Palais-Smale Condition Research Articles

Infinitely many small-energy solutions of a nonlinear boundary value problem with Dirichlet-to-Neumann operator

In this study, we investigate the following nonlinear boundary value problem{−Δu+u=|u|p−2u,inΩ∂u∂ν=Du,on∂Ω, where 2<p<+∞ if N=2 and 2<p≤2⁎ if N≥3, Ω is a bounded domain in RN (N≥2) with C∞ smooth boundary, u:Ω→R, Δu is the Laplacian operator, ν is the unit outer normal vector at ∂Ω, and D is the Dirichlet-to-Neumann operator. We prove that this equation has a sequence of solutions {um} that satisfies limm→∞⁡‖um‖Lp(Ω)=0 and liminfm→∞‖um‖L∞(Ω)≥1. To prove this, a new critical point theorem without the usual Palais-Smale condition is used.

Solvability of a Class of Fractional Advection–Dispersion Coupled Systems

The purpose of this study is to provide some criteria for the existence and multiplicity of solutions for a class of fractional advection–dispersion coupled systems with nonlinear Sturm–Liouville conditions and instantaneous and non-instantaneous impulses. Specifically, the existence is derived through the Nehari manifold method, and the proof of multiplicity is based on Bonanno and Bisci’s critical point theorem, which does not require proof that the functional satisfies the Palais–Smale condition. Finally, to illustrate the obtained results, an example is provided.

On a class of generalized Choquard system in fractional Orlicz-Sobolev spaces

We study a category of non-homogeneous Choquard systems involving the fractional ϕi(⋅)-Laplacian operator. By establishing suitable hypotheses on the Choquard nonlinearities and the continuous function ϕi, we confirm the existence of nontrivial solutions to the problem at hand. This achievement occurs within the structure of the fractional Orlicz-Sobolev space. Our research makes use of the mountain pass theorem, which circumvents the necessity for the Palais-Smale condition, and strategically leverages the Hardy-Littlewood-Sobolev inequality to enhance our analysis.

Existence and multiplicity of solutions for fractional differential equations with p-Laplacian at resonance

In this article, we investigate the existence and multiplicity of solutions for a fractional differential equations with p-Laplacian equation at resonance in the \(\psi\)-fractional space \(H^{\alpha, \beta; \psi}_p\). In addition, we show that the energy functional satisfies the Palais-Smale condition. For more information see https://ejde.math.txstate.edu/Volumes/2024/34/abstr.html

Multiple homoclinic solutions for nonsmooth second-order differential systems

In the present paper, we obtain infinitely many pairs of homoclinic solutions for a class of nonsmooth second-order differential systems when the energy functional associated is not continuously differentiable and does not satisfy the Palais–Smale condition.

Existence of solutions for singular elliptic problems with singular nonlinearities and critical Caffarelli-Kohn-Nirenberg exponent

In this article, we consider a singular elliptic problem with singular nonlinearities and critical Caffarelli-Kohn-Nirenberg exponent. By using variational methods and Palais-Smale condition, we show the existence of at least two nontrivial solutions. The result depends crucially on the parameters \(a,b,N,\beta,\gamma,\lambda,\mu\). For more information see https://ejde.math.txstate.edu/Volumes/2023/54/abstr.html

Dinkelbach Type Approximation Algorithms for Nonlinear Fractional Optimization Problems

In this paper we establish some approximation versions of the classical Dinkelbach algorithm for nonlinear fractional optimization problems in Banach spaces. We start by examining what occurs if at any step of the algorithm, the generated point desired to be a minimizer can only be determined with a given error. Next we assume that the step error tends to zero as the algorithm advances. The last version of the algorithm we present is making use of Ekeland’s variational principle for generating the sequence of minimizer-like points. In the final part of the article we deliver some results in order to achieve a Palais-Smale type compactness condition that guarantees the convergence of our Dinkelbach-Ekeland algorithm.

Remarks on compactness conditions and their application

We review typical compactness conditions used in variational techniques and some of their properties, and the relationships between them. In particular, we provide some new insights into results related to the Palais-Smale and Cerami conditions, and their comparison. See also https://ejde.math.txstate.edu/special/02/c3/abstr.html

Existence of solutions for a double-phase variable exponent equation without the Ambrosetti-Rabinowitz condition

Abstract The article deals with the existence of a pair of nontrivial nonnegative and nonpositive solutions for a nonlinear weighted quasilinear equation in R N {{\mathbb{R}}}^{N} , which involves a double-phase general variable exponent elliptic operator A {\bf{A}} . More precisely, A {\bf{A}} has behaviors like ∣ ξ ∣ q ( x ) − 2 ξ {| \xi | }^{q\left(x)-2}\xi if ∣ ξ ∣ | \xi | is small and like ∣ ξ ∣ p ( x ) − 2 ξ {| \xi | }^{p\left(x)-2}\xi if ∣ ξ ∣ | \xi | is large. Existence is proved by the Cerami condition instead of the classical Palais-Smale condition, so that the nonlinear term f ( x , u ) f\left(x,u) does not necessarily have to satisfy the Ambrosetti-Rabinowitz condition.

On variational methods for interval-valued functions with some applications

In this paper, we introduce the concept of sequential weak lower semicontinuity of interval-valued functions defined on a Banach space. We focus on the direct methods of the calculus of variations for interval-valued functions and extend some well-known results of F.E. Browder from scalar results to interval-valued cases. Moreover, we obtain some new results on the Palais–Smale condition and the coercivity for interval-valued functions. Finally, we apply the obtained results to interval-valued integral functions, to interval-valued optimal control problems, and to variational problems for interval-valued functions.

Caffarelli–Kohn–Nirenberg inequality for biharmonic equations with inhomogeneous term and Rellich potential

In this article, multiplicity of nontrivial solutions for an inhomogeneous singular biharmonic equation with Rellich potential are studied. Firstly, a negative energy solution of the studied equations is achieved via the Ekeland's variational principle and Caffarelli–Kohn–Nirenberg inequality. Then by applying Mountain pass theorem lack of Palais–Smale conditions, the second solution with positive energy is also obtained.

A local Palais-Smale condition and existence of solitary waves for a class of nonhomogeneous generalized Kadomtsev-Petviashvili equations

&lt;abstract&gt;&lt;p&gt;This paper is concerned with a class of nonhomogeneous generalized Kadomtsev-Petviashvili equations&lt;/p&gt; &lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ \bigg\{ \begin{array}{rl} &amp;amp; u_t + (|u|^{p-2}u)_x + u_{xxx} +h_x(x-\tau t, y) +\beta \nabla_y v = 0, \\ &amp;amp; v_x = \nabla_y u.\end{array} $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt; &lt;p&gt;By proving a local Palais-Smale condition, we manage to prove the existence of solitary waves with the help of a variational characterization on the smallest positive constant of an anisotropic Sobolev inequality (Huang and Rocha, J. Inequal. Appl., 2018,163). The novelty is to give an &lt;bold&gt;explicit estimate&lt;/bold&gt; on the sufficient condition of $ h $ to get the existence of solitary waves.&lt;/p&gt;&lt;/abstract&gt;

On a class of supercritical [formula omitted]-Laplacian problems

We derive a new existence result for a class of N-Laplacian problems where the classical N-Laplacian is replaced by an operator which admits some coefficients depending on the solution itself. Even if such coefficients make the variational approach more difficult, a suitable supercritical growth for the nonlinear term is allowed. Our proof, which exploits the interaction between two different norms, is based on a weak version of the Cerami–Palais–Smale condition and a proper decomposition of the ambient space. Then, a suitable generalization of the Ambrosetti–Rabinowitz Mountain Pass Theorem allows us to establish the existence of at least one nontrivial bounded solution.

Multiplicity and concentration results for fractional Kirchhoff equation with magnetic field

ABSTRACT In the paper, we considered the fractional Kirchhoff equation with magnetic field ( a ε 2 s + b ε 4 s − 3 [ u ] A / ε 2 ) ( − Δ ) A / ε s u + V ( x ) u = f ( | u | 2 ) u in R 3 , where ε is a small parameter, a, b are positive constants, s ∈ ( 3 4 , 1 ) , A : R 3 → R 3 is a Hölder continuous magnetic potential with exponent α ∈ ( 0 , 1 ] , [ u ] A / ε 2 = ∬ R 6 | u ( x ) − ε ı ( x − y ) ⋅ A ε ( x + y 2 ) u ( y ) | 2 | x − y | 3 + 2 s d x d y . With proper assumptions on V and f, we develop a new approach to prove the Palais–Smale condition for the corresponding energy functional and establish the multiplicity and concentration of solutions by using perturbation techniques and Ljusternik–Schnirelmann theory.

The regularized free fall II. Homology computation via heat flow

In [BOV20] Barutello, Ortega, and Verzini introduced a non-local functional which regularizes the free fall. This functional has a critical point at infinity and therefore does not satisfy the Palais-Smale condition. In this article we study the $L^2$ gradient flow which gives rise to a non-local heat flow. We construct a rich cascade Morse complex which has one generator in each degree $k\ge 1$. Calculation reveals a rather poor Morse homology having just one generator. In particular, there must be a wealth of solutions of the heat flow equation. These can be interpreted as solutions of the Schr\"odinger equation after a Wick rotation.

Existence of Pareto Solutions for Vector Polynomial Optimization Problems with Constraints

This paper deals with a vector polynomial optimization problem over a basic closed semi-algebraic set. By invoking some powerful tools from real semi-algebraic geometry, we first introduce the concept called tangency varieties; obtain the relationships of the Palais–Smale condition, Cerami condition, M-tameness, and properness related to the considered problem, in which the condition of Mangasarian–Fromovitz constraint qualification at infinity plays an essential role in deriving these relationships. At last, according to the obtained connections, we establish the existence of Pareto solutions to the problem in consideration and give some examples to illustrate our main findings.

Generalized critical Kirchhoff-type potential systems with Neumann Boundary conditions

In this paper, we consider a class of quasilinear stationary Kirchhoff type potential systems with Neumann Boundary conditions, which involves a general variable exponent elliptic operator with critical growth. Under some suitable conditions on the nonlinearities, we establish existence and multiplicity of solutions for the problem by using the concentration-compactness principle of Lions for variable exponents and the mountain pass theorem without the Palais–Smale condition.

Existence results for nonlinear Schrodinger equations involving the fractional (p,q)-Laplacian and critical nonlinearities

In this article, we consider the existence of ground state positive solutions for nonlinear Schrodinger equations of the fractional (p,q)-Laplacian with Rabinowitz potentials defined in \(R^n\), $$ ( -\Delta ) _p^{s_1}u+( -\Delta )_q^{s_2}u+V( \epsilon x) ( | u|^{p-2}u+| u| ^{q-2}u) =\lambda f( u) +\sigma | u| ^{q_{s_2}^{\ast }-2}u. $$ We prove existence by confining different ranges of the parameter \(\lambda\) under the subcritical or critical nonlinearities caused by \(\sigma=0\) or 1, respectively. In particular, a delicate calculation for the critical growth is provided so as to avoid the failure of a global Palais-Smale condition for the energy functional.&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2021/100/abstr.html

Non-autonomous weighted elliptic equations with double exponential growth

Abstract We consider the existence of solutions of the following weighted problem: { L : = - d i v ( ρ ( x ) | ∇ u | N - 2 ∇ u ) + ξ ( x ) | u | N - 2 u = f ( x , u ) i n B u &gt; 0 i n B u = 0 o n ∂ B , \left\{ {\matrix{{L: = - div\left( {\rho \left( x \right){{\left| {\nabla u} \right|}^{N - 2}}\nabla u} \right) + \xi \left( x \right){{\left| u \right|}^{N - 2}}} \hfill &amp; {u = f\left( {x,u} \right)} \hfill &amp; {in} \hfill &amp; B \hfill \cr {} \hfill &amp; {u &gt; 0} \hfill &amp; {in} \hfill &amp; B \hfill \cr {} \hfill &amp; {u = 0} \hfill &amp; {on} \hfill &amp; {\partial B,} \hfill \cr } } \right. where B is the unit ball of ℝ N , N #62; 2, ρ ( x ) = ( log e | x | ) N - 1 \rho \left( x \right) = {\left( {\log {e \over {\left| x \right|}}} \right)^{N - 1}} the singular logarithm weight with the limiting exponent N − 1 in the Trudinger-Moser embedding, and ξ(x) is a positif continuous potential. The nonlinearities are critical or subcritical growth in view of Trudinger-Moser inequalities of double exponential type. We prove the existence of positive solution by using Mountain Pass theorem. In the critical case, the function of Euler Lagrange does not fulfil the requirements of Palais-Smale conditions at all levels. We dodge this problem by using adapted test functions to identify this level of compactness.

A new critical point theorem and small magnitude solutions of magnetic Schrödinger equations with Landau levels

In this paper, we consider the following magnetic Schrödinger equation−ΔAu+V(x)u=Enu+|u|p−2u,x∈R2, where 2<p<∞, A(x)=(bx22,−bx12), x=(x1,x2)∈R2, En is an eigenvalue of −ΔA with infinitely multiplicity, and V is a non-zero and nonnegative function in Lp/(p−2)(R2,R). We prove that this equation has a sequence of non-zero solutions whose L∞ norms tend to zero along this sequence. To prove this, a new critical point theorem without the Palais-Smale condition is established.