Nehari-type Ground State Solutions Research Articles

On the planar Kirchhoff-type problem involving supercritical exponential growth

Abstract This article is concerned with the following nonlinear supercritical elliptic problem: − M ( ‖ ∇ u ‖ 2 2 ) Δ u = f ( x , u ) , in B 1 ( 0 ) , u = 0 , on ∂ B 1 ( 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 B 1 ( 0 ) {B}_{1}\left(0) is the unit ball in R 2 {{\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 t t , which behaves as exp [ ( β 0 + ∣ x ∣ α ) t 2 ] \exp {[}({\beta }_{0}+| x\hspace{-0.25em}{| }^{\alpha }){t}^{2}] and exp ( β 0 t 2 + ∣ x ∣ α ) \exp ({\beta }_{0}{t}^{2+| x{| }^{\alpha }}) with β 0 {\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 liminf t → ∞ t f ( x , t ) exp [ ( β 0 + ∣ x ∣ α ) t 2 ] {\mathrm{liminf}}_{t\to \infty }\frac{tf\left(x,t)}{\exp {[}({\beta }_{0}+| \hspace{-0.25em}x\hspace{-0.25em}{| }^{\alpha }){t}^{2}]} and liminf t → ∞ t f ( x , t ) exp ( β 0 t 2 + ∣ 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 ) = 1 M(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.

Nehari-Type Ground State Solutions for Superlinear Elliptic Equations with Variable Exponent in $${\mathbb {R}}^N$$

Nehari-Type Ground State Solutions for Superlinear Elliptic Equations with Variable Exponent in $${\mathbb {R}}^N$$

Nehari type ground state solutions for periodic Schrödinger–Poisson systems with variable growth

In this paper, we deal with the variable growth Schrödinger–Poisson Systems in : where and , and are periodic in x. We use the non-Nehari manifold approach to establish the existence of the Nehari type ground state solutions, under the condition: for all , t>0, and the constant , where as and as t<1. In particular, some new inequalities and tricks are used to overcome the difficulties caused by the variable exponent.

Nehari-type ground state solutions for a Choquard equation with doubly critical exponents

AbstractThis paper deals with the following Choquard equation with a local nonlinear perturbation:−Δu+u=Iα∗|u|α2+1|u|α2−1u+f(u),x∈R2;u∈H1(R2),$$\begin{array}{} \displaystyle \left\{ \begin{array}{ll} - {\it\Delta} u+u=\left(I_{\alpha}*|u|^{\frac{\alpha}{2}+1}\right)|u|^{\frac{\alpha}{2}-1}u +f(u), &amp; x\in \mathbb{R}^2; \\ u\in H^1(\mathbb{R}^2), \end{array} \right. \end{array}$$whereα∈ (0, 2),Iα: ℝ2→ ℝ is the Riesz potential andf∈ 𝓒(ℝ, ℝ) is of critical exponential growth in the sense of Trudinger-Moser. The exponentα2+1$\begin{array}{} \displaystyle \frac{\alpha}{2}+1 \end{array}$is critical with respect to the Hardy-Littlewood-Sobolev inequality. We obtain the existence of a nontrivial solution or a Nehari-type ground state solution for the above equation in the doubly critical case, i.e. the appearance of both the lower critical exponentα2+1$\begin{array}{} \displaystyle \frac{\alpha}{2}+1 \end{array}$and the critical exponential growth off(u).

Nehari-type ground state solutions for Schrödinger equations with Hardy potential and critical nonlinearities

ABSTRACTIn this paper, we develop a direct approach to find Nehari-type ground state solutions for the following Schrödinger equation with inverse square potential and critical Sobolev exponent where , (the critical Sobolev exponent), , and . Under the conditions and and some other mild assumptions on V and f, we prove the existence of a positive ground state solution of Nehari-type to the above problem in the asymptotically periodic case and asymptotically constant case, respectively, with satisfying for some .

Nehari-type ground state solutions for asymptotically periodic fractional Kirchhoff-type problems in mathbb{R}^{N}

In this paper, we studied the following fractional Kirchhoff-type equation: \t\t\t(a+b∫RN|(−△)α2u|2dx)(−△)αu+V(x)u=f(x,u),x∈RN,\\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}$$\\biggl(a+b \\int_{\\mathbb{R}^{N}} \\bigl\\vert (-\\triangle)^{\\frac{\\alpha }{2}}u \\bigr\\vert ^{2}\\,\\mathrm{d}x \\biggr) (-\\triangle)^{\\alpha }u+V(x)u=f(x,u), \\quad x\\in{\\mathbb{R}}^{N}, $$\\end{document} where a, b are positive constants, alphain(0,1), Nin (2alpha,4alpha), (-triangle)^{alpha} is the fractional Laplacian operator, V(x) and f(x,u) are periodic or asymptotically periodic in x. Under some weaker conditions on the nonlinearity, we obtain the existence of ground state solutions for the above problem in periodic case and asymptotically periodic case, respectively. In particular, our results unify both asymptotically cubic and super-cubic nonlinearities, which are new even for alpha=1.

Nehari-type ground state solutions for Kirchhoff type problems in ℝN

ABSTRACTIn the present paper, by employing Non-Nehari manifold method, we make use of a weak condition on the existence of Nehari-type ground state solutions for the following Kirchhoff-type problem (0.1) which weakens both the variant Ambrosetti–Rabinowitz condition and the variant Nehari-type condition related to Problem (0.1). Furthermore, we allow to be sign-changing, which is different from the previous literature.

Nehari-type ground state solutions for Schrödinger equations including critical exponent

In this paper, we study the following system of nonlinear Schrödinger equations: { − △ u + a ( x ) u = | u | p − 2 u + λ ( x ) v , x ∈ R N , − △ v + b ( x ) v = | v | 2 ∗ − 2 v + λ ( x ) u , x ∈ R N , where N ≥ 3 , 2 < p < 2 ∗ and 2 ∗ = 2 N / ( N − 2 ) is the critical Sobolev exponent. Under assumptions that a ( x ) , b ( x ) , λ ( x ) ∈ C ( R N , R ) are all 1-periodic in each of x 1 , x 2 , … , x N and λ 2 ( x ) < a ( x ) b ( x ) , we prove that the above system has a Nehari-type ground state solution when 0 < a ( x ) < μ 0 for some μ 0 ∈ ( 0 , 1 ) .

Nehari-Type Ground State Positive Solutions for Superlinear Asymptotically Periodic Schrödinger Equations

We deal with the existence of Nehari-type ground state positive solutions for the nonlinear Schrödinger equation-Δu+Vxu=fx, u, x ∈ RN, u ∈ H1 RN. Under a weaker Nehari condition, we establish some existence criteria to guarantee that the above problem has Nehari-type ground state solutions by using a more direct method in two cases: the periodic case and the asymptotically periodic case.