Berestycki Lions Type Research Articles

Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions

Abstract This article is concerned with the following Kirchhoff equation: − a + b ∫ R 3 ∣ ∇ u ∣ 2 d x Δ u = g ( u ) + h ( x ) in R 3 , -\left(a+b\mathop{\int }\limits_{{{\mathbb{R}}}^{3}}{| \nabla u| }^{2}{\rm{d}}x\right)\Delta u=g\left(u)+h\left(x)\hspace{1em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{3}, where a a and b b are positive constants and h ≠ 0 h\ne 0 . Under the Berestycki-Lions type conditions on g g , we prove that the equation has at least two positive solutions by using variational methods. Furthermore, we obtain the existence of ground state solutions.

Existence of normalized solutions to Choquard equation with general mixed nonlinearities

We study the existence of normalized solutions to the following Choquard equation with F being a Berestycki-Lions type function { − Δ u + λu = ( I α ∗ F ( u ) ) f ( u ) , in R N , ∫ R N | u | 2 d x = ρ 2 , where N ≥ 3 , 0 $ ]]> ρ > 0 is assigned, α ∈ ( 0 , N ) , I α is the Riesz potential, and λ ∈ R is an unknown parameter that appears as a Lagrange multiplier. Here, the general nonlinearity F contains the L 2 -subcritical and L 2 -supercritical mixed case, the Hardy-Littlewood-Sobolev lower critical and upper critical cases.

Bound state solutions for quasilinear Schrödinger equations with Hardy potential

This paper is concerned with the existence of bound state solutions for quasilinear Schrödinger equations with Hardy potential and Berestycki–Lions type conditions. By using the variational methods, we get the existence results which is a complement to the ones in Hu et al. (2022).

On the Pohozaev identity for the fractional p$p$‐Laplacian operator in RN$\mathbb {R}^N$

AbstractIn this paper, we show the existence of a nontrivial weak solution for a nonlinear problem involving the fractional ‐Laplacian operator and a Berestycki–Lions type nonlinearity. This solution satisfies a Pohozaev identity. Moreover, we prove that any sufficiently smooth solution fulfills the Pohozaev identity.

On solutions for a class of Klein–Gordon equations coupled with Born–Infeld theory with Berestycki–Lions conditions on $ \mathbb{R}^3 $

<abstract><p>In this paper, the existence of multiple solutions for a class of Klein–Gordon equations coupled with Born–Infeld theory was investigated. The potential and the primitive of the nonlinearity in this kind of elliptic equations are both allowed to be sign-changing. Besides, we assumed that the nonlinearity satisfies the Berestycki–Lions type conditions. By employing Ekeland's variational principle, mountain pass theorem, Pohožaev identity, and various other techniques, two nontrivial solutions were obtained under some suitable conditions.</p></abstract>

Multiple solutions for quasi-linear elliptic equations with Berestycki-Lions type nonlinearity

<abstract><p>We studied the modified nonlinear Schrödinger equation</p> <p><disp-formula> <label>0.1</label> <tex-math id="E0.1"> \begin{document}$ \begin{equation} -\Delta u-\frac12\Delta(u^2)u = g(u)+h(x), \quad u\in H^1({\mathbb{R}}^N), \end{equation} $\end{document} </tex-math></disp-formula></p> <p>where $ N\geq3 $, $ g\in C({\mathbb{R}}, {\mathbb{R}}) $ is a nonlinear function of Berestycki-Lions type, and $ h\not\equiv 0 $ is a nonnegative function. When $ \|h\|_{L^2({\mathbb{R}}^N)} $ is suitably small, we proved that (0.1) possesses at least two positive solutions by variational approach, one of which is a ground state while the other is of mountain pass type.</p></abstract>

Multiplicity of solutions for a scalar field equation involving a fractional p-Laplacian with general nonlinearity

We prove the existence of infinitely many radially symmetric solutions to the problem ( − Δ p ) s u = g ( u ) in R N , u ∈ W s , p ( R N ) , where s ∈ ( 0 , 1 ) , 2 ≤ p < ∞ , sp ≤ N , 2 ≤ N ∈ N and ( − Δ p ) s is the fractional p-Laplacian operator. We treat both the cases sp = N and sp<N. The nonlinearity g is a function of Berestycki–Lions type with critical exponential growth if sp = N and critical polynomial growth if sp<N. We also prove the existence of a ground state solution for the same problem.

Ground state solutions for Choquard equations with Bessel potential

Firstly, we study the existence, regularity and decay of positive solutions for the elliptic system { − Δ u + u = vg ( u ) in R N , − Δv + v = G ( u ) in R N , where N ≥ 3 , g satisfies Berestycki-Lions type conditions. In Sobolev space, the system can be transformed into the Choquard equations with Bessel potential − Δ u + u = ( g 2 ⋆ G ( u ) ) g ( u ) in R N , where g 2 is the fundamental solution of the linear operator − Δ + I on R N . Moreover, by assuming that g is nondecreasing, we obtain the radial symmetry and monotonic property of positive solutions by moving plane method, and the existence of ground state solutions by non-Nehari manifold method.

Existence and asymptotic behavior of solutions for Choquard equations with singular potential under Berestycki–Lions type conditions

We prove the existence and asymptotic behavior of solutions to the following problem: − Δ u + V ( x ) u − g ( x ) u = ( I α ∗ F ( u ) ) f ( u ) , x ∈ R N ; u ∈ H 1 ( R N ) , where g ( x ) : = μ | x | is called the Coulomb potential, g ( x ) : = β | x | 2 is called the Hardy potential (the inverse-square potential). μ , β &gt; 0 are parameters, I α : R N ⟶ R is the Riesz potential. Moreover, the nonlinearity f satisfies Berestycki–Lions type conditions which are introduced by Moroz and Van Schaftingen (Trans. Amer. Math. Soc. 367 (2015) 6557–6579). When μ ∈ ( 0 , α ( N − 2 ) / 2 ( α + 1 ) ) and β ∈ ( 0 , α ( N − 2 ) 2 / 4 ( 2 + α ) ), under some mild assumptions on V, we establish the existence and asymptotic behavior of solutions. Particularly, our results extend some relate ones in the literature.

Multiple solutions for nonhomogeneous Klein–Gordon–Maxwell system with Berestycki–Lions conditions

This paper is concerned with a kind of nonhomogeneous Klein–Gordon–Maxwell system −Δu−(2ω+ϕ)ϕu=g(u)+h(x),inR3,Δϕ=(ω+ϕ)u2,inR3,where ω>0 is a constant and g satisfies the Berestycki–Lions type conditions. By using Ekelands variational principle, the mountain pass theorem, Pohožaev identity and some new tricks, two nontrivial solutions can be obtained here.

Existence of a positive solution for some quasilinear elliptic equations in [formula omitted

We study the existence of a positive solution of the following quasilinear elliptic equations in RN:−Δpu=g(u),u∈W1,p(RN), where N≥2, 1<p≤N and Δpu=div(|∇u|p−2∇u) is the p-Laplacian operator. We assume that g(s) satisfies Berestycki-Lions type condition. We give a new concentration compactness type result for a Palais-Smale sequence with an additional property related to Pohozaev identity. Keys of our proof are Tartar's inequality and Brezis-Lieb Lemma.

Existence and asymptotic behavior of solutions for a quasilinear Schrödinger equation with Hardy potential

In this paper, we discuss the quasilinear Schrödinger equation with a Hardy potential (Pμ)−Δu−μu|x|2−Δ(u2)u=g(u),x∈RN∖{0},u∈H1(RN),where N≥3, μ<μ̄=(N−2)24, 1|x|2 is called the Hardy potential and g∈C(R,R) satisfies the Berestycki–Lions type conditions. When 0<μ<(N−2)24, we show that the above problem has a positive and radial solution ū∈H1(RN). At the same time, we prove that the solution ū together with its derivatives up to order 2 have exponential decay at infinity and the solution has the possibility of blow-up at the origin. When μ<0, we obtain a solution ũ of (Pμ) in the radial space Hr1(RN) and we also prove that the solution ũ together with its derivatives up to order 2 have exponential decay at infinity and the solution has the possibility of blow-up at the origin. Furthermore, we construct a family of solutions which converge to a solution of the limiting problem as μ→0.

A Global Minimization Trick to Solve Some Classes of Berestycki–Lions Type Problems

In this paper we show an abstract theorem that can be used to prove the existence of solution for a class of elliptic equation considered in Berestycki–Lions (Arch Ration Mech Anal 82:313–345, 1983) and related problems. Moreover, we use the abstract theorem to show that a class of zero mass problems has multiple solutions, which is a novelty for this type of problem.

A Berestycki–Lions type result for a class of degenerate elliptic problems involving the Grushin operator

In this work we study the existence of nontrivial solution for the following class of semilinear degenerate elliptic equations \[ -\Delta_{\gamma} u + a(z)u = f(u) \quad \mbox{in}\ \mathbb{R}^{N}, \]where $\Delta _{\gamma }$ is known as the Grushin operator, $z:=(x,y)\in \mathbb {R}^{m}\times \mathbb {R}^{k}$ and $m+k=N\geqslant 3$, $f$ and $a$ are continuous function satisfying some technical conditions. In order to overcome some difficulties involving this type of operator, we have proved some compactness results that are crucial in the proof of our main results. For the case $a=1$, we have showed a Berestycki–Lions type result.

Bound state solutions for a class of Nonlinear Elliptic Equations with Hardy potential and Berestycki–Lions type conditions

By using the variational methods, a class of Nonlinear Elliptic Equations with the Hardy potential and Berestycki-Lions type conditions is studied and a bound state solution is obtained.

Ground state solutions for Kirchhoff-type problems with convolution nonlinearity and Berestycki–Lions type conditions

Ground state solutions for Kirchhoff-type problems with convolution nonlinearity and Berestycki–Lions type conditions

Ground state solutions for nonlinear Choquard equation with singular potential and critical exponents

In this paper, we study the ground state solutions to the following Choquard equation involving singular potential:−Δu+V(|x|)u=(Iα⁎F(u))f(u),x∈RN, where N⩾3, α∈(0,N), Iα is the Riesz potential, V is a singular potential with parameter θ∈(max⁡{0,4−N},2)∪(2,2N−2)∪(2N−2,∞), F is the primitive of f and f satisfies critical growth in sense of the Hardy-Littlewood-Sobolev inequality. Under different range of θ and almost necessary conditions on the nonlinearity f in the spirit of Berestycki-Lions type conditions, we divide this paper into three parts. By virtue of two different kinds of Lions-type theorem and Nehari manifold, some existence results are established.

Standing waves for the pseudo-relativistic Hartree equation with Berestycki-Lions nonlinearity

We study the following class of pseudo-relativistic Hartree equations − ε 2 Δ + m 2 u + V ( x ) u = ε μ − N ( | x | − μ ⁎ F ( u ) ) f ( u ) in R N , where the nonlinearity satisfies general hypotheses of Berestycki-Lions type. By using the method of penalization arguments, we prove the existence of a family of localized positive solutions that concentrate at the local minimum points of the indefinite potential V ( x ) , as ε → 0 .

Multiplicity and concentration results for local and fractional NLS equations with critical growth

Goal of this paper is to study the following singularly perturbed nonlinear Schrödinger equation $$ \varepsilon^{2s}(- \Delta)^s v+ V(x) v= f(v), \quad x \in \mathbb{R}^N, $$ where $s \in (0,1)$, $N \geq 2$, $V \in C(\mathbb{R}^N,\mathbb{R})$ is a positive potential and $f$ is assumed critical and satisfying general Berestycki-Lions type conditions. When $\varepsilon>0$ is small, we obtain existence and multiplicity of semiclassical solutions, relating the number of solutions to the cup-length of a set of local minima of $V$; in particular, we improve the result in [37]. Furthermore, these solutions are proved to concentrate in the potential well, exhibiting a polynomial decay. Finally, we prove the previous results also in the limiting local setting $s=1$ and $N\geq 3$, with an exponential decay of the solutions.

Existence ground state solutions for a quasilinear Schrödinger equation with Hardy potential and Berestycki–Lions type conditions

In this paper, we discuss the quasilinear Schrödinger equation with a Hardy potential −Δu+V(x)u−μu|x|2−Δ(u2)u=g(u),x∈RN∖{0},where N≥3, 0≤μ<μ̄=(N−2)24, 1|x|2 is called the Hardy potential. By using Jeanjean’s monotonicity trick, we admit a class of ground state solutions for the above problem, under the general Berestycki–Lions type assumptions on the nonlinearity g, as well as some weak assumptions on the potential V.