Minimax Level Research Articles

Ground state solutions for the Hamilton–Choquard elliptic system with critical exponential growth

In this paper, we study the following Hamilton–Choquard type elliptic system: − Δ u + u = ( I α ∗ F ( v ) ) f ( v ) , x ∈ R 2 , − Δ v + v = ( I β ∗ F ( u ) ) f ( u ) , x ∈ R 2 , where I α and I β are Riesz potentials, f : R → R possessing critical exponential growth at infinity and F ( t ) = ∫ 0 t f ( s ) d s. Without the classic Ambrosetti–Rabinowitz condition and strictly monotonic condition on f, we will investigate the existence of ground state solution for the above system. The strongly indefinite characteristic of the system, combined with the convolution terms and critical exponential growth, makes such problem interesting and challenging to study. With the help of a proper auxiliary system, we employ an approximation scheme and the non-Nehari manifold method to control the minimax levels by a fine threshold, and succeed in restoring the compactness for the critical problem. Existence of a ground state solution is finally established by the concentration compactness argument and some detailed estimates.

Mountain-pass-type solutions for Schrödinger equations in R2 with unbounded or vanishing potentials and critical exponential growth nonlinearities

Abstract In this article, we consider the existence of solutions for nonlinear elliptic equations of the form − Δ u + V ( ∣ x ∣ ) u = Q ( ∣ x ∣ ) f ( u ) , x ∈ R 2 , -\Delta u+V\left(| x| )u=Q\left(| x| )f\left(u),\hspace{1em}x\in {{\mathbb{R}}}^{2}, where the nonlinear term f ( s ) f\left(s) has critical exponential growth which behaves like e α s 2 {e}^{\alpha {s}^{2}} , the radial potentials V , Q : R + → R V,Q:{{\mathbb{R}}}^{+}\to {\mathbb{R}} are unbounded, singular at the origin or decaying to zero at infinity. By combining the variational methods, Trudinger-Moser inequality, and some new approaches to estimate precisely the minimax level of the energy functional, we prove the existence of a Mountain-pass-type solution for the above problem under some weak assumptions.

On concave perturbations of a periodic elliptic problem in R2 involving critical exponential growth

Abstract In this paper, we consider the existence of solutions for nonlinear elliptic equations of the form (0.1) − Δ u + V ( x ) u = f ( x , u ) + λ a ( x ) ∣ u ∣ q − 2 u , x ∈ R 2 , -\Delta u+V\left(x)u=f\left(x,u)+\lambda a\left(x)| u{| }^{q-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{2}, where λ > 0 \lambda \gt 0 , q ∈ ( 1 , 2 ) q\in \left(1,2) , a ∈ L 2 / ( 2 − q ) ( R 2 ) a\in {L}^{2\text{/}\left(2-q)}\left({{\mathbb{R}}}^{2}) , V ( x ) V\left(x) , and f ( x , t ) f\left(x,t) are 1-periodic with respect to x x , and f ( x , t ) f\left(x,t) has critical exponential growth at t = ∞ t=\infty . By combining the variational methods, Trudinger-Moser inequality, and some new techniques with detailed estimates for the minimax level of the energy functional, we prove the existence of a nontrivial solution for the aforementioned equation under some weak assumptions. Our results show that the presence of the concave term (i.e. λ > 0 \lambda \gt 0 ) is very helpful to the existence of nontrivial solutions for equation (0.1) in one sense.

Schrödinger equations in [formula omitted] with critical exponential growth and concave nonlinearities

In this paper, we consider the existence of solutions for nonlinear elliptic equations of the form(0.1)−Δu+V(|x|)u=Q(|x|)f(u)+λg(|x|,u),x∈R2, where the nonlinear term f(s) has critical exponential growth which behaves like eαs2, g(r,s) is a concave term on s, the radial potentials V,Q:R+→R are unbounded, singular at the origin or decaying to zero at infinity and λ>0 is a parameter. Based on the known Trudinger-Moser inequality in H0,rad1(B1), we establish a new version of Trudinger-Moser inequality in the working space of the associated with the energy functional related to the above problem. By combining the variational methods, Trudinger-Moser inequality and some new approaches to estimate precisely the minimax level of the energy functional, we prove the existence of a nontrivial solution for the above problem under some weak assumptions. Our results show that the presence of the concave term (i.e. λ>0) is very helpful to the existence of nontrivial solutions for Eq. (0.1) in one sense.

Ground state solutions for planar periodic Kirchhoff type equation with critical exponential growth

In this paper, we prove the following nonlocal Kirchhoff problem of the type has a Nehari‐type ground state solution when and are periodic on and has critical exponential growth in the sense of Trudinger–Moser inequality on . We develop some new approaches to estimate precisely the minimax level of the energy functional and to recover the compactness of Cerami sequences of the associated Euler–Lagrange functional. With this in hand, we can overcome difficulties arising from the appearance of the nonlocal term and the nonlinearity which is of critical growth of Trudinger–Moser type in whole Euclidean space .

Strongly indefinite Choquard equation in ℝ2 with critical exponential growth

In this paper, we show the existence of nontrivial solutions for the following planar Choquard equation: where is the Riesz potential. We mainly consider the case that is 1‐periodic, has critical exponential growth at infinity and 0 lies in a gap of the spectrum of . There are few works in the literature concerned with such problem due to the compactness issue, appearance of the convolution term , and the difficulties aroused by the strongly indefinite characteristic and the critical exponential growth of . By employing an approximation scheme and some fine estimates of the mountain pass minimax levels in combination with related variational arguments, we show nontrivial solution for the above strongly indefinite problem.

On critical N‐Kirchhoff type equations involving Trudinger–Moser nonlinearity

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: where is a smooth bounded domain, is a Kirchhoff function, and f(x, s) has critical exponential growth on s which behaviors as 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 . 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 and Goyal et al. (2016) (Theorem 1.3) for N ≥ 2.

Mountain-pass type solutions for the Chern-Simons-Schrödinger equation with critical exponential growth

In this paper, we consider the nonlinear Chern-Simons-Schrödinger equations where is the so-called Chern-Simons term, has critical exponential growth which behaves like . By combing the variational methods, Trudinger-Moser inequality and some new approaches to estimate precisely the minimax level of the energy functional, we prove the existence of a mountain-pass type solution for the above problem under some weak assumptions. Our results improve and extend the previous results.

Ground states for planar Hamiltonian elliptic systems with critical exponential growth

This paper focuses on the study of ground states and nontrivial solutions for the following Hamiltonian elliptic system:{−Δu+V(x)u=f1(x,v),x∈R2,−Δv+V(x)v=f2(x,u),x∈R2, where V∈C(R2,(0,∞)) and f1,f2:R2×R→R have critical exponential growth. The strongly indefinite features together with the critical exponent bring some new difficulties in our analysis. In this paper, we develop a direct approach and use an approaching argument to seek Cerami sequences for the energy functional and estimate the minimax levels of such sequences. In particular, under some general assumptions imposed on the nonlinearity fi, we obtain the existence of ground states and nontrivial solutions for the above system as well as the following system in bounded domain,{−Δu=f1(v),x∈Ω,−Δv=f2(u),x∈Ω,u=0,v=0,x∈∂Ω. Our results improve and extend the related results of de Figueiredo-do Ó-Ruf (2004) [20]; (2011) [21], of Lam-Lu (2014) [30], and of de Figueiredo-do Ó-Zhang (2020) [22].

Improved results on planar Kirchhoff-type elliptic problems with critical exponential growth

In this paper, we prove the existence of nontrivial solutions and Nehari-type ground-state solutions for the following Kirchhoff-type elliptic equation: $$\begin{aligned} {\left\{ \begin{array}{ll} -m(\Vert \nabla u\Vert _2^2)\Delta u=f(x,u), \;\;&{} \text{ in } \ \ \Omega ,\\ u= 0, \;\;&{} \text{ on } \ \ \partial \Omega , \end{array}\right. } \end{aligned}$$ where $$\Omega \subset \mathbb {R}^2$$ is a smooth bounded domain, $$m:\mathbb {R}^+\rightarrow \mathbb {R}^+$$ 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.

Concentration-compactness for singular nonlocal Schrödinger equations with oscillatory nonlinearities

The paper is dedicated to the theory of concentration-compactness principles for inhomogeneous fractional Sobolev spaces. This subject for the local case has been studied since the publication of the celebrated works due to P.-L. Lions, which laid the broad foundations of the method and outlined a wide scope of its applications. Our study is based on the analysis of the profile decomposition for the weak convergence following the approach of dislocation spaces, introduced by K. Tintarev and \hbox{K.-H. Fieseler}. As an application, we obtain existence of nontrivial and nonnegative solutions and ground states for fractional Schrodinger equations for a wide class of possible singular potentials, not necessarily bounded away from zero. We consider possible oscillatory nonlinearities for both cases, subcritical and critical which are superlinear at the origin, without the classical Ambrosetti and Rabinowitz growth condition. In some of our results we prove existence of solutions by means of compactness of Palais-Smale sequences of the associated functional at the mountain pass level. To this end we study and provide the behavior of the weak profile decomposition convergence under the related functionals. Moreover, we use a Pohozaev type identity in our argument to compare the minimax levels of the energy functional with the ones of the associated limit problem. Motivated by this fact, in our work we also prove that this kind of identities hold for a larger class of potentials and nonlinearities for the fractional framework.

Foliated Schwarz symmetry for the nodal solution at the second minimax level

We study the second minimax level $$\lambda _2$$ of the eigenvalue problem for the scalar field equation in $${\mathbb {R}}^N$$ . By using the tool of polarization, we prove that every nodal solution at the second minimax level is foliated Schwarz symmetric. As a consequence, we prove an open problem of Perera and Tintarev (Annali di Matematica Pura ed Applicata (4) 194(1):131–144, 2015).

A critical nonlinear fractional elliptic equation with saddle-like potential in ℝN

In this paper, we study the existence of positive solution for the following class of fractional elliptic equation ϵ2s(−Δ)su+V(z)u=λuq−2u+u2s∗−2uinRN, where ϵ, λ > 0 are positive parameters, q∈(2,2s∗),2s∗=2NN−2s,N>2s,s∈(0,1),(−Δ)su is the fractional Laplacian, and V is a saddle-like potential. The result is proved by using minimizing method constrained to the Nehari manifold. A special minimax level is obtained by using an argument made by Benci and Cerami.

(N,q)-Laplacian problems with critical Trudinger–Moser nonlinearities

We obtain nontrivial solutions of a $(N,q)$-Laplacian problem with a critical Trudinger-Moser nonlinearity in a bounded domain. In addition to the usual difficulty of the loss of compactness associated with problems involving critical nonlinearities, this problem lacks a direct sum decomposition suitable for applying the classical linking theorem. We show that every Palais-Smale sequence at a level below a certain energy threshold admits a subsequence that converges weakly to a nontrivial critical point of the variational functional. Then we prove an abstract critical point theorem based on a cohomological index and use it to construct a minimax level below this threshold.

On a class of critical (p, q)-Laplacian problems

We obtain nontrivial solutions of a critical (p, q)-Laplacian problem in a bounded domain. In addition to the usual difficulty of the loss of compactness associated with problems involving critical Sobolev exponents, this problem lacks a direct sum decomposition suitable for applying the classical linking theorem. We show that every Palais–Smale sequence at a level below a certain energy threshold admits a subsequence that converges weakly to a nontrivial critical point of the variational functional. Then we prove an abstract critical point theorem based on a cohomological index and use it to construct a minimax level below this threshold.

A nodal solution of the scalar field equation at the second minimax level

We prove the existence of a sign-changing eigenfunction at the second minimax level of the eigenvalue problem for the scalar field equation under a slow decay condition on the potential near infinity. The proof involves constructing a set consisting of sign-changing functions that is dual to the second minimax class. We also obtain a nonradial sign-changing eigenfunction at this level when the potential is radial.

A multiplicity result for the scalar field equation

Abstract We prove the existence of N - 1 distinct pairs of nontrivial solutions of the scalar field equation in ℝ N under a slow decay condition on the potential near infinity, without any symmetry assumptions. Our result gives more solutions than the existing results in the literature when N ≥ 6. When the ground state is the only positive solution, we also obtain the stronger result that at least N - 1 of the first N minimax levels are critical, i.e., we locate our solutions on particular energy levels with variational characterizations. Finally we prove a symmetry breaking result when the potential is radial. To overcome the difficulties arising from the lack of compactness we use the concentration compactness principle of Lions, expressed as a suitable profile decomposition for critical sequences.

On the second minimax level of the scalar field equation and symmetry breaking

We study the second minimax level λ2 of the eigenvalue problem for the scalar field equation in R N. We prove the existence of an eigenfunction at the level λ2 when the potentialnearinfinityapproachestheconstantlevelfrombelownofasterthane −e |x| .Wealso consider questions about the nodality of eigenfunctions at this level and establish symmetry breaking at the levels 2 ,..., N.

Quasilinear asymptotically periodic elliptic equations with critical growth

The existence of a nontrivial solution for a class of asymptotically periodic equations involving the p -Laplacian operator in the euclidean space is established. The solutions are obtained by an application of the Mountain Pass Theorem and the Concentration-Compactness Principle. The lack of compactness due to the fact that the domain is unbounded is compensated by comparing the minimax level associated with the original problem and the minimax level associated with the periodic limit problem.

Subcritical perturbations of a singular quasilinear elliptic equation involving the critical Hardy–Sobolev exponent

In this work we improve some known results for a singular operator and also for a wide class of lower-order terms by proving a multiplicity result. The proof is made by applying the generalized mountain-pass theorem due to Ambrosetti and Rabinowitz. To do this, we show that the minimax levels are in a convenient range by combining a special class of approximating functions, due to Gazzola and Ruf, with the concentrating functions of the best Sobolev constant.