Subcritical Exponential Growth Research Articles

Existence of solutions for a nonlocal elliptic system with critical and subcritical exponential growth

In this paper, the existence of nontrivial solutions to the following nonlocal elliptic system: is studied by variational method, where is a bounded open subset of with smooth boundary , and the nonlinearity and have critical or subcritical exponential growth in the sense of Trudinger–Moser inequality.

Combined effects in planar quasilinear Schrödinger equations with superlinear reaction

In this paper, we prove the existence of nontrivial solutions for the following planar quasilinear Schrödinger equation: − Δ u + V ( x ) u − Δ ( u 2 ) u = g ( u ) , x ∈ R 2 , where V ∈ C ( R 2 , [ 0 , ∞ ) ) and g ∈ C ( R , R ) is of subcritical exponential growth satisfying some mild conditions. In particular, by means of the Trudinger–Moser inequality, we give a different method from the one of the polynomial growth nonlinearities to prove the Brézis–Lieb split property when f has subcritical exponential growth. Our result extends and complements the one of Chen–Rădulescu–Tang–Zhang (Rev. Mat. Iberoam. 36 (2020) 1549–1570) dealing with the higher dimensions N ⩾ 3 to the dimension N = 2.

On a weighted elliptic equation of N-Kirchhoff type with double exponential growth

Abstract In this work, we study the weighted Kirchhoff problem − g ∫ B σ ( x ) ∣ ∇ u ∣ N d x div ( σ ( x ) ∣ ∇ u ∣ N − 2 ∇ u ) = f ( x , u ) in B , u > 0 in B , u = 0 on ∂ B , \left\{\begin{array}{ll}-g\left(\mathop{\displaystyle \int }\limits_{B}\sigma \left(x)| \nabla u\hspace{-0.25em}{| }^{N}{\rm{d}}x\right){\rm{div}}\left(\sigma \left(x)| \nabla u\hspace{-0.25em}{| }^{N-2}\nabla u)=f\left(x,u)& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}B,\\ u\gt 0& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}B,\\ u=0& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial B,\end{array}\right. where B B is the unit ball of R N {{\mathbb{R}}}^{N} , σ ( x ) = log e ∣ x ∣ N − 1 \sigma \left(x)={\left(\log \left(\frac{e}{| x| }\right)\right)}^{N-1} , the singular logarithm weight in the Trudinger-Moser embedding, and g g is a continuous positive function on R + {{\mathbb{R}}}^{+} . The nonlinearity is critical or subcritical growth in view of Trudinger-Moser inequalities. We first obtain the existence of a solution in the subcritical exponential growth case with positive energy by using minimax techniques combined with the Trudinger-Moser inequality. In the critical case, the associated energy does not satisfy the condition of compactness. We provide a new condition for growth, and we stress its importance to check the compactness level.

Schrödinger–Poisson system with zero mass and convolution nonlinearity in R 2

In this paper, we prove the existence of nontrivial solutions and ground state solutions for the following planar Schrödinger–Poisson system with zero mass − Δ u + ϕ u = ( I α ∗ F ( u ) ) f ( u ) , x ∈ R 2 , Δ ϕ = u 2 , x ∈ R 2 , where α ∈ ( 0 , 2 ), I α : R 2 → R is the Riesz potential, f ∈ C ( R , R ) is of subcritical exponential growth in the sense of Trudinger–Moser. In particular, some new ideas and analytic technique are used to overcome the double difficulties caused by the zero mass case and logarithmic convolution potential.

New existence results on planar quasilinear Schrödinger equations with subcritical exponential growth

In this paper, we study the following quasilinear Schrödinger equation −Δu+V(x)u−Δ(u2)u=g(x,u),x∈R2,where V(x) and g(x,u) are 1-periodic in xi (i=1,2). By using a change of variable and a version of Trudinger–Moser inequality, we obtain the existence of nontrivial solutions and infinitely many pairs of geometrically distinct solutions. Our results extend the well-known results in the literature.

On the planar Choquard equation with indefinite potential and critical exponential growth

In the present paper, we study the following planar Choquard equation:{−Δu+V(x)u=(Iα⁎F(u))f(u),x∈R2,u∈H1(R2), where V(x) is an 1-periodic function, Iα:R2→R is the Riesz potential and f(t) behaves like ±eβt2 as t→±∞. A direct approach is developed in this paper to deal with the problems with both critical exponential growth and strongly indefinite features when 0 lies in a gap of the spectrum of the operator −△+V. In particular, we find nontrivial solutions for the above equation with critical exponential growth, and establish the existence of ground states and geometrically distinct solutions for the equation when the nonlinearity has subcritical exponential growth. Our results complement and generalize the known ones in the literature concerning the positive potential V to the general sign-changing case, such as, the results of de Figueiredo-Miyagaki-Ruf (1995) [16], of Alves-Cassani-Tarsi-Yang (2016) [4], of Ackermann (2004) [1], and of Alves-Germano (2018) [5].

Ground state solutions for planar Schrödinger-Poisson system involving subcritical and critical exponential growth with convolution nonlinearity

We study a class of planar Schrödinger-Poisson systems{−Δu+u+ϕu=(Iα⁎∫0uf(t)dt)f(u),x∈R2,Δϕ=u2,x∈R2, where Iα is the Riesz potential of order α∈(0,2). Under some suitable assumptions on f(u) with subcritical exponential growth, we consider the existence of positive Nehari-Pohoz̆aev type ground state solutions by using some new analytic techniques. Also, we obtain a positive ground state solution of Nehari type with the help of the non-Nehari manifold method when f(u) has critical exponential growth. Our results generalize and improve the ones in Du and Weth (2017) [20], Alves and Figueiredo (2019) [2], and some other related literatures as α→0+.

Dirichlet problems involving the 1-Laplacian

The present paper concerns Dirichlet problem for a class of non-linear partial differential equations involving 1-Laplacian. The non-linear term holds the sub-critical exponential growth and guarantees the solution is non-trivial. The difficulty provided by the degeneration of 1-Laplacian has been overcame by a replacement with a suitable vector field. The underlying minimization problem has been investigated to verify the existence result to the concern problem.

Nonlocal Kirchhoff Problems with Singular Exponential Nonlinearity

In this paper, we first develop the fractional Trudinger–Moser inequality in singular case and then we use it to study the existence and multiplicity of solutions for a class of perturbed fractional Kirchhoff type problems with singular exponential nonlinearity. Under some suitable assumptions, the existence of two nontrivial and nonnegative solutions is obtained by using the mountain pass theorem and Ekeland’s variational principle as the nonlinear term satisfies critical or subcritical exponential growth conditions. Moreover, the existence of ground state solutions for the aforementioned problems without perturbation and without the Ambrosetti–Rabinowitz condition is investigated.

Fractional Adams–Moser–Trudinger type inequality on Heisenberg group

The aim of this paper is to establish singular fractional Adams–Moser–Trudinger inequality for both bounded and unbounded domains in the Heisenberg group Hn. We first establish fractional Adams–Moser–Trudinger type inequality on domain Ω⊂Rn with finite measure (Theorem 1.12) and then using this inequality and Hardy–Littlewood–Sobolev inequality adapted to the result of R.O’Neil(1963), we establish singular fractional Adams–Moser–Trudinger type inequality on domain Ω⊂Rn with finite measure (Theorem 1.13). We also establish singular fractional Adams–Moser–Trudinger type inequality in Hn, using the approach of N.Lam and G.Lu (2012, 2013). The main idea of our approach is that any function in higher order fractional Sobolev space in Heisenberg group can be represented in terms of Riesz potential and then using techniques from harmonic analysis and kernel properties of the associated operator, we establish fractional Adams–Moser–Trudinger type inequality in Hn. In this paper, our approach is quite simple and free from symmetrization arguments. As an applications of our theorems, we establish the existence of solution to the following class of problems Tαu=f(ξ,u)|ξ|a+b(ξ)|u|γ−1uinΩ,u=0inHn∖Ω,where Ω is a bounded subset of Hn of class C0,1 with bounded boundary and 0≤a<Q,f satisfies either the subcritical exponential growth or critical exponential growth condition and b is a small L2-perturbation, that is, there exists a small η>0 with 0<‖b‖L2(Ω)<η,0≤γ<1 and α=Q2.

Ground state solutions for a class of gauged Schrödinger equations with subcritical and critical exponential growth

We study a class of gauged nonlinear Schrödinger equations where ω,λ&gt;0 and Under some suitable assumptions on f with critical exponential growth, we obtain a positive ground state solution by the non‐Nehari manifold method. When f(u) has subcritical exponential growth, we prove the existence of a positive ground state solution by using a new approach. Our results generalize and improve the ones in Ji‐Fang [J. Math. Anal. Appl. 450 (2017) 578‐591], Byeon et al [J. Funct. Anal. 263 (2012) 1575‐1608], and some other related literatures.

Spike solutions for nonlinear Schrödinger equations in 2D with vanishing potentials

We consider $$\varepsilon $$-perturbed nonlinear Schrodinger equations of the form $$\begin{aligned} - \varepsilon ^2\Delta u + V(x)u = Q(x)f(u) \quad \text {in } \mathbb {R}^2, \end{aligned}$$where V and Q behave like $$(1+|x|)^{-\alpha }$$ with $$\alpha \in (0,2)$$ and $$(1+|x|)^{-\beta }$$ with $$\beta \in (\alpha , + \infty )$$, respectively. When f has subcritical exponential growth—by means of a weighted Trudinger–Moser-type inequality and the mountain pass theorem in weighted Sobolev spaces—we prove the existence of nontrivial mountain pass solutions, for any $$\varepsilon >0$$, and in the semi-classical limit, these solutions concentrate at a global minimum point of $$\mathcal A=V/Q$$. Our existence result holds also when f has critical growth, for any $$\varepsilon >0$$.

On a nonlocal asymmetric Kirchhoff problem

This work is devoted to study the existence of nontrivial solutions to nonlocal asymmetric problems involving the [Formula: see text]-Laplacian. [Formula: see text] where [Formula: see text] is a bounded domain with smooth boundary, [Formula: see text] is a Kirchhoff function, [Formula: see text] and [Formula: see text] is of subcritical polynomial or subcritical exponential growth. Moreover, the existence of nontrivial solutions for the above problem is obtained by using variational methods combined with the Moser–Trudinger inequality. Our interest then is to study [Formula: see text] without the analogue of Ambrosetti–Rabinowitz superquadratic condition ([Formula: see text] condition for short) in the positive semi-axis. To the best of our best knowledge, our results are new even in the asymmetric Kirchhoff Laplacian and [Formula: see text]-Laplacian cases.

The cone Moser–Trudinger inequalities and applications

In this paper, we first study the cone Moser–Trudinger inequalities and their best exponents on both bounded and unbounded domains R + 2 . Then, using the cone Moser–Trudinger inequalities, we study the asymptotic behavior of Cerami sequences and the existence of weak solutions to the nonlinear equation − Δ B u = f ( x , u ) , in x ∈ int ( B ) , u = 0 , on ∂ B , where Δ B is an elliptic operator with conical degeneration on the boundary x 1 = 0, and the nonlinear term f has the subcritical exponential growth or the critical exponential growth.

Compact embedding results of Sobolev spaces and existence of positive solutions to quasilinear equations

In this paper, we study mainly the existence of multiple positive solutions for a quasilinear elliptic equation of the following form on RN, when N≥2,(0.1)−ΔNu+V(x)|u|N−2u=λ|u|r−2u+f(x,u). Here, V(x)>0:RN→R is a suitable potential function, r∈(1,N), f(x,u) is a continuous function of N-superlinear and subcritical exponential growth without having the Ambrosetti–Rabinowitz condition, while λ>0 is a constant. A suitable Moser–Trudinger inequality and the compact embedding WV1,N(RN)↪Lr(RN) are proved to study problem (0.1). Moreover, the compact embedding HV1(RN)↪LKt(RN) is also analyzed to investigate the existence of a positive ground state to the following nonlinear Schrödinger equation(0.2)−Δu+V(x)u=K(x)g(u) with potentials vanishing at infinity in a measure-theoretic sense when N≥3.

Multiple solutions for biharmonic equations with improved subcritical polynomial growth and subcritical exponential growth

The main purpose of this paper is to establish the existence of three nontrivial solutions for a class of fourth-order elliptic equations with subcritical polynomial growth and subcritical exponential growth by using the minimax method and Morse theory.

On a class of Hamiltonian systems with Trudinger-Moser nonlinearities

In the present paper, we study existence of nontrivial weak solutions for a class of Hamiltonian systems of type \begin{align} \left \{ \begin{array}{ll} -\Delta u +b(x) u = w_{1}(x)f_{1}(v), & v > 0; \\ -\Delta v +b(x) v =w_{2}(x)f_{2}(u), & u > 0 , \end{array} \right . \end{align} where $b:\mathbb{R}^2 \rightarrow \mathbb{R}$ is a continuous potential which may change sign and the nonlinearity $f_{i}:\mathbb{R} \rightarrow \mathbb{R}$ has critical or subcritical exponential growth in the sense of Trudinger-Moser's inequality, for $i=1,2$. The main results are proved by using variational methods through strongly indefinite functionals.

Biharmonic equations with improved subcritical polynomial growth and subcritical exponential growth

The main purpose of this paper is to establish the existence of two nontrivial solutions and the existence of infinitely many solutions for a class of fourth-order elliptic equations with subcritical polynomial growth and subcritical exponential growth by using a suitable version of the mountain pass theorem and the symmetric mountain pass theorem.

A Biharmonic Equation in ℜ4 Involving Nonlinearities with Subcritical Exponential Growth

Abstract In this paper we consider a biharmonic equation of the form Δ2u+V(x)u = f (u) in the whole four-dimensional space ℜ4. Assuming that the potential V satisfies some symmetry conditions and is bounded away from zero and that the nonlinearity f is odd and has subcritical exponential growth (in the sense of an Adams’ type inequality), we prove a multiplicity result. More precisely we prove the existence of infinitely many nonradial sign-changing solutions and infinitely many radial solutions in H2(ℜ4). The main difficulty is the lack of compactness due to the unboundedness of the domain ℜ4 and in this respect the symmetries of the problem play an important role.

Solitary Waves for Quasilinear Schrödinger Equations Arising in Plasma Physics

Abstract We study the quasilinear Schrödinger equation izt = −∆z +W(x)z − η(|z|2)z − k[∆p(|z|2)]p′(|z|2)z in ℝ2, where W : ℝ2 →ℝ is a positive potential and the nonlinearity η : ℝ2 × ℝ → ℝ has critical or sub-critical exponential growth. Quasilinear Schrödinger equations of this type have been studied as models of several physical phenomena such as superfluid film equation, in the theory of Heisenberg ferromagnets and magnons, in dissipative quantum mechanics and in condensed matter theory. In a suitable Orlicz space together with Trudinger-Moser inequality we establish an existence of standing wave solutions for this problem. The second order nonlinearity considered in this paper corresponds to the superfluid equation in plasma physics.