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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 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 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]

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 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 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]

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]