Kirchhoff type equations in [formula omitted

In this paper, we study the multiplicity of solutions with a prescribed L2-norm for a class of nonlinear Kirchhoff type problems in R3−(a+b∫R3|∇u|2)Δu−λu=|u|p−2u, where a,b>0 are constants, λ∈R, p∈(143,6). To get such solutions we look for critical points of the energy functional Ib(u)=a2∫R3|∇u|2+b4(∫R3|∇u|2)2−1p∫R3|u|p restricted on the following set Sr(c)={u∈Hr1(R3):‖u‖L2(R3)2=c}, c>0. For the value p∈(143,6) considered, the functional Ib is unbounded from below on Sr(c). By using a minimax procedure, we prove that for any c>0, there are infinitely many critical points {unb}n∈N+ of Ib restricted on Sr(c) with the energy Ib(unb)→+∞(n→+∞). Moreover, we regard b as a parameter and give a convergence property of unb as b→0+.

Existence of positive ground state solutions for Kirchhoff type problems

In this paper, we study the existence of positive ground state solutions for the nonlinear Kirchhoff type problem {−(a+b∫R3|∇u|2)△u+V(x)u=f(u)in R3,u∈H1(R3),u>0in R3, where a,b>0 are constants, f∈C(R,R) is subcritical near infinity and superlinear near zero and satisfies the Berestycki–Lions condition. By using an abstract critical point theorem established by Jeanjean and a new global compactness lemma, we show that the above problem has at least a positive ground state solution. Our result generalizes the results of Li and Ye (2014) concerning the nonlinearity f(u)=|u|p−1u with p∈(2,5).

Ground states for the nonlinear Kirchhoff type problems

In this paper we study the existence of ground states for the asymptotically periodic Kirchhoff type problems with critical growth and three times growth. The proof is based on the method of Nehari manifold and concentration compactness principle. In particular, we improve the method of Nehari manifold for Kirchhoff type equations with three times growth.

Existence of Solutions to a Class of Kirchhoff-Type Equation with a General Subcritical Nonlinearity

This paper is devoted to the following nonlinear Kirchhoff-type problem $$-\bigg(a + b\int_{\Omega}|\nabla u|^{2}\,{d}x\bigg)\Delta u = \lambda f(x,u)$$ with the Dirichlet boundary value. We show that the Kirchhoff-type problem has at least a weak nontrivial solution for all \({\lambda > 0}\) under suitable assumptions on the nonlinear term f with more general growth condition. The main tools are variational method, critical point theory and some analysis techniques.

Existence of positive solutions for nonlinear Kirchhoff type problems in with critical Sobolev exponent

AbstractIn this paper, we study the following nonlinear problem of Kirchhoff type with critical Sobolev exponent: urn:x-wiley:01704214:media:mma3000:mma3000-math-0001 where a, b &gt; 0 are constants. Under certain assumptions on the sign‐changing function f(x,u), we prove the existence of positive solutions by variational methods.Our main results can be viewed as a partial extension of a recent result of He and Zou in [Journal of Differential Equations, 2012] concerning the existence of positive solutions to the nonlinear Kirchhoff problem urn:x-wiley:01704214:media:mma3000:mma3000-math-0002 where ϵ &gt; 0 is a parameter, V (x) is a positive continuous potential, and with 4 &lt; p &lt; 6 and satisfies the Ambrosetti–Rabinowitz type condition. Copyright © 2013 John Wiley &amp; Sons, Ltd.

Positive solution for a superlinear Kirchhoff type problem with a parameter

This paper is devoted to a nonlinear Kirchhoff type problem depending on a real function and a nonnegative parameter on a smooth bounded domain of RN, N≥2. We show that if the nonlinearity is subcritical and superlinear at zero and infinity, then there exists a positive value such that the Kirchhoff type problem has at least one positive solution for any value of the parameter less than the positive value. The main tools are variational methods, critical point theory and iterative techniques.

Existence and concentration behavior of positive solutions for a Kirchhoff equation in [formula omitted

We study the existence, multiplicity and concentration behavior of positive solutions for the nonlinear Kirchhoff type problem { − ( ε 2 a + ε b ∫ R 3 | ∇ u | 2 ) Δ u + V ( x ) u = f ( u ) in R 3 , u ∈ H 1 ( R 3 ) , u > 0 in R 3 , where ε > 0 is a parameter and a , b > 0 are constants; V is a positive continuous potential satisfying some conditions and f is a subcritical nonlinear term. We relate the number of solutions with the topology of the set where V attains its minimum. The results are proved by using the variational methods.

Exponential Decay for the Solutions of Wave Equations of Kirchhoff Type with Strong Damping

We study some decay estimates of the energy of the following nonlinear problem of Kirchhoff type:.....