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In this paper, we consider the following fourth‐order equation of Kirchhoff type Δ2u−(a+b‖∇u‖22)Δu=λu+|u|p−2uinℝd$$ {\Delta}^2u-\left(a+b{\left\Vert \nabla u\right\Vert}_2^2\right)\Delta u=\lambda u+{\left|u\right|}^{p-2}u\kern0.30em \mathrm{in}\kern6.05pt {\mathrm{\mathbb{R}}}^d $$ under the normalized constraint ‖u‖2=m$$ {\left\Vert u\right\Vert}_2=m $$, where a,b,m>0$$ a,b,m>0 $$ are constants, d≥1$$ d\ge 1 $$ and 2

In this paper, we consider the modified Kirchhoff type equation, that is, the Kirchhoff type equation with a quasilinear term (1) $$\begin{equation} \begin{cases} -\left(a+b\displaystyle\int_{\mathbb{R}^3}|\nabla u|^2{\rm d}x\right)\Delta u+V(|x|)u-\dfrac{1}{2}\Delta(|u|^2)u\\\noalign{\vskip2.6pt} \quad=|u|^{p-2}u\quad\mbox{in }\mathbb{R}^3,\\ u\to 0 \qquad as\ |x|\to\infty, \end{cases} \end{equation}$$ { − (a + b ∫ R 3 | ∇u | 2 d x) Δu + V (| x |) u − 1 2 Δ (| u | 2) u = | u | p − 2 u in R 3 , u → 0 as | x | → ∞ , where a, b>0, $ p\in (4,22^*) $ p ∈ (4 , 22 ∗) and $ V(|x|) $ V (| x |) is a radial potential function and bounded below by a positive number. The appearance of nonlocal term $ b\int _{\mathbb {R}^3}|\nabla u|^2{\rm d}x\Delta u $ b ∫ R 3 | ∇u | 2 d x Δu and quasilinear term $ \frac {1}{2}\Delta (|u|^2)u $ 1 2 Δ (| u | 2) u makes the variational functional of (1) totally different from the classical Schrödinger equation. By introducing the Miranda theorem, via a construction and gluing method, for any given integer $ k\geq ~1 $ k ≥ 1 , we prove that Equation (1) admits a radial nodal solution $ U_{k}^b $ U k b having exactly k nodes. Moreover, the energy of $ U_{k}^b $ U k b is monotonically increasing in k and for any sequence $ \{b_n\} $ { b n } , and up to a subsequence, $ U_{k}^{b_n} $ U k b n converges strongly to some $ U_{k}^0\neq ~0 $ U k 0 ≠ 0 as $ b_n\to 0_+ $ b n → 0 + , which is a nodal solution with exactly k nodes to the local quasilinear Schrödinger equation (2) $$\begin{equation} \begin{cases} -a\Delta u+V(|x|)u-\dfrac{1}{2}\Delta(|u|^2)u=|u|^{p-2}u\quad\mbox{in }\mathbb{R}^3,\\\noalign{\vskip2.6pt} u\to 0 \qquad as\ |x|\to\infty. \end{cases} \end{equation}$$ { − a Δu + V (| x |) u − 1 2 Δ (| u | 2) u = | u | p − 2 u in R 3 , u → 0 as | x | → ∞. These results improve and generalize the previous results in the literature from the local quasilinear Schrödinger equation to the nonlocal quasilinear Kirchhoff equation. [ABSTRACT FROM AUTHOR]

In this paper, we studied the existence of multiple normalized solutions to the following Kirchhoff type equation:$ \begin{equation*} \begin{cases} -\left(a\varepsilon^2+b\varepsilon\int_{\mathbb{R}^3}|\nabla u|^2dx\right)\Delta u+V(x)u = \mu u+f(u) & {\rm{in}}\;\mathbb{R}^3, \\ \int_{\mathbb{R}^3}|u|^2dx = m\varepsilon^3 , u\in H^1(\mathbb{R}^3) , \end{cases} \end{equation*} $where $ a $, $ b $, $ m > 0 $, $ \varepsilon $ is a small positive parameter, $ V $ is a nonnegative continuous function, $ f $ is a continuous function with $ L^2 $-subcritical growth and $ \mu\in\mathbb{R} $ will arise as a Lagrange multiplier. Under the suitable assumptions on $ V $ and $ f $, the existence of multiple normalized solutions was obtained by using minimization techniques and the Lusternik-Schnirelmann theory. We pointed out that the number of normalized solutions was related to the topological richness of the set where the potential $ V $ attained its minimum value.

In this paper, we study the existence and concentration behavior of normalized positive solutions to the following L 2 -supercritical Kirchhoff type equation - a ε 2 + b ε ∫ R 3 | ∇ v | 2 d x Δ v + λ v = K (x) | v | p - 2 v , x ∈ R 3 , ∫ R 3 | v | 2 d x = m ε 3 , v ∈ H 1 (R 3) , where a , b , m > 0 , p ∈ 14 3 , 6 , ε is a small positive parameter, K is a positive continuous function possessing a local maximum point and λ ∈ R will arise as a Lagrange multiplier. We construct a family of positive normalized solutions v ε ∈ H 1 (R 3) which concentrates around the local maximum of K as ε → 0 by using a combination of the variational approach and a penalization technique. Moreover, we also give the expression of the approximation solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study multi-bump solutions of the following Kirchhoff type equation: - M ∫ R 2 | ∇ u | 2 d x Δ u + μ V (x) + h (x) u = λ f (u) in R 2 , where M is continuous with inf R + M > 0 , V ≥ 0 and its zero set has several disjoint bounded components, μ , λ are positive parameters, f has exponential critical growth. When V decays to zero at infinity, we use variational methods to obtain the existence and concentration behavior of multi-bump solutions. [ABSTRACT FROM AUTHOR]

In this paper, we studied the existence of multiple normalized solutions to the following Kirchhoff type equation: { − (a ε 2 + b ε ∫ R 3 | ∇ u | 2 d x) Δ u + V (x) u = μ u + f (u) i n R 3 , ∫ R 3 | u | 2 d x = m ε 3 , u ∈ H 1 (R 3) , where a , b , m > 0 , ε is a small positive parameter, V is a nonnegative continuous function, f is a continuous function with L 2 -subcritical growth and μ ∈ R will arise as a Lagrange multiplier. Under the suitable assumptions on V and f , the existence of multiple normalized solutions was obtained by using minimization techniques and the Lusternik-Schnirelmann theory. We pointed out that the number of normalized solutions was related to the topological richness of the set where the potential V attained its minimum value. [ABSTRACT FROM AUTHOR]

Let a > 0 , b > 0 and V (x) ≥ 0 be a coercive function in R 2 . We study the solutions with normalized L 2 -norm for the following Kirchhoff type equation - a + b ∫ R 2 | ∇ u | 2 d x Δ u + V (x) u = β | u | 2 u + λ u on a suitable weighted Sobolev space H = u ∈ H 1 (R 2) : ∫ R 2 V (x) u 2 d x < ∞. Our aim is to investigate the limit behaviors of the solutions with normalized L 2 -norm for this equation as (a , b) → (0 , 0) . Moreover, the uniqueness of the solution with normalized L 2 -norm for this equation is also discussed for a, b close to 0 [ABSTRACT FROM AUTHOR]

In this paper, we study the following fractional Kirchhoff type equation \begin{equation*} \begin{cases} \left(a+b\displaystyle\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy\right)(-\Delta)_p^su=|u|^{q-2}u\ln |u|^2+\frac{\lambda}{u^\gamma}, &\rm \mathrm{in}\ \Omega, \\ u>0, &\rm \mathrm{in}\ \Omega, \\ u=0, &\rm \mathrm{in}\ \mathbb{R}^N\backslash \Omega, \end{cases} \end{equation*} where $\Omega$ $\subset$ $\mathbb{R}^N$ is a bounded domain with Lipschitz boundary, $00, b\geq0$, $N>ps$, $2p0$ is a real parameter. By using the critical point theory for nonsmooth functionals and analytic techniques, the existence and multiplicity of positive solutions are obtained.

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In this paper, we study a class of Kirchhoff type equations with concave and convex nonlinearities and steep potential well. Firstly, we obtain a positive energy solution u b , λ + by a truncated functional. Furthermore, the concentration behavior of u b , λ + is also explored on the set V−1 (0) as λ → ∞. Secondly, we also give the existence of a negative solution u b , λ − via Ekeland variational principle. Finally, we show a nonexistence result of the nontrivial solutions. [ABSTRACT FROM AUTHOR]

In this article, we study the following general Kirchhoff type equation: −M∫R3∣∇u∣2dxΔu+u=a(x)f(u)inR3,-M\left(\mathop{\int }\limits_{{{\mathbb{R}}}^{3}}| \nabla u{| }^{2}{\rm{d}}x\right)\Delta u+u=a\left(x)f\left(u)\hspace{1em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{3}, where infR+M>0{\inf }_{{{\mathbb{R}}}^{+}}M\gt 0 and ff is a superlinear subcritical term. By using the Pohozǎev manifold, we obtain the existence of high energy solutions of the aforementioned equation without the well-known Ambrosetti-Rabinowitz type condition.

We investigate the following nonlinear system $ \begin{cases} -(a+ b\int _{\Omega}|\nabla u|^{2}dx)\Delta u+\phi u = \lambda u+\mu|u|^{2}u, \; \ x\in\Omega, \\ -\Delta\phi = u^{2}, \; \ x\in\Omega, \\ u = \phi = 0, \; \ x\in \partial\Omega, \end{cases} $ with $ a, b > 0 $, $ \lambda, \mu\in\mathbb{R} $, and $ \Omega\subset \mathbb{R}^{3} $ is bounded with smooth boundary. Let $ \lambda_{1} > 0 $ be the first eigenvalue of $ (-\Delta u, H^{1}_{0}(\Omega)) $. We get that for certain $ \widetilde{\mu} > 0 $ there exists at least one least energy sign-changing solution for the above system if $ \lambda < a\lambda_{1} $ and $ \mu > \widetilde{\mu} $. In addition, we remark that the nonlinearity $ \lambda u+\mu|u|^{2}u $ does not satisfy the growth conditions.

We investigate the following nonlinear system { − (a + b ∫ Ω | ∇ u | 2 d x) Δ u + ϕ u = λ u + μ | u | 2 u , x ∈ Ω , − Δ ϕ = u 2 , x ∈ Ω , u = ϕ = 0 , x ∈ ∂ Ω , with a , b > 0 , λ , μ ∈ R , and Ω ⊂ R 3 is bounded with smooth boundary. Let λ 1 > 0 be the first eigenvalue of (− Δ u , H 0 1 (Ω)). We get that for certain μ ~ > 0 there exists at least one least energy sign-changing solution for the above system if λ < a λ 1 and μ > μ ~ . In addition, we remark that the nonlinearity λ u + μ | u | 2 u does not satisfy the growth conditions. [ABSTRACT FROM AUTHOR]

This paper is dedicated to studying a (p;q)-Laplacian-Kirchhoff type equation. We prove the existence of three bounded solutions (one positive, one negative, and one nodal with precisely two nodal domains) by applying the Nehari manifold along with a quantitative deformation lemma and truncation technique. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence and multiplicity of weak solutions for discrete Kirchhoff type equations with Dirichlet boundary conditions. We first establish the existence of a nontrivial weak solution and three weak solutions, by using variational methods, mainly based on the mountain pass theorem of A. Ambrosetti and P.H. Rabinowitz and critical point theory. [ABSTRACT FROM AUTHOR]

In this paper, we will study the following nonlinear Kirchhoff-type equation: - a + λ ∫ R 3 | ∇ u | 2 d x ▵ u + V (x) u = f (u) , x ∈ R 3 , u ∈ H 1 (R 3) , u > 0 , x ∈ R 3 , where a , λ are positive constants, f ∈ C (R , R) is subcritical near infinity and superlinear near zero, and f also satisfies general Ambrosetti-Rabinowitz condition. Besides, V is a positive continuous potential satisfying 0 < V 0 ≤ V (x) ≤ lim inf | x | → ∞ V (x) = V ∞ . Under the conditions of f and V, it is not easy to obtain bounded Palais–Smale sequences, so there we use a cut-off technique to overcome. Moreover, in order to conquer the difficulties of the lack of compactness due to the unboundedness of the domain R 3 and V is a non-constant function, we use a new global compactness lemma to show the convergence of Palais–Smale sequences. Therefore, by the above methods, we prove the existence of positive solutions to the above equation for all λ > 0 small. [ABSTRACT FROM AUTHOR]

For the Kirchhoff type equation - a + b ∫ R 3 ∇ u 2 d x Δ u + V (x) u = (I α ∗ | u | p ) | u | p - 2 u in R 3 , where a , b > 0 , 0 < α < 3 , 2 < p < 3 + α and I α is the Riesz potential, we establish the existence of a positive ground state solution by using Nehari manifold technique and concentration compactness argument. The main novelty in our context is that the potential V exhibits a mixed behavior, i.e., V is periodic in some directions while tends to a positive constant in the remaining ones. [ABSTRACT FROM AUTHOR]

In this article, we deal with the following fractional $ p $-Kirchhoff type equation $ \begin{equation*} \begin{cases} M\left( \int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy\right)(-\Delta)_p^su=\frac{|u|^{p_\alpha^*-2}u}{|x|^\alpha}+\frac{\lambda}{|x|^\beta} , &\rm \mathrm{in}\ \ \Omega, \\ u>0, &\rm \mathrm{in}\ \ \Omega, \\ u=0, &\rm \mathrm{in}\ \ \mathbb{R}^N\backslash \Omega, \end{cases} \end{equation*} $ where $ \Omega\subset \mathbb{R}^N $ is a smooth bounded domain containing $ 0 $, $ (-\Delta)_p^s $ denotes the fractional $ p $-Laplacian, $ M(t)=a+bt^{k-1} $ for $ t\geq0 $ and $ k>1 $, $ a, b>0 $, $ \lambda>0 $ is a parameter, $ 0

We study the following Kirchhoff type equation:

In this paper, we consider the following Kirchhoff-type equation: $ -\left(a+b\int_{ \mathbb{R}^3}|\nabla u|^2dx\right)\Delta u+u = |u|^{p-1}u,\quad {\rm{in }}\; \mathbb{R}^3, $ where $ a $, $ b > 0 $, $ p \in (1, 5) $. By considering a minimization problem on a special constraint set, we prove that the above problem has at least one sign-changing solution for any $ p \in (1, 5) $. Our results (especially $ p \in (1, 3] $) can be regarded as an improvement on the existing results.

In this paper we study the existence and the multiplicity of nontrivial weak solutions for a fourth order variable exponent Kirchhoff type problem involving p(x)-biharmonic operator with changing sign weight and with no flux boundary condition. By using variational approach and the theory of variable exponent Sobolev spaces, we determine an interval of parameters for which this problem admits at least two nontrivial weak solutions. [ABSTRACT FROM AUTHOR]

In this paper, we aim to understand the characteristics of dynamical behaviour for a higher order Kirchhoff type systems with logarithmic nonlinearities. Based on the potential well method, the main ingredient of this study is to construct several conditions for initial data leading to the solution global existence in case of E (0) < d. On the other hand, we proved that if the solution lies in a smaller set compared with the stable set, we can estimate the decay rate of energy. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following Kirchhoff type equation - a + b ∫ R 3 | ∇ u | 2 Δ u + V (x) u = f (u) , x ∈ R 3 , where a , b > 0 and f ∈ C (R , R) , and the potential V ∈ C 1 (R 3 , R) is positive, bounded and satisfies suitable decay assumptions. By using a perturbation approach together with a new version of global compactness lemma of Kirchhoff type, we prove the existence and multiplicity of bound state solutions for the above problem with a general nonlinearity. We especially point out that neither the Ambrosetti-Rabinowitz condition with μ > 4 nor any monotonicity assumption on f is required. Moreover, the potential V may not be radially symmetry or coercive. As a prototype, the nonlinear term involves the power-type nonlinearity f (u) = | u | p - 2 u for p ∈ (2 , 6) . [ABSTRACT FROM AUTHOR]

In this paper, we consider a Neumann problem of Kirchhoff type equation \begin{equation*} \begin{cases} \displaystyle-\left(a+b\int_{\Omega}|\nabla u|^2dx\right)\Delta u+u= Q(x)|u|^4u+\lambda P(x)|u|^{q-2}u, &\rm \mathrm{in}\ \ \Omega, \\ \displaystyle\frac{\partial u}{\partial v}=0, &\rm \mathrm{on}\ \ \partial\Omega, \end{cases} \end{equation*} where $\Omega$ $\subset$ $\mathbb{R}^3$ is a bounded domain with a smooth boundary, $a,b>0$, $10$ is a real parameter, $Q(x)$ and $P(x)$ satisfy some suitable assumptions. By using the variational method and the concentration compactness principle, we obtain the existence and multiplicity of nontrivial solutions.

In this paper, we study the following semilinear Kirchhoff type equation: − ϵ 2 a + ϵ b ∫ R 3 | ∇ u | 2 Δ u + V (x) u = f (u) + u 5 , u > 0 , u ∈ H 1 (R 3) , where ϵ > 0 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 ϵ → 0. The proof is based on the method of Nehari manifold, penalization techniques and Ljusternik–Schnirelmann category theory. [ABSTRACT FROM AUTHOR]

The Neumann boundary value problem about a class of Kirchhoff type equations with sign? changing potential terms was studied. By means of the variational method and the decomposition process for the underly? ing space, the energy functional was proved to satisfy the mountain pass geometry. Then,the energy functional (PS) sequence was proved to have a strongly convergent subsequence. Finally,the existence of two nontrivial solutions was obtained by Ekeland’s variational principle and the mountain pass lemma. [ABSTRACT FROM AUTHOR]

In this paper, we consider the initial boundary value problem of a coupled viscoelastic Kirchhoff-type equations with degenerate damping:... Firstly, we prove a local existence theorem by using the Faedo-Galerkin approximations. Then, we study blow up of solutions when initial energy is positive. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we study the following Kirchhoff type problem \[ -\left(a+b\int_{\mathbb{R}^3} K(x)|\nabla u|^2dx\right)\hbox{div}(K(x)\nabla u)=\lambda K(x)|x|^{\beta}|u|^{q-2}u+K(x)|u|^{4}u,\quad x\in \mathbb{R}^3, \] where $K(x)=\exp({|x|^{\alpha}/4})$ with $\alpha\geq2$, $\beta=(\alpha-2)(6-q)/4$ and the parameters $a, b, \lambda >0$. When $6-\frac{4}{\alpha}0$. When $20$ small enough. In the proof we use variational methods.

Let a> 0,b > 0 and V(x)≥0 be a coercive function in R². We study the following constrained minimization problem on a suitable weighted Sobolev space H: ..., where Eab(u) is a Kirchhoff type energy functional defined on H by ... . It is known that, for some a*>0, ea(b) has no minimizer if b=0 and a≥a, but ea(b) has always a minimizer for any a≥0 if b>0. The aim of this paper is to investigate the limit behaviors of the minimizers of ea(b) as b→0+. Moreover, the uniqueness of the minimizers of ea(b) is also discussed for b close to 0. [ABSTRACT FROM AUTHOR]

In this article, we study the Kirchhoff type problem $$ -\Big(a+\epsilon\int_{\mathbb{R}^3} K(x)|\nabla u|^2dx\Big)\hbox{div} (K(x)\nabla u)=\lambda K(x)f(x)|u|^{q-2}u+K(x)|u|^{4}u, $$ where $x\in \mathbb{R}^3$, $10$ is small enough, and the parameters $a, \lambda >0$. Under some assumptions on $f(x)$, we establish the existence of two nonnegative nontrivial solutions and obtain uniform lower estimates for extremal values of the problem via variational methods.

In this study, we study a Kirchhoff type problem involving singular and critical nonlinearities. With aid of variational methods and concentration compactness principle, we prove that the problem admits a weak solution.

Abstract In this paper, we study the following nonlinear Kirchhoff type equation: − ( a + b ∫ R N | ∇ u | 2 d x ) △ u + V u = f ( u ) + h ( x ) , x ∈ R N , $$\begin{aligned} - \biggl(a+b \int_{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}\,dx \biggr) \triangle u+Vu=f(u)+h(x),\quad x\in\mathbb{R}^{N}, \end{aligned}$$ where a, b, V are positive constants, N = 2 $N=2$ or 3. Under appropriate assumptions on f and h, we get that the equation has two positive solutions by using variational methods.

We consider the existence of positive solutions to the following Kirchhoff type equation with critical exponent −a+b∫ℝ3|∇u|2dxΔu=λV(x)u+K(x)u5,u∈D1,2(ℝ3),where a>0, b>0, and λ>0 is a parameter. Under some assumptions, two positive solutions are obtained by means of variational methods. It is worth emphasizing that for the local minimal solution, we use an entirely new method, and its some properties are also obtained. [ABSTRACT FROM AUTHOR]

In this paper we address the following Kirchhoff type problem - Δ (g (| ∇ u | 2 2) u + u r) = a u + b u p in Ω , u > 0 in Ω , u = 0 on ∂ Ω , in a bounded and smooth domain Ω in R N . By using change of variables and bifurcation methods, we show, under suitable conditions on the parameters a, b, p, r and the non-linearity g, the existence of positive solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the Kirchhoff type equation − a + b ∫ R 3 | ∇ u | 2 d x Δ u + V (x) u = f (x , u) , x ∈ R 3 , u ∈ H 1 (R 3) , where a, b>0 are constants and V (x) is a given positive potential. The nonlinearity covers the pure power type f (x , u) = Q (x) | u | p − 2 u with 2

Abstract We are concerned with ground-state solutions for the following Kirchhoff type equation with critical nonlinearity: { − ( ε 2 a + ε b ∫ R 3 | ∇ u | 2 ) Δ u + V ( x ) u = λ W ( x ) | u | p − 2 u + | u | 4 u in R 3 , u > 0 , u ∈ H 1 ( R 3 ) , $$\textstyle\begin{cases} - ({\varepsilon^{2}}a + \varepsilon b\int_{{\mathbb{R}^{3}}} {{{ \vert {\nabla u} \vert }^{2}}} )\Delta u + V(x)u = \lambda W(x){ \vert u \vert ^{p - 2}}u + { \vert u \vert ^{4}}u\quad {\text{in }}{\mathbb{R}^{3}} ,\\ u > 0, \quad\quad u \in{H^{1}}({\mathbb{R}^{3}}) , \end{cases} $$ where ε is a small positive parameter, a , b > 0 $a,b>0$ , λ > 0 $\lambda > 0$ , 2 < p ≤ 4 $2 < p \le4$ , V and W are two potentials. Under proper assumptions, we prove that, for ε > 0 $\varepsilon > 0$ sufficiently small, the above problem has a positive ground-state solution u ε ${u_{\varepsilon}}$ 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 ε ${u_{\varepsilon}}$ is concentrated around a set which is related to the set where the potential V ( x ) $V(x)$ attains its global minima or the set where the potential W ( x ) $W(x)$ attains its global maxima as ε → 0 $\varepsilon \to0$ .

In this paper, we establish two critical points theorems for functional which is not defined on the whole space, but C 1 on a closed subset. As applications, we study the existence and multiplicity of solutions for a class of Kirchhoff type equations with bounded nonlocal term. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of weak solutions to a class of nonlinear elliptic Navier boundary value problem involving the p(x)-Kirchhoff type triharmonic operator. By means of a variational approach and the theory of the variable exponent Sobolev spaces, we establish conditions ensuring the existence of weak solutions for the problem. [ABSTRACT FROM AUTHOR]

In the paper, we consider the following non-autonomous Kirchhoff type equation − (a + b ∫ R 3 | ∇ u | 2 d x) △ u + V (x) u = g (u) , x ∈ R 3 , u > 0 , x ∈ R 3 , u ∈ H 1 (R 3) , where a , b are positive constants, V satisfies mild assumptions and g verifies the Berestycki–Lions conditions. By using the variational methods, we show the equation has a ground state solution. [ABSTRACT FROM AUTHOR]

Abstract With the help of the Nehari manifold, a Kirchhoff type equation involving two potentials was considered. Under new assumptions, a ground state solution was obtained. [ABSTRACT FROM AUTHOR]

In this work, we study the following Kirchhoff type problem -(a+b∫Ω|∇u|pdx)Δpu=g(x)u-γ+λf(x,u),inΩ,u=0,on∂Ω,where p≥2, Ω is a regular bounded domain in RN, (N≥3). Firstly, for p>2, we prove under some appropriate conditions on the singularity and the nonlinearity the existence of nontrivial weak solution to this problem. For p=2, we show, under supplementary condition, the positivity of this solution. Moreover, in the case λ=0 we prove an uniqueness result. We use the variational method to prove our main results. [ABSTRACT FROM AUTHOR]

In this article, we study the existence and multiplicity of positive solutions for a class of Kirchhoff type problems involving changing-sign potential and critical growth terms. Using the concentration compactness principle and Nehari manifold, we obtain the existence and multiplicity of nonzero non-negative solutions.

In this article, we study a Kirchhoff type problem with nonlinear Neumann boundary conditions on a bounded domain. By using variational methods, we prove the existence of infinitely many solutions.

This article concerns the existence of sign-changing solutions to nonlocal Kirchhoff type problems of the form $$ -\Big(a+b\int_\Omega|\nabla u|^2dx\Big)\Delta u=f(x,u) \text{ in }\Omega,\quad u=0 \text{ on }\partial\Omega, $$ where $\Omega$ is a bounded domain in $\mathbb{R}^N$ ($N=1,2,3$) with smooth boundary, $a>0$, $b\geq0$, and $f:\overline{\Omega}\times\mathbb{R}\to\mathbb{R}$ is a continuous function. We first establish a new sign-changing version of the symmetric mountain pass theorem and then apply it to prove the existence of a sequence of sign-changing solutions with higher and higher energy.