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We are concerned with the Kirchhoff problem −M∫Ω|∇u|2Δu=−λ|u|q−2u+au+|u|p−2uinΩ,u=0on∂Ω,$$ \left\{\begin{array}{ll}-M\left({\int}_{\Omega}{\left|\nabla u\right|}^2\right)\Delta u=-\lambda {\left|u\right|}^{q-2}u+ au+{\left|u\right|}^{p-2}u& \mathrm{in}\kern0.20em \Omega, \\ {}u=0& \mathrm{on}\kern0.20em \mathrm{\partial \Omega },\end{array}\right. $$ where N≥1,Ω⊂ℝN$$ N\ge 1,\Omega \subset {\mathrm{\mathbb{R}}}^N $$ is a bounded domain with a regular boundary ∂Ω,a,λ>0$$ \mathrm{\partial \Omega },a,\lambda >0 $$, and 1

This paper discusses a class of Kirchhoff boundary value problem that involve the fractional T(·)-Laplacian operator of the form where fl is a smooth bounded domain in RN, (N 2 2),0

In this paper, we study the existence of ground state solution for the following Kirchhoff equation with combined nonlinearities \[ -\left(a+b\displaystyle\int_{\mathbb{R}^N} |\nabla u|^2\right)\Delta u=\lambda u+|u|^{p-2}u+\mu |u|^{q-2}u \ \hbox{in}\ \mathbb{R}^N, \quad 1\leq N\leq 3 \] − (a + b ∫ R N | ∇u | 2) Δu = λu + | u | p − 2 u + μ | u | q − 2 u in R N , 1 ≤ N ≤ 3 with prescribed mass $ \int _{\mathbb {R}^N} u^2=c^2 $ ∫ R N u 2 = c 2 , where $ a{ \gt }0,b{ \gt }0,c{ \gt }0 $ a > 0 , b > 0 , c > 0 , $ \mu { \lt }0 $ μ < 0 , $ 2{ \lt }q\leq 2+\frac {8}{N}{ \lt }p{ \lt }2^{*} $ 2 < q ≤ 2 + 8 N < p < 2 ∗ . Under certain assumptions on the parameter μ, we obtain the existence of normalized solution $ (\tilde {u},\lambda _c)\in S_c \times \mathbb {R} $ (u ~ , λ c) ∈ S c × R , where $ S_c=\{ u\in H^1(\mathbb {R}^N): \int _{\mathbb {R}^N} u^2=c^2 \} $ S c = { u ∈ H 1 (R N) : ∫ R N u 2 = c 2 }. [ABSTRACT FROM AUTHOR]

In this present paper, we concern investigating nonlinear Kirchhoff-type problems subject to Dirichlet boundary conditions, incorporating nonlocal terms and logarithmic nonlinearity in the ϕ -Hilfer fractional spaces with the η (·) -Laplacian operator by means of the do Mountain Pass Theorem, Fountain Theorem and Dual Fountain Theorem. [ABSTRACT FROM AUTHOR]

This paper deals with the following critical Choquard equation with a Kirchhoff type perturbation in bounded domains, \[ \begin{cases} -(1+b\|u\|^{2})\Delta u=\left(\int_{\Omega}\frac{u^{2}(y)}{|x-y|^{4}}\,{\rm d}y\right)u +\lambda u & {\rm in}\ \Omega,\\ u=0 & {\rm on}\ \partial\Omega, \end{cases} \] { − (1 + b ‖ u ‖ 2) Δu = (∫ Ω u 2 (y) | x − y | 4 d y) u + λu in Ω , u = 0 on ∂Ω , where $ \Omega \subset \mathbb {R}^{N}(N\geq 5) $ Ω ⊂ R N (N ≥ 5) is a smooth bounded domain and $ \|\cdot \| $ ‖ ⋅ ‖ is the standard norm of $ H_{0}^{1}(\Omega) $ H 0 1 (Ω). Under the suitable assumptions on the constant $ b\geq 0 $ b ≥ 0 , we prove the existence of solutions for $ 0 \lt \lambda \leq \lambda _{1} $ 0 < λ ≤ λ 1 , where $ \lambda _{1} \gt 0 $ λ 1 > 0 is the first eigenvalue of $ -\Delta $ − Δ on Ω. Moreover, we prove the multiplicity of solutions for $ \lambda \gt \lambda _{1} $ λ > λ 1 and b>0 in suitable intervals. [ABSTRACT FROM AUTHOR]

Considering the solutions of a class of noncooperative Kirchhoff-type p (x) -Laplacian elliptic systems with nonlinear boundary conditions, we derive a sequence of solutions utilizing both the variational method and limit index theory under certain underlying assumptions. The novelty of this study is that we verify the (P S) c * condition using another method, diverging from the approaches cited in the previous literature. [ABSTRACT FROM AUTHOR]

In this work, we study the existence of the least energy sign-changing solution for the following degenerate Kirchhoff-type problem involving the fractional N/s-Laplacian with logarithmic and both subcritical and critical exponential nonlinearities: \[ \begin{cases} \displaystyle M\left(\int_{\mathbb R^{2N}}\frac{|u(x) - u(y)|^{\frac{N}{s}}}{|x - y|^{2N}}\,{\rm d}x\,{\rm d}y \right)(-\Delta)_{N/s}^s u \\ \quad = |u|^{q-2}u\ln |u|^2 + \mu f(u), & {\rm in}\ \Omega,\\ u=0, & {\rm in}\ \mathbb R^N\setminus \Omega, \end{cases} \] { M (∫ R 2 N | u (x) − u (y) | N s | x − y | 2 N d x d y) (− Δ) N / s s u = | u | q − 2 uln ⁡ | u | 2 + μf (u) , in Ω , u = 0 , in R N ∖ Ω , where $ \Omega \subset \mathbb {R}^N $ Ω ⊂ R N is a bounded domain. The proof is based on constrained variational method, fractional Trudinger–Moser inequality, quantitative deformation lemma and Brouwer's degree theory in Nehari sets. To be more precise, the least energy sign-changing solution is obtained by minimizing the energy functional on the sign-changing Nehari manifold. [ABSTRACT FROM AUTHOR]

This article aims to study the existence of weak solutions for the Kirchhoff type double‐phase problem where x∈ℝN$$ x\in {\mathrm{\mathbb{R}}}&#x0005E;N $$ and λ$$ \lambda $$ is a real parameter. The main distinctive feature of this problem lies in the second nonlinearity on the right‐hand side, which can be in the supercritical. Additionally, we encounter the Kirchhoff term, which might become zero at the origin. By imposing certain assumptions, we establish the existence and multiplicity results. [ABSTRACT FROM AUTHOR]

In this paper, we study the following nonautonomous Kirchhoff problem: −1+b∫ℝN|∇u|2dxΔu+V(x)u=a(x)|u|p−2u+λ|u|q−2u,x∈ℝN,u∈H1ℝN,$$ \left\{\begin{array}{l}-\left(1&#x0002B;b{\int}_{{\mathrm{\mathbb{R}}}&#x0005E;N}{\left&#x0007C;\nabla u\right&#x0007C;}&#x0005E;2\mathrm{d}x\right)\Delta u&#x0002B;V(x)u&#x0003D;a(x){\left&#x0007C;u\right&#x0007C;}&#x0005E;{p-2}u&#x0002B;\lambda {\left&#x0007C;u\right&#x0007C;}&#x0005E;{q-2}u,x\in {\mathrm{\mathbb{R}}}&#x0005E;N,\\ {}u\in {H}&#x0005E;1\left({\mathrm{\mathbb{R}}}&#x0005E;N\right),\end{array}\right. $$where N≥2$$ N\ge 2 $$, p∈(1,2∗)$$ p\in \left(1,{2}&#x0005E;{\ast}\right) $$, q∈[2,2∗)$$ q\in \left[2,{2}&#x0005E;{\ast}\right) $$, b$$ b $$ is a positive constant, λ≥0$$ \lambda \ge 0 $$ is a parameter, and the potential functions a,V$$ a,V $$ belong to C(ℝN,(0,+∞))$$ C\left({\mathrm{\mathbb{R}}}&#x0005E;N,\left(0,&#x0002B;\infty \right)\right) $$. The existence of radially symmetric and positive solution to the above problem is first established for all λ≥0$$ \lambda \ge 0 $$ when a,V$$ a,V $$ are radially symmetric and p,q∈(4,2∗)$$ p,q\in \left(4,{2}&#x0005E;{\ast}\right) $$, and the range of q$$ q $$ can be extended to q∈(2,2∗)$$ q\in \left(2,{2}&#x0005E;{\ast}\right) $$ with the aid of a coercive type assumption on V$$ V $$. Moreover, we show the existence of infinitely many solutions with high energies via the fountain theorem under more general assumption on V$$ V $$ which allows it to be sign‐changing. When p∈(1,2)$$ p\in \left(1,2\right) $$ and q∈(2,2∗)$$ q\in \left(2,{2}&#x0005E;{\ast}\right) $$, we show that the above problem possesses infinitely many solutions with negative critical values for small λ$$ \lambda $$ provided that the function V−1$$ {V}&#x0005E;{-1} $$ belongs to a suitable space. In particular, by imposing a hypothesis on the potential V$$ V $$ controlling its growth at infinity, we obtain a nonradial solution via the mountain pass theorem and the principle of symmetric criticality. [ABSTRACT FROM AUTHOR]

In this paper, we are devoted to studying two classes of non-autonomous Kirchhoff problems with the potential term. To be precise, we assume the potential term satisfies different assumptions and the nonlinearity enjoys exponential critical growth. Based on the Trudinger–Moser inequality, we derive the existence of normalized solutions for two classes of Kirchhoff problems. Our results are new and improve and complement some related literature. [ABSTRACT FROM AUTHOR]

In this work, we solve a Kirchhoff type boundary problem governed by the nonlocal fractional p-Laplacian operator M ∬ R 2 N | u (x) - u (y) | p K (x - y) d x d y L K p u = f (x , u) , in Ω , u = 0 on R N \ Ω , where L K p is a non-local operator with singular kernel K, Ω is an open bounded subset of R N with smooth boundary ∂ Ω , M is a continuous function and the nonlinearity f is a Caratheodory function which does not verify the Ambrosetti–Rabinowitz type condition. Through some adequate assumptions, we prove the existence of nontrivial weak solutions to our problem by applying new skills and variational methods. [ABSTRACT FROM AUTHOR]

In the present paper, we apply a global branch approach to study the existence, non-existence and multiplicity of positive normalized solutions $ (\lambda_c, u_c)\in \mathbb{R}\times H^1(\mathbb{R}^N) $ to the following Kirchhoff problem$ -\left(a+b\int_{\mathbb{R}^N}|\nabla u|^2dx\right)\Delta u+\lambda u = g(u)\; \hbox{in}\; \mathbb{R}^N, \;N\geq 1 $satisfying the normalization constraint $ \int_{\mathbb{R}^N}u^2 = c, $ which appears in free vibrations of elastic strings. The parameters $ a, b>0 $ are prescribed as well as the mass $ c>0 $. Due to the presence of the non-local term $ b\int_{\mathbb{R}^N}|\nabla u|^2dx \Delta u $, such problems lack the mountain pass geometry in the higher dimension case $ N\geq 5 $. Our result seems to be the first attempt in this aspect. [ABSTRACT FROM AUTHOR]

In the present paper, we prove the existence of the solutions (λ, u) ∈ ℝ × H1(ℝ3) to the following Kirchhoff equations with the Hartree-type nonlinearity under the general mass supercritical settings, {-(a+b∫ℝ3|∇u|2dx)Δu-λu=[Iα*(K(x)F(u))]K(x)f(u),u∈H1(ℝ3),\left\{ {\matrix{{ - \left( {a + b\int\limits_{{\mathbb{R}^3}} {{{\left| {\nabla u} \right|}^2}dx} } \right)\Delta u - \lambda u = \left[ {{I_\alpha }*\left( {K\left( x \right)F\left( u \right)} \right)} \right]K\left( x \right)f\left( u \right),} \hfill \cr {u \in {H^1}\left( {{\mathbb{R}^3}} \right),} \hfill \cr } } \right. where a, b > 0 are prescribed, Iα = |x|α−3 is the riesz potential where α ∈ (0, 3), K ∈ 𝒞1(ℝ3, ℝ+) satisfies an explicit assumption and f ∈𝒞 (ℝ, ℝ) satisfies some weak conditions, we develop some new tricks for dealing with the Hartree-type term to overcome the difficulties produced by the appearance of non-constant potential K(x). This paper extends and promotes the previous results on prescribed L2-norm solutions of the Kirchhoff-type equation.

In this paper the authors study the existence of positive radial solutions to the Kirchhoff type problem involving the \(p\)-Laplacian \[-\Big(a+b\int_{\Omega_e}|\nabla u|^p dx\Big)\Delta_p u=\lambda f\left(|x|,u\right),\ x\in \Omega_e,\quad u=0\ \text{on} \ \partial\Omega_e,\] where \(\lambda \gt 0\) is a parameter, \(\Omega_e = \lbrace x\in\mathbb{R}^N : |x|\gt r_0\rbrace\), \(r_0\gt 0\), \(N \gt p \gt 1\), \(\Delta_p\) is the \(p\)-Laplacian operator, and \(f\in C(\left[ r_0, +\infty\right)\times\left[0,+\infty\right),\mathbb{R})\) is a non-decreasing function with respect to its second variable. By using the Mountain Pass Theorem, they prove the existence of positive radial solutions for small values of \(\lambda\).

In the present paper, we study the existence of normalized solutions (u c , λ c) ∈ H 1 (R 3) × R to the following Kirchhoff problem - a + b ∫ R 3 | ∇ u | 2 d x Δ u + V (x) u + λ u = g (u) + | u | 4 u in R 3 , satisfying the normalization constraint ∫ R 3 u 2 d x = c , where a , b , c > 0 are prescribed constants, and the nonlinearities g(s) are very general and of mass super-critical. Under some suitable assumptions on V(x) and g(u), we will prove that the above problem has a ground state normalized solutions for any given c > 0 , by studying a constraint problem on a Nehari–Pohozaev manifold. [ABSTRACT FROM AUTHOR]

We study a class of $ p(x) $-Kirchhoff problems which is seldom studied because the nonlinearity has nonstandard growth and contains a bi-nonlocal term. Based on variational methods, especially the Mountain pass theorem and Ekeland's variational principle, we obtain the existence of two nontrivial solutions for the problem under certain assumptions. We also apply the Symmetric mountain pass theorem and Clarke's theorem to establish the existence of infinitely many solutions. Our results generalize and extend several existing results. [ABSTRACT FROM AUTHOR]

In present paper, we study the normalized solutions (λ c , u c) ∈ R × H 1 (R N) to the following Kirchhoff problem − (a + b ∫ R N | ∇ u | 2 d x) Δ u + λ u = g (u) in R N , 1 ≤ N ≤ 3 satisfying the normalization constraint ∫ R N u 2 = c , which appears in free vibrations of elastic strings. The parameters a , b > 0 are prescribed as is the mass c > 0. The nonlinearities g (s) considered here are very general and of mass super-critical. Under some suitable assumptions, we can prove the existence of ground state normalized solutions for any given c > 0. After a detailed analysis via the blow up method, we also make clear the asymptotic behavior of these solutions as c → 0 + as well as c → + ∞. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the (p, q)-Kirchhoff type problem - 1 + a ∫ R N | ∇ u | p d x Δ p u - 1 + b ∫ R N | ∇ u | q d x Δ q u + V ε (x) | u | p - 2 u + | u | q - 2 u = f (u) in R N , where ε > 0 is a small parameter, a , b > 0 , 1 < p < q < N < 2 q , Δ s u = div (| ∇ u | s - 2 ∇ u) , with s ∈ { p , q } , is the s-Laplacian, V : R N → R is a continuous potential, V ε (x) = V (ε x) and f : R → R is a continuous non-linearity without satisfying usual Ambrosetti–Rabinowitz condition. By using Nehari manifold techniques and perturbation methods, we establish the existence of positive ground state solution, and by those methods combined with Ljusternik–Schnirelmann theory, we investigate the multiplicity and concentration of solutions. In particular, our results extend and improve the works of He and Zou (J Differ Equ 252:1813–1834, 2012). [ABSTRACT FROM AUTHOR]

We consider a Kirchhoff problem of Brezis–Nirenberg type in a smooth bounded domain of R 4 with Dirichlet boundary conditions. Our approach, novel in this framework and based upon approximation arguments, allows us to cope with the interaction between the higher order Kirchhoff term and the critical nonlinearity, typical of the dimension four. We derive several existence results of positive solutions, complementing and improving earlier results in the literature. In particular, we provide explicit bounds of the parameters b and λ coupled, respectively, with the higher order Kirchhoff term and the subcritical nonlinearity, for which the existence of solutions occurs. [ABSTRACT FROM AUTHOR]

This article is concerned with the following nonlinear supercritical elliptic problem: −M(‖∇u‖22)Δu=f(x,u),inB1(0),u=0,on∂B1(0),\left\{\begin{array}{ll}-M(\Vert \nabla u{\Vert }_{2}^{2})\Delta u=f\left(x,u),& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{B}_{1}\left(0),\\ u=0,& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial {B}_{1}\left(0),\end{array}\right. where B1(0){B}_{1}\left(0) is the unit ball in R2{{\mathbb{R}}}^{2}, M:R+→R+M:{{\mathbb{R}}}^{+}\to {{\mathbb{R}}}^{+} is a Kirchhoff function, and f(x,t)f\left(x,t) has supercritical exponential growth on tt, which behaves as exp[(β0+∣x∣α)t2]\exp {[}({\beta }_{0}+| x\hspace{-0.25em}{| }^{\alpha }){t}^{2}] and exp(β0t2+∣x∣α)\exp ({\beta }_{0}{t}^{2+| x{| }^{\alpha }}) with β0{\beta }_{0}, α>0\alpha \gt 0. Based on a deep analysis and some detailed estimate, we obtain Nehari-type ground state solutions for the above problem by variational method. Moreover, we can determine a fine upper bound for the minimax level under weaker assumption on liminft→∞tf(x,t)exp[(β0+∣x∣α)t2]{\mathrm{liminf}}_{t\to \infty }\frac{tf\left(x,t)}{\exp {[}({\beta }_{0}+| \hspace{-0.25em}x\hspace{-0.25em}{| }^{\alpha }){t}^{2}]} and liminft→∞tf(x,t)exp(β0t2+∣x∣α){\mathrm{liminf}}_{t\to \infty }\frac{tf\left(x,t)}{\exp ({\beta }_{0}{t}^{2+| x{| }^{\alpha }})}, respectively. Our results generalize and improve the ones in G. M. Figueiredo and U. B. Severo (Ground state solution for a Kirchhoff problem with exponential critical growth, Milan J. Math. 84 (2016), no. 1, 23–39.) and Q. A. Ngó and V. H. Nguyen (Supercritical Moser-Trudinger inequalities and related elliptic problems, Calc. Var. Partial Differ. Equ. 59 (2020), no. 2, Paper No. 69, 30.) for M(t)=1M(t)=1. In particular, if the weighted term ∣x∣α| x\hspace{-0.25em}{| }^{\alpha } is vanishing, we can obtain the ones in S. T. Chen, X. H. Tang, and J. Y. Wei (2021) (Improved results on planar Kirchhoff-type elliptic problems with critical exponential growth, Z. Angew. Math. Phys. 72 (2021), no. 1, Paper No. 38, Theorem 1.3 and Theorem 1.4) immediately.

The objective of this work is to investigate a nonlocal problem involving singular and critical nonlinearities: ([ u ] s , p p) σ − 1 (− Δ) p s u = λ u γ + u p s ∗ − 1 in Ω , u > 0 , in Ω , u = 0 , in R N ∖ Ω , where Ω is a bounded domain in R N with the smooth boundary ∂ Ω , 0 < s < 1 < p < ∞ , N > s p , 1 < σ < p s ∗ / p , with p s ∗ = N p N − p s , (− Δ) p s is the nonlocal p-Laplace operator and [ u ] s , p is the Gagliardo p-seminorm. We combine some variational techniques with a truncation argument in order to show the existence and the multiplicity of positive solutions to the above problem. [ABSTRACT FROM AUTHOR]

In the present paper, we prove the existence of the solutions (λ, u) ∈ R × H¹ (R³ ) to the following Kirchhoff equations with the Hartree-type nonlinearity under the general mass supercritical settings, − (a + b Z R³ |∇u| 2 dx) ∆u − λu = [Iα ∗ (K(x)F(u))] K(x)f(u), u ∈ H¹ (R³) where a, b > 0 are prescribed, Iα = |x| α−3 is the riesz potential where α ∈ (0, 3), K ∈ C 1 (R³, R+) satisfies an explicit assumption and f ∈ C(R, R) satisfies some weak conditions, we develop some new tricks for dealing with the Hartree-type term to overcome the difficulties produced by the appearance of non-constant potential K(x). This paper extends and promotes the previous results on prescribed L² -norm solutions of the Kirchhoff-type equation. [ABSTRACT FROM AUTHOR]

In this paper the authors study the existence of positive radial solutions to the Kirchhoff type problem involving the p-Laplacian − ( a + b ∫Ωe |∇u|pdx) ∆pu = λf (|x|, u), x ∈ Ωe, u = 0 on ∂Ωe, where λ > 0 is a parameter, Ωe = {x ∈ RN : |x| > r0}, r0 > 0, N > p > 1, ∆p is the p-Laplacian operator, and f ∈ C([r0, +∞) × [0, +∞), R) is a non-decreasing function with respect to its second variable. By using the Mountain Pass Theorem, they prove the existence of positive radial solutions for small values of λ. [ABSTRACT FROM AUTHOR]

In the present paper, we deal with the existence of nontrivial solutions and Nehari‐type ground state solutions to the Kirchhoff type elliptic equations of the form: −M‖∇u‖NNΔNu=f(x,u),inΩ,u=0,on∂Ω,where Ω⊂ℝN(N≥2) is a smooth bounded domain, M:ℝ+→ℝ+ is a Kirchhoff function, and f(x, s) has critical exponential growth on s which behaviors as eα0s2 with α0 > 0. Based on a deep analysis and some detailed estimate, we can determine a fine upper bound for the minimax level under weaker assumption on lim inft→∞tf(x,t)expα0tN/(N−1). Our results generalize and improve the ones in Chen et al. (2021) (Lemma 3.1) and Figueiredo and Severo (2016) (Theorems 1.3 and 1.4) for N=2 and Goyal et al. (2016) (Theorem 1.3) for N ≥ 2. [ABSTRACT FROM AUTHOR]

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In this paper, we consider a class of Kirchhoff problems as follows (1) − a + b ∫ R N | ∇ u | 2 Δ u = (1 + ε K (x)) u 2 ∗ − 1 , u > 0 i n R N , where a, b>0 are given constants, ϵ is a small parameter and 2 ∗ = 2 N / (N − 2) , N ≥ 3. We first prove the nondegeneracy of positive solutions to Equation (1) when ε = 0. Then, as an application, using a perturbed argument and finite-dimensional reduction method, we prove the existence of positive solutions of Equation (1) for ϵ small. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence of normalized solutions to the following Kirchhoff-type equation - a + b ∫ R 3 | ∇ u | 2 d x Δ u - λ u = K (x) f (u) , x ∈ R 3 ; u ∈ H 1 (R 3) , where a , b > 0 , λ is unknown and appears as a Lagrange multiplier, K ∈ C (R 3 , R +) with 0 < lim | y | → ∞ K (y) ≤ inf R 3 K , and f ∈ C (R , R) satisfies general L 2 -supercritical or L 2 -subcritical conditions. We introduce some new analytical techniques in order to exclude the vanishing and the dichotomy cases of minimizing sequences due to the presence of the potential K and the lack of the homogeneity of the nonlinearity f. This paper extends to the nonautonomous case previous results on prescribed L 2 -norm solutions of Kirchhoff problems. [ABSTRACT FROM AUTHOR]

In this paper, we consider a type of Kirchhoff problem as follows − (a + b ∫ R N | ∇ u | 2 ) Δ u = (1 + ε K (x)) u 2 ∗ − 1 , u > 0 , i n R N , where a, b>0 are given constants, ε > 0 is a small parameter and 2 ∗ = 2 N / (N − 2) , (N ≥ 3). We show that if K (x) has k critical points near which K (x) satisfies some expansion assumption, then by Lyapunov–Schmidt reduction method, we construct multi-peak solutions for ε > 0 small, which concentrate at the k critical points of K (x). [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of nontrivial solutions and Nehari-type ground-state solutions for the following Kirchhoff-type elliptic equation: - m (‖ ∇ u ‖ 2 2) Δ u = f (x , u) , in Ω , u = 0 , on ∂ Ω , where Ω ⊂ R 2 is a smooth bounded domain, m : R + → R + is a Kirchhoff function, and f has critical exponential growth in the sense of Trudinger–Moser inequality. We develop some new approaches to estimate precisely the minimax level of the energy functional and prove the existence of Nehari-type ground-state solutions and nontrivial solutions for the above problem. Our results improve and extend the previous results. In particular, we give a more precise estimation than the ones in the existing literature about the minimax level, and also give a simple proof of a known inequality due to P.L. Lions. [ABSTRACT FROM AUTHOR]

We consider a class of fourth-order elliptic equations of Kirchhoff type with critical growth in ℝN. By using constrained minimization in the Nehari manifold, we establish sufficient conditions for the existence of nodal (that is, sign-changing) solutions. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to study the existence of solutions for the following nonhomogeneous singular Kirchhoff problem involving the p (x) -biharmonic operator: { M (t) (Δ p (x) 2 u + a (x) | u | p (x) − 2 u) = g (x) u − γ (x) ∓ λ f (x , u) , in Ω , Δ u = u = 0 , on ∂ Ω , where Ω ⊂ R N , (N ≥ 3) be a bounded domain with C 2 boundary, λ is a positive parameter, γ : Ω ‾ ⟶ (0 , 1) be a continuous function, p ∈ C (Ω ‾) with 1 < p − : = inf x ∈ Ω p (x) ≤ p + : = sup x ∈ Ω p (x) < N 2 , as usual, p ∗ (x) = N p (x) N − 2 p (x) , g ∈ L p ∗ (x) p ∗ (x) + γ (x) − 1 (Ω) . We assume that M (t) is a continuous function with t : = ∫ Ω 1 p (x) (| Δ u | p (x) + a (x) | u | p (x)) d x , and assumed to verify assertions (M1)-(M3) in Sect. 3, moreover f (x , u) are assumed to satisfy assumptions (f1)-(f6). In the proofs of our results we use variational techniques and monotonicity arguments combined with the theory of the generalized Lebesgue Sobolev spaces. [ABSTRACT FROM AUTHOR]

This paper is concerned by the study of the existence of nonnegative and nonpositive solutions for a nonlocal quasilinear Kirchhoff problem by using the Mountain Pass lemma technique. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of three solutions for a Kirchhoff equation involving the nonlocal fractional p‐Laplacian considering Sobolev and Hardy nonlinearities at subcritical and critical growths. The proof is based on mountain pass theorem and constrained minimization in Nehari sets. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we devote ourselves to investigating a nonlocal problem involving singularity and asymptotically linear nonlinearities. By using the variational and perturbation methods, we obtain the existence of two positive solutions which improve the existing result in the literature. [ABSTRACT FROM AUTHOR]

The aim of this paper is to discuss the existence and multiplicity of solutions for the following fractional p-Kirchhoff type problem involving the critical Sobolev exponent and discontinuous nonlinearity: ... where M(t)=a+btθ-1 for t≥0, a≥0,b>0,θ>1, (-Δ)sp is the fractional p-Laplacian with 0s=Np/(N-ps) is the critical Sobolev exponent, λ>0 is a parameter, and f:RN×R→R is a function. Under suitable assumptions on f, we show that there exists λ0>0 such that the above equation admits at least one nontrivial nonnegative solution provided λ<λ0 by using the nonsmooth critical point theory for locally Lipschitz functionals. Furthermore, for any k∊N, there exists λk>0 such that the above equation has k pairs of nontrivial solutions if λ<λk. The main feature is that our paper covers the degenerate case, that is the coefficient of (-Δ)sp may be zero at zero. Moreover, the existence results are obtained when f is discontinuous. Thus, our results are new even in the semilinear case. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of solutions for the Kirchhoff problem M(∫ℝ3|∇u|2dx+∫ℝ3(λa(x)+1)u2dx)(-Δu+(λa(x)+1)u)=f(u)$M\Biggl (\int _{\mathbb {R}^{3}}|\nabla u|^{2}\, dx + \int _{\mathbb {R}^{3}} (\lambda a(x)+1)u^{2}\, dx\Biggl ) (- \Delta u + (\lambda a(x)+1)u) = f(u)$ in ℝ3, u ∈ H1(ℝ3) Assuming that the nonnegative function a(x) has a potential well with int (a -1({0})) consisting of k disjoint components Ω1,Ω2,...,Ωk and the nonlinearity f(t) has a subcritical growth, we are able to establish the existence of positive multi-bump solutions by using variational methods.

We propose a numerical scheme to obtain an approximate solution of a nonlocal elliptic Kirchhof-type problem. We first reduce the problem to a nonlinear finite dimensional system by a Legendre–Galerkin spectral method and then solve it by an iterative process. Convergence of the iterative process and an error estimation of the approximate solution is provided. Numerical experiments are conducted to illustrate the performance of the proposed method. [ABSTRACT FROM AUTHOR]

This paper concerns with the existence of solutions for the following fractional Kirchhoff problem with critical nonlinearity: where (−Δ) is the fractional Laplacian operator with 0 < s < 1, 2* = 2 N/( N − 2 s), N > 2 s, p ∈ (1, 2*), θ ∈ [1, 2*/2), h is a nonnegative function and λ a real positive parameter. Using the Ekeland variational principle and the mountain pass theorem, we obtain the existence and multiplicity of solutions for the above problem for suitable parameter λ > 0. Furthermore, under some appropriate assumptions, our result can be extended to the setting of a class of nonlocal integro-differential equations. The remarkable feature of this paper is the fact that the coefficient of fractional Laplace operator could be zero at zero, which implies that the above Kirchhoff problem is degenerate. Hence our results are new even in the Laplacian case. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence of solutions for a Kirchhoff-type system with Dirichlet boundary condition and nonlinearities having exponential critical growth. Our approach is based on the Trudinger-Moser inequality and on a minimax theorem. [ABSTRACT FROM AUTHOR]

We explore the multiplicity of weak solutions to a nonlocal equation ingoverned by a fractional power of thep-Laplacian combined with a Kirchhoff-type term. The nonlinearity involves two real parameters, one of which is of perturbing nature, and can overpower the critical fractional exponent. Our arguments rely upon the variational apparatus; the lack of compactness of the Sobolev embeddings is circumvented by the use of convenient weight functions. [ABSTRACT FROM AUTHOR]

In this article we prove necessary and sufficient conditions for the existence of local and global bifurcations of relative equilibria of the 8-vortex problem. Applying the equivariant bifurcation theory we prove that from the families of regular concentric squares forming relative equilibria, bifurcate relative equilibria whose shape is more complicated. [ABSTRACT FROM AUTHOR]

We show the existence of a nodal solution (sign-changing solution) for a Kirchhoff equation of the type [ABSTRACT FROM AUTHOR]

In this paper, by the Leray-Schauder principle, we study the existence of solutions for the following p -Laplacian Kirchhoff equation u t − (1 + ‖ ∇ u ‖ L p (Ω) p) Δ p u + g (u) = 0 on a bounded domain Ω ⊂ R N with initial–boundary conditions. [ABSTRACT FROM AUTHOR]