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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

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 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 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 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.

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]

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]

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]

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.

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]

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]

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]