Class Of Kirchhoff Type Problems Research Articles

Bounded Variation Solution for a Class of Kirchhoff Type Problem Involving the 1-Laplacian Operator

Bounded Variation Solution for a Class of Kirchhoff Type Problem Involving the 1-Laplacian Operator

On a class of Kirchhoff type problems with singular exponential nonlinearity

We study the following singular Kirchhoff type problem 
 \[\left( P\right) \left\{
 \begin{array} [c]{c}
 -m\left({\displaystyle\int\limits_{\Omega}}\left\vert \nabla u\right\vert ^{2}dx\right) \Delta u=h\left( u\right)
 \frac{e^{\alpha u^{2}}}{\left\vert x\right\vert ^{\beta}}\text{ \ \ \ in} \Omega,\\
 u=0 \text{on}\; \partial\Omega
 \end{array} \right.
 \]
 where $\Omega\subset\mathbb{R}^{2}$ is a bounded domain with smooth boundary and $0\in\Omega,$ $\beta\in\left[ 0,2\right)$, $\alpha>0$ and $m$ is a continuous function
 on $\mathbb{R}^{+}.$ Here, $h$ is a suitable preturbation of $e^{\alpha u^{2}}$ as $u\rightarrow\infty.$ In this paper, we prove the existence of solutions of
 $(P)$. Our tools are Trudinger-Moser inequality with a singular weight and the mountain pass theorem

Existence of nonnegative nontrivial solutions for Kirchhoff type problems with variable exponent

This paper is devoted to study a class of Kirchhoff type problems with variable exponent. By means of perturbation technique, variational methods and a priori estimation, the existence of nonnegative nontrivial solutions to this problem is obtained.

Existence of two solutions for Kirchhoff-double phase problems with a small perturbation without (AR)-condition

This paper deals with a class of Kirchhoff type problems on a double phase setting with a small perturbation. It is well known that (AR)-condition is an important technique to apply the Mountain Pass theorem. A legitimate question arises, can we ensure the existence of weak solutions in the case where (AR)-condition is not satisfied? Our goal is to give a positive answer to this question. We present a new sufficient assumption weaker than (AR)-condition for which the considered problem admits at least two weak solutions. The proof rely on variational arguments based on the Mountain Pass Theorem with Cerami condition. Our results extend the results already proven by other authors.

Existence and multiplicity of positive solutions for a class of Kirchhoff type problems with singularity and critical exponents

<abstract><p>In this paper, we study the multiplicity results of positive solutions for a class of Kirchhoff type problems with singularity and critical exponents. Combining with the Nehari method and variational method, we prove the existence of positive ground state solutions. Furthermore, we obtain a relationship between the number of positive solutions and the topology of the global maximum set of $ Q(x) $.</p></abstract>

Infinitely many nodal solutions with a prescribed number of nodes for the Kirchhoff type equations

In this paper, we investigate the existence of multiple radial sign-changing solutions with the nodal characterization for a class of Kirchhoff type problems{−(a+b|∇u|L22)Δu+V(|x|)u=K(|x|)f(u)in RN,u∈H1(RN), where N=1,2,3,a,b>0, V,K are radial and bounded away from below by positive numbers. Under some weak assumptions on f∈C0(R;R), by taking advantage of the Gersgorin disc's theorem and Miranda theorem, we develop some new analytic techniques and prove that this problem admits infinitely many nodal solutions {Ukb} having a prescribed number of nodes k, whose energy is strictly increasing in k. Moreover, the asymptotic behaviors of Ukb as b→0+ are established. These results improve and generalize the previous results in the literature.

Existence of solutions for a class of critical Kirchhoff type problems involving Caffarelli-Kohn-Nirenberg inequalities

In this paper, we study the existence of a nontrival weak solution for a class of Kirchhoff type problems with singular potentials and critical exponents. The proofs are essentially based on an appropriated truncated argument, Caffarelli-Kohn-Nirenberg inequalities, combined with a variant of the concentration compactness principle. We also get a priori estimates of the obtained solution

On a p⋅-biharmonic problem of Kirchhoff type involving critical growth

ABSTRACT We establish a concentration-compactness principle for the Sobolev space that is a tool for overcoming the lack of compactness of the critical Sobolev imbedding. Using this result, we obtain several existence and multiplicity results for a class of Kirchhoff type problems involving -biharmonic operator and critical growth.

Existence of Solutions for a Class of Kirchhoff-Type Equation via Young Measures

In this article, our aim is to obtain weak solutions for a class of Kirchhoff-type problems by using the theory of Young measures.

Existence and multiplicity of positive solutions for a class of Kirchhoff type problems with nonlinear boundary conditions

In this work, we peruse a class of Kirchhoff type problems with nonlinear boundary conditions on a bounded domain, and earn the uniqueness and existence of positive solutions for a class of those problems by the variational methods. Also, we earn the multiplicity of positive solutions of those problems by using the Nehari manifold.

Existence of solution for a class of nonvariational Kirchhoff type problem via dynamical methods

In this paper, we use dynamical methods to establish the existence of nontrivial solution for the following class of Kirchhoff type problem (P)−1+b∫Ω|∇u|2dxΔu=f(u)+h(u,∇u),x∈Ω,u=0,x∈∂Ω, where Ω⊂RN(N=2,3) is a smooth bounded domain, b≥0, f:R→R and h:R×RN→[0,+∞) are C1-functions that satisfy some technical conditions.

Some Results for a Class of Kirchhoff-Type Problems with Hardy–Sobolev Critical Exponent

We study a class of Kirchhoff equations $$\begin{aligned} {\left\{ \begin{array}{ll} -\left( a+b\displaystyle \int _{\Omega }|\nabla u|^2\mathrm{d}x\right) \Delta u=\displaystyle \frac{u^{3}}{|x|}+\lambda u^{q},&{}\hbox {in } \Omega , \\ u=0, &{}\hbox {on } \partial \Omega , \end{array}\right. } \end{aligned}$$ where $$\Omega \subset {\mathbb {R}}^{3}$$ is a bounded domain with smooth boundary and $$0\in \Omega $$ , $$a,b,\lambda >0,0<q<1.$$ By the variational method, two positive solutions are obtained. Moreover, when $$b>\frac{1}{A_{1}^{2}}$$ ( $$A_{1}>0$$ is the best Sobolev–Hardy constant), using the critical point theorem, infinitely many pairs of distinct solutions are obtained for any $$\lambda >0.$$

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.

Existence and multiplicity of positive solutions for a class of singular Kirchhoff type problems with sign‐changing potential

In this paper, the existence and multiplicity of positive solutions are obtained for a class of Kirchhoff type problems with two singular terms and sign‐changing potential by the Nehari method.

Multiplicity of positive solutions for critical fractional Kirchhoff type problem with concave-convex nonlinearity

This paper is devoted to study a class of Kirchhoff type problem with critical fractional exponent and concave nonlinearity. By means of variational methods, the multiplicity of the positive solutions to this problem is obtained.

Multiplicity of Solutions for Kirchhoff-Type Problem with Two-Superlinear Potentials

In this paper we consider a class of Kirchhoff-type problem with 2-superlinear potentials. The existence of one positive solution and one negative solution will be established by using iterative technique and the Mountain Pass theorem, and a sign changing solution will be obtained by combining iterative technique and the Nehari method.

The existence and concentration of ground-state solutions for a class of Kirchhoff type problems in {mathbb{R}^{3}} involving critical Sobolev exponents

We are concerned with ground-state solutions for the following Kirchhoff type equation with critical nonlinearity: \t\t\t{−(ε2a+εb∫R3|∇u|2)Δu+V(x)u=λW(x)|u|p−2u+|u|4uin R3,u>0,u∈H1(R3),\\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}$$\\textstyle\\begin{cases} - ({\\varepsilon^{2}}a + \\varepsilon b\\int_{{\\mathbb{R}^{3}}} {{{ \\vert {\\nabla u} \\vert }^{2}}} )\\Delta u + V(x)u = \\lambda W(x){ \\vert u \\vert ^{p - 2}}u + { \\vert u \\vert ^{4}}u\\quad {\\text{in }}{\\mathbb{R}^{3}} ,\\\\ u > 0, \\quad\\quad u \\in{H^{1}}({\\mathbb{R}^{3}}) , \\end{cases} $$\\end{document} where ε is a small positive parameter, a,b>0, lambda > 0, 2 < p le4, V and W are two potentials. Under proper assumptions, we prove that, for varepsilon > 0 sufficiently small, the above problem has a positive ground-state solution {u_{varepsilon}} by using a monotonicity trick and a new version of global compactness lemma. Moreover, we use another global compactness method due to Gui (Commun. Partial Differ. Equ. 21:787-820, 1996) to show that {u_{varepsilon}} is concentrated around a set which is related to the set where the potential V(x) attains its global minima or the set where the potential W(x) attains its global maxima as varepsilon to0.

Ground state sign-changing solutions for asymptotically 3-linear Kirchhoff-type problems

This paper is devoted to the existence of ground state sign-changing solutions for a class of Kirchhoff-type problems(Section.Display) where is a bounded domain with a smooth boundary , , and satisfies asymptotically linear growth that is very different from super-3-linear growth in previous literatures. Without assuming the standard Variant Nehari-type condition related to (0.1), we prove that (0.1) possesses one ground state sign-changing solution , and show that its energy is strictly larger than twice that of the ground-state solutions of Nehari-type. Furthermore, we establish the convergence property of as the parameter .

Multiple positive solutions for Kirchhoff type problems involving concave-convex nonlinearities

In this paper, we are interested in looking for multiple solutions for a class of Kirchhoff type problems with concave-convex nonlinearities. Under the combined effect of coefficient functions of concave-convex nonlinearities, by the Nehari method, we obtain two solutions, and one of them is a ground state solution. Under some stronger conditions, we point that the two solutions are positive solutions by the strong maximum principle.

The existence and nonexistence results of ground state nodal solutions for a Kirchhoff type problem

In this paper, we investigate the existence and nonexistence of ground state nodal solutions to a class of Kirchhoff type problems \begin{document} $ -\left( a+b\int_{\Omega }{|}\nabla u{{|}.{2}}dx \right)\vartriangle u=\lambda u+|u{{|}.{2}}u, u\in H_{0}.{1}(\Omega ), $ \end{document} where $a, b>0$, $\lambda 0$ such that the Kirchhoff type problem possesses at least one ground state nodal solution $u_b$ for all $0 21 ], X.H. Tang and B.T. Cheng (2016)[ 22 ].