Bounded Palais Smale Sequences Research Articles

Nonlinear scalar field equations with Berestycki–Lions’ nonlinearity on large domains

We prove the existence of solutions for the following semilinear elliptic equation: $$\begin{aligned} - \Delta u = g(u) \quad \text {in } \Omega , \quad u \in H^1_0(\Omega ). \end{aligned}$$ Here $$\Omega $$ is a suitable large domain and g satisfies the completely same conditions as Berestycki–Lions’ conditions. Those conditions of g are known as “almost sufficient and necessary conditions” to the existence of nontrivial solutions of the equations defined in $$\mathbb {R}^N$$ . The main difficulty to prove the existence of solutions of the equation is that we can not obtain bounded Palais–Smale sequences. To overcome this difficulty, we modify the corresponding functional, which is a new idea introduced in our previous paper.

Nonlinear scalar field equations with general nonlinearity

Consider the nonlinear scalar field equation (0.1)−Δu=f(u)inRN,u∈H1(RN),where N≥3 and f satisfies the general Berestycki–Lions conditions. We are interested in the existence of positive ground states, of nonradial solutions and in the multiplicity of radial and nonradial solutions. Very recently Mederski (2017) made a major advance in that direction through the development, in an abstract setting, of a new critical point theory for constrained functionals. In this paper we propose an alternative, more elementary approach, which permits to recover Mederski’s results on (0.1). The keys to our approach are an extension to the symmetric mountain pass setting of the monotonicity trick, and a new decomposition result for bounded Palais–Smale sequences.

Ground State Solutions for Quasilinear Schrödinger Equations with Critical Growth and Lower Power Subcritical Perturbation

Abstract We study the following generalized quasilinear Schrödinger equation: - ( g 2 ⁢ ( u ) ⁢ ∇ ⁡ u ) + g ⁢ ( u ) ⁢ g ′ ⁢ ( u ) ⁢ | ∇ ⁡ u | 2 + V ⁢ ( x ) ⁢ u = h ⁢ ( u ) , x ∈ ℝ N , -(g^{2}(u)\nabla u)+g(u)g^{\prime}(u)|\nabla u|^{2}+V(x)u=h(u),\quad x\in% \mathbb{R}^{N}, where N ≥ 3 {N\geq 3} , g : ℝ → ℝ + {g\colon\mathbb{R}\rightarrow\mathbb{R}^{+}} is an even differentiable function such that g ′ ⁢ ( t ) ≥ 0 {g^{\prime}(t)\geq 0} for all t ≥ 0 {t\geq 0} , h ∈ C 1 ⁢ ( ℝ , ℝ ) {h\in C^{1}(\mathbb{R},\mathbb{R})} is a nonlinear function including critical growth and lower power subcritical perturbation, and the potential V ⁢ ( x ) : ℝ N → ℝ {V(x)\colon\mathbb{R}^{N}\rightarrow\mathbb{R}} is positive. Since the subcritical perturbation does not satisfy the (AR) condition, the standard variational method cannot be used directly. Combining the change of variables and the monotone method developed by Jeanjean in [L. Jeanjean, On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on 𝐑 N {\mathbf{R}}^{N} , Proc. Roy. Soc. Edinburgh Sect. A 129 1999, 4, 787–809], we obtain the existence of positive ground state solutions for the given problem.

Compactness properties for geometric fourth order elliptic equations with application to the Q-curvature flow

Abstract We prove the compactness of solutions of general fourth order elliptic equations which are L 1 L^{1} -perturbations of the Q-curvature equation on compact Riemannian 4-manifolds. Consequently, we prove the global existence and convergence of the Q-curvature flow on a generic class of Riemannian 4-manifolds. As a by-product, we give a positive answer to an open question by A. Malchiodi [J. reine angew. Math. 594 (2006), 137–174] on the existence of bounded Palais–Smale sequences for the Q-curvature problem when the Paneitz operator is positive with trivial kernel.

Positive and negative solutions for a class of Kirchhoff type problems on unbounded domain

In this paper, we study the existence of multiple positive and negative solutions for a class of Kirchhoff type problems on an unbounded domain. Using the variational methods, we show that our problem admits at least two positive and two negative solutions under different conditions. A cut-off functional is utilized to obtain the bounded Palais–Smale sequences.

Existence of positive solutions to Kirchhoff type problems with zero mass

The existence of positive solutions depending on a nonnegative parameter λ to Kirchhoff type problems with zero mass is proved by using variational method, and the new result does not require usual compactness conditions. A priori estimate and a Pohozaev type identity are used to obtain the bounded Palais–Smale sequences for constant coefficient nonlinearity, while a cut-off functional and Pohozaev type identity are utilized to obtain the bounded Palais–Smale sequences for the variable-coefficient case.

Standing waves for quasilinear Schrödinger equations

We study the existence of standing waves for a class of quasilinear Schrödinger equations −Δu+V(x)u−[Δ(u2)]u=μ|u|q−2u+|u|r−2u,x∈RN, where N≥3, 2<q<4, 4<r<2(2∗)≔4N/(N−2) and μ≥0. V is a positive potential. One main difficulty in dealing with this problem seems to be that of obtaining the boundedness of a (PS) sequence for the corresponding functional. We overcome this difficulty by using Jeanjean’s result [L. Jeanjean, On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer type problem, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 787–809]. Our results answer some questions raised in [J. M. do Ó, U. Severo, Quasilinear Schrödinger equations involving concave and convex nonlinearities, Comm. Pure Appl. Anal. 8 (2009) 621–644].

Existence of a positive solution to Kirchhoff type problems without compactness conditions

The existence of a positive solution to a Kirchhoff type problem on RN is proved by using variational methods, and the new result does not require usual compactness conditions. A cut-off functional is utilized to obtain the bounded Palais–Smale sequences.

Bounded Palais–Smale sequences for non-differentiable functions

The existence of bounded Palais–Smale sequences (briefly BPS) for functionals depending on a parameter belonging to a real interval and which are the sum of a locally Lipschitz continuous term and of a convex, proper, lower semicontinuous function, is obtained when the parameter runs in a full measure subset of the given interval. Specifically, for this class of non-smooth functions, we obtain BPS related to mountain pass and to global infima levels. This is done by developing a unifying approach, which applies to both cases and relies on a suitable deformation lemma.

On the existence of a solution for elliptic system related to the Maxwell–Schrödinger equations

In this paper we study the existence and the multiplicity of standing wave solutions for the nonlinear Schrödinger equation coupled with the Maxwell equations. It seems difficult to obtain a boundedness of a Palais–Smale sequence for the functional associated with this system. We overcome this difficulty by Jeanjean’s result [L. Jeanjean, On the existence of bounded Palais–Smale sequence and application to a Landesman–Lazer type problem set on R N , Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 787–809], which generalizes Struwe’s argument [M. Struwe, Variational Methods, 2nd ed., Springer, 1996; M. Struwe, The existence of a surface of constant mean curvature with free boundaries, Acta Math. 160 (1988) 19–64] and Zou’s result [W. Zou, Variant fountain theorem and their applications, Manuscripta Math. 104 (2001) 343–358].

Weak Linking Theorems and Schrödinger Equations with Critical Sobolev Exponent

In this paper we establish a variant and generalized weak linking theorem, which contains more delicate result and insures the existence of bounded Palais–Smale sequences of a strongly indefinite functional. The abstract result will be used to study the semilinear Schrodinger equation , where N ≥ 4; V,K,g are periodic in xj for 1 ≤ j ≤ N and 0 is in a gap of the spectrum of -Δ + V ; K>0 . If for an appropriate constant c , we show that this equation has a nontrivial solution.

On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on ℝN

Using the ‘monotonicity trick’ introduced by Struwe, we derive a generic theorem. It says that for a wide class of functionals, having a mountain-pass (MP) geometry, almost every functional in this class has a bounded Palais-Smale sequence at the MP level. Then we show how the generic theorem can be used to obtain, for a given functional, a special Palais–Smale sequence possessing extra properties that help to ensure its convergence. Subsequently, these abstract results are applied to prove the existence of a positive solution for a problem of the formWe assume that the functional associated to (P) has an MP geometry. Our results cover the case where the nonlinearity f satisfies (i) f(x, s)s−1 → a ∈)0, ∞) as s →+∞; and (ii) f(x, s)s–1 is non decreasing as a function of s ≥ 0, a.e. x → ℝN.

Bounded Palais-Smale mountain-pass sequences

Let I(λ, ·), λ ε ℝ, be a family of C 1-functionals having mountain-pass geometry. Under hypotheses which do not ensure that the mountain-pass level c(λ) is a monotone function of λ, it is shown that I(λ) has a bounded Palais-Smale sequence at level c(λ), for almost every λ.