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This paper is concerned with the following Schrödinger–Kirchhoff equation $$\begin{align*} & -\left(a+b\int_{\mathbb{R}^{3}}\left\vert \nabla u\right\vert ^{2}\,{\rm d}x\right) \Delta u+V(x)u\\ & \quad =k(x)|u|^{q-2}u-h(x)|u|^{p-2}u,\quad u\in H^1(\mathbb{R}^{3}), \end{align*}$$ − (a + b ∫ R 3 | ∇ u | 2 d x) Δu + V (x) u = k (x) | u | q − 2 u − h (x) | u | p − 2 u , u ∈ H 1 (R 3) , where a and b are positive constants, $ 1 \lt q \lt 2 \lt p \lt +\infty $ 1 < q < 2 < p < + ∞. Under some suitable assumptions on $ V(x) $ V (x) , $ k(x) $ k (x) and $ h(x) $ h (x) , we obtain the existence of multiple solutions for this problem via a variant of Clark's theorem. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of infinitely many solutions for a class of quasilinear elliptic m-polyharmonic Kirchhoff equations where the nonlinear function has a quasicritical growth at infinity and without assuming the Ambrosetti and Rabinowitz type condition. The new aspect consists in employing the notion of a Schauder basis to verify the geometry of the symmetric mountain pass theorem. Furthermore, we introduce a positive quantity $ \lambda _M $ λ M similar to the first eigenvalue of the m-polyharmonic operator to find a mountain pass solution, and also to discuss the sublinear case under large growth conditions at infinity and at zero. Our results are an improvement and generalization of the corresponding results obtained by Colasuonno-Pucci (Nonlinear Analysis: Theory, Methods and Applications, 2011) and Bae-Kim (Mathematical Methods in the Applied Sciences, 2020). [ABSTRACT FROM AUTHOR]

In this article, we consider a Δ α 1 , β 1 α , β -Laplacian system with critical nonlinearity in R N - ε 2 Δ α 1 , β 1 α , β u + b (X) u = g (X) | u | 2 ~ ∗ - 2 u + F u (X , u , v) , X ∈ R N , - ε 2 Δ α 1 , β 1 α , β v + b (X) v = g (X) | v | 2 ~ ∗ - 2 v + F v (X , u , v) , X ∈ R N , u (X) , v (X) → 0 as | X | → ∞ , where Δ α 1 , β 1 α , β is the subelliptic operator of the type Δ α 1 , β 1 α , β : = Δ x + Δ y + x 2 α y 2 β x α 1 + y β 1 2 Δ z ; x ∈ R N 1 ; y ∈ R N 2 , z ∈ R N 3 ; N = N 1 + N 2 + N 3 , α , β , α 1 , β 1 ≥ 0 , N ~ : = N 1 + N 2 + N 3 (1 + α + α 1 + β + β 2) , 2 ~ ∗ = 2 N ~ / (N ~ - 2) , X = (x , y , z) , (N ~ > 2). Under some proper conditions, we obtain the existence of standing wave solutions (u ε , v ε) which tend to the trivial solutions as ε → 0 . Moreover, we get m pairs of solutions for the above system under some extra assumptions. Our results improve and supplement some existing relevant results. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of multiple solutions in the Bessel Potential space for a new class of nonlinear fractional Schr¨odinger-Poisson systems involving the distributional Riesz fractional derivative. To reach our goal, we use the symmetric mountain pass theorem under some suitable assumptions on nonlinearity f(x, u) and potential V (x). [ABSTRACT FROM AUTHOR]

In this paper we study the generalized Lagrangian system with a small perturbation. We assume the main term in the system to have a maximum, but do not suppose any condition for perturbation term. Then we prove the existence of a periodic solution via Ekeland’s principle. Moreover, we prove a convergence theorem for periodic solutions of perturbed systems. [ABSTRACT FROM AUTHOR]

Using the variational methods, this paper is concerned with the existence and non-existence results to a class of elliptic equation with singular term on compact Riemannian manifold. In the context of complete, non-compact and non parabolic manifold, we obtain the non-existence result of solution for this type of elliptic problems. [ABSTRACT FROM AUTHOR]

In this paper, we study a higher order Kirchhoff problem with variable exponent of type M∫Ω|Dru|m(x)m(x)dxΔm(x)ru=f(x,u)inΩ,Dαu=0,on∂Ω,for eachα∈ℝNwith|α|≤r−1,$$ \left\{\begin{array}{ll}M\left({\int}_{\Omega}\frac{{\left&#x0007C;{\mathcal{D}}_ru\right&#x0007C;}&#x0005E;{m(x)}}{m(x)} dx\right){\Delta}_{m(x)}&#x0005E;ru&#x0003D;f\left(x,u\right)& \mathrm{in}\kern0.30em \Omega, \\ {}{D}&#x0005E;{\alpha }u&#x0003D;0,\kern0.30em & \mathrm{on}\kern0.30em \mathrm{\partial \Omega },\kern0.30em \mathrm{for}\ \mathrm{each}\kern0.4em \alpha \in {\mathrm{\mathbb{R}}}&#x0005E;N\kern0.4em \mathrm{with}\kern0.4em \mid \alpha \mid \le r-1,\end{array}\right. $$where Ω⊂ℝN$$ \Omega \subset {\mathrm{\mathbb{R}}}&#x0005E;N $$ is a smooth bounded domain, r∈ℕ∗,m∈C(Ω‾),1

By applying Clark's theorem as altered by Liu and Wang and the truncation method, we obtain a sequence of solutions for a Schrödinger–Poisson system − Δ u + V (x) u + ϕ u = f (u) in R 3 , − Δ ϕ = u 2 in R 3 with negative energy. A similar result is also obtained for the Schrödinger-Kirchhoff equation as follows: − 1 + ∫ R N ∇ u 2 Δ u + V (x) u = f (u) u ∈ H 1 (R N) . [ABSTRACT FROM AUTHOR]

We consider a class of noncooperative Schrödinger–Kirchhof–type system, which involves a general variable exponent elliptic operator with critical growth. Under certain suitable conditions on the nonlinearities, we establish the existence of infinitely many solutions for the problem by using the limit index theory, a version of concentration–compactness principle for weighted‐variable exponent Sobolev spaces and the principle of symmetric criticality of Krawcewicz and Marzantowicz. [ABSTRACT FROM AUTHOR]

We consider two types of nonlinear eigenvalue problems involving Laplace and p-Laplace operators (p > 2) . The main result establishes the existence of at least two nontrivial weak solutions in the case of the perturbed equation and the existence of a continuous spectrum in the case of the (p, 2)-equation. In both cases, variational methods play a central role in our arguments. [ABSTRACT FROM AUTHOR]

In this paper, we study the following system of fractional p(x)-Laplacian equations: (- Δ) p 1 (.) s u = λ V (x) | u | r 1 (x) - 2 u + ∂ F ∂ u (x , u , v) in Ω , (- Δ) p 2 (.) s v = λ K (x) | v | r 2 (x) - 2 v + ∂ F ∂ v (x , u , v) in Ω , u = v = 0 in ∂ Ω , where N ≥ 2 , s ∈ (0 , 1) , λ > 0 is a parameter, p 1 , p 2 ∈ C ( Ω ¯ 2 , R) , r 1 , r 2 ∈ C (Ω ¯ , R) , F : Ω × R 2 ⟶ R a Carathéodory function and V, K are two bounded functions. Under suitable nonstandard growth conditions on the reaction terms and using variational techniques, we establish the existence and multiplicity of solutions for the above problem. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and multiplicity of solutions for the Schrödinger–Bopp–Podolsky system { - Δ ⁢ u + V ⁢ (x) ⁢ u + ϕ ⁢ u = f ⁢ (u) + λ ⁢ | u | 4 ⁢ u in ⁢ ℝ 3 , - Δ ⁢ ϕ + a 2 ⁢ Δ 2 ⁢ ϕ = 4 ⁢ π ⁢ u 2 in ⁢ ℝ 3 , where x ∈ ℝ 3 , a > 0 , V ⁢ (x) ∈ 풞 ⁢ (ℝ 3 , ℝ) . Using variational methods and the symmetric mountain pass theorem, we establish the existence of multiple solutions for this system. [ABSTRACT FROM AUTHOR]

In this paper, the following modified fourth-order Schrödinger equation α Δ 2 u - Δ u + V (x) u - u Δ (u 2) = g (u) , in R N , and quasilinear Schrödinger equation with α = 0 are discussed. The nonlinearity is subquadratic, i.e., lim | t | → ∞ g (t) t 2 = 0 , and the potential V is indefinite in sign. By variational methods, we will prove the existence of multiple solutions if α ≠ 0 and N ≤ 6 or α = 0 and N ≥ 3 . [ABSTRACT FROM AUTHOR]

The existence of infinitely many weak solutions for a class of quasilinear elliptic pi(x)$$ {p}_i(x) $$‐Kirchhoff system is investigated by the symmetric mountain pass theorem. Furthermore, for the case pi(x)≡Const$$ {p}_i(x)\equiv Const $$ where i=1,2,...,n$$ i&amp;#x0003D;1,2,\dots, n $$, I introduce a positive quantity λM$$ {\lambda}_M $$ to find a mountain pass weak solution and to consider the sublinear case under large growth conditions at infinity and zero. [ABSTRACT FROM AUTHOR]

We consider the coupled propagation of an optical field and its second harmonic in a quadratic nonlinear medium governed by a coupled system of Schrodinger equations. We prove the existence of ring‐profiled optical vortex solitons appearing as solutions to a constrained minimization problem and as solutions to a min–max problem. In the case of the constrained minimization problem, solutions are shown to be positive with undetermined wave propagation constants, but in the min–max approach, the wave propagation constants can be prescribed. The quadratic nonlinearity introduces some interesting properties not commonly observed in other coupled systems in the context of nonlinear optics, such as the system not accepting any semi‐trivial solutions, meaning that optical solitons cannot be observed when, say, one of the beams is off. Additionally, the second harmonic always remains positive. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the multiplicity of solutions for a class of noncooperative Schrödinger systems in R N . The systems involves a variable exponent elliptic operators with critical nonlinearity. By applying the Limit Index Theory developed by Li (Nonlinear Anal 25, 1371–1389, 1995) and utilizing a version of the concentration-compactness principle and the principle of symmetric criticality of Krawcewicz and Marzantowicz, we obtain a sequence of solutions under appropriate assumptions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the multiplicity of homoclinic solutions for the following damped vibration problems $$ \ddot{x}(t)+B\dot{x}(t)-A(t)x(t)+H_{x}(t,x(t))=0,$$ where $A(t)\in (\mathbb{R},\mathbb{R}^{N})$ is a symmetric matrix for all $t\in \mathbb{R}$, $B=[b_{ij}]$ is an antisymmetric $N \times N$ constant matrix, and $H(t,x)\in C^{1}(\mathbb{R}\times B_{\delta},\mathbb{R})$ is only locally defined near the origin in $x$ for some $\delta>0$. With the nonlinearity $H(t,x)$ being partially sub-quadratic at zero, we obtain infinitely many homoclinic solutions near the origin by using a Clark's theorem.

The Darboux theorem, one of the fundamental results in analysis, states that the derivative of a real (not necessarily continuously) differentiable function defined on a compact interval has the intermediate value property, that is, attains each value between the derivatives at the endpoints. The Bolzano intermediate value theorem, which implies Darboux's theorem when the derivative is continuous, states that a continuous real‐valued function f$$ f $$ defined on [−1,1]$$ \left[-1,1\right] $$ satisfying f(−1)<0$$ f\left(-1\right)<0 $$ and f(1)>0$$ f(1)>0 $$, has a zero, that is, f(x)=0$$ f(x)&#x0003D;0 $$ for at least one number −1

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. [ABSTRACT FROM AUTHOR]

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, β, γ, λ, µ. [ABSTRACT FROM AUTHOR]

In this paper, we use variational methods to study the existence and multiplicity of non negative solutions for a p-Kirchhoff equation involving critical Sobolev exponent. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider the existence of multiple solutions for discrete boundary value problems involving the mean curvature operator by means of Clark’s Theorem, where the nonlinear terms do not need any asymptotic and superlinear conditions at 0 or at infinity. Further, the existence of a positive solution has been considered by the strong comparison principle. As an application, some examples are given to illustrate the obtained results.

In this paper, by using the concentration-compactness principle of Lions for variable exponents found in [Bonder JF, Silva A. Concentration-compactness principal for variable exponent space and applications. Electron J Differ Equ. 2010;141:1–18.] and the Mountain Pass Theorem without the Palais–Smale condition given in [Rabinowitz PH. Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., Vol. 65, Amer. Math. Soc., Providence, RI, 1986.], we obtain the existence and multiplicity solutions u = (u 1 , u 2 , .... u n) , for a class of Kirchhoff-Type Potential Systems with critical exponent, namely { − M i ( A i (u i)) div ( B i (∇ u i)) = | u i | s i (x) − 2 u i + λ F u i (x , u) in Ω , u = 0 on ∂ Ω , where Ω is a bounded smooth domain in R N (N ≥ 2) , and B i (∇ u i) = a i ( | ∇ u i | p i (x) ) | ∇ u i | p i (x) − 2 ∇ u i. The functions M i , A i , a i and a i ( 1 ≤ i ≤ n) are given functions, whose properties will be introduced hereafter, λ is the positive parameter, and the real function F belongs to C 1 (Ω × R n) , F u i denotes the partial derivative of F with respect to u i . Our results extend, complement and complete in several ways some of many works in particular [Chems Eddine N. Existence of solutions for a critical (p1(x),... , pn(x))-Kirchhoff-type potential systems. Appl Anal. 2020.]. We want to emphasize that a difference of some previous research is that the conditions on a i (.) are general enough to incorporate some differential operators of great interest. In particular, we can cover a general class of nonlocal operators for p i (x) > 1 for all x ∈ Ω ¯ . [ABSTRACT FROM AUTHOR]

We investigate the existence of a weak nontrivial solution for the following problem. Our analysis is generally bathed on discussions of variational based on the Mountain Pass theorem and some recent theories one the generalized Lebesgue-Sobolev space. This paper guarantees the existence of at least one weak nontrivial solution for our problem. More precisely, by applying Ambrosetti and Rabinowitz’s mountain pass theorem and under appropriate conditions, we show that there exists a positive number such that our problem has at least one nontrivial weak solution.

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. [ABSTRACT FROM AUTHOR]

In this paper, using variational methods and critical point theory, we establish the existence of two and infinitely many solutions for a fractional Kirchhoff type problem driven by a nonlocal operator of elliptic type in a fractional Orlicz-Sobolev space with homogeneous Dirichlet boundary conditions. Some examples are presented to demonstrate the application of our main results. [ABSTRACT FROM AUTHOR]

Given an objective function (differentiable or nondifferentiable), an important optimization problem is to find the global minimizer of the given function and a natural issue is to identify such functions with the global property. In this paper, we study real-valued functions defined on a Hilbert space and provide several sufficient conditions to ensure the existence of the global minimizer of such functions. For a proper lower semicontinuous function, we prove that a global minimizer can be guaranteed if it is bounded below, has the primal-lower-nice property and satisfies the generalized Palais-Smale condition. This result can cover the classic differentiable case whose proof depends heavily on the global existence of the ordinary differential equation. Several examples are constructed to show that the global minimizer may be violated if any of three conditions above is dropped. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of multiple solutions for discrete boundary value problems involving the mean curvature operator by means of Clark's Theorem, where the nonlinear terms do not need any asymptotic and superlinear conditions at 0 or at infinity. Further, the existence of a positive solution has been considered by the strong comparison principle. As an application, some examples are given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider the existence of solutions of the following Kirchhoff-type problem: { − ( a + b ∫ R 3 | ∇ u | 2 d x ) Δ u + V ( x ) u = f ( x , u ) , in R 3 , u ∈ H 1 ( R 3 ) , $$\begin{aligned} \textstyle\begin{cases} - (a+b\int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx )\Delta u+ V(x)u=f(x,u) , & \text{in }\mathbb{R}^{3}, \\ u\in H^{1}(\mathbb{R}^{3}),\end{cases}\displaystyle \end{aligned}$$ where a , b > 0 $a,b>0$ are constants, and the potential V ( x ) $V(x)$ is indefinite in sign. Under some suitable assumptions on f, the existence of solutions is obtained by Morse theory.

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. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of homoclinic solution for a class of damped vibration problem x ¨ (t) + (q (t) I N × N + B) x ˙ (t) + 1 2 q (t) B - A (t) x (t) + H x (t , x (t)) = f (t). For every k ∈ N , we obtain the 2kT-periodic solution x k by a standard minimizing argument. By taking the limit of { x k } , we get a solution x 0 of this problem. We prove that x 0 satisfies x 0 → 0 and x ˙ 0 → 0 as t → ± ∞ , and therefore x 0 is a homoclinic solution of the problem. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a version of Ekeland's variational principle for interval-valued functions by means of the Dancs-Hegedüs-Medvegyev theorem (Dancs et al. (1983) [14]). We also derive two versions of Ekeland's variational principle involving the generalized Hukuhara Gâteaux differentiability of interval-valued functions as well as a version of Ekeland's variational principle for interval-valued bifunctions. Finally, we apply these new versions of Ekeland's variational principle to fixed point theorems, to interval-valued optimization problems, to the interval-valued Mountain Pass Theorem, to noncooperative interval-valued games, and to interval-valued optimal control problems described by interval-valued differential equations. [ABSTRACT FROM AUTHOR]

In this paper, by using the concentration–compactness principle of Lions for variable exponent spaces and the Mountain Pass Theorem, we obtain the existence of nontrivial solutions for a class of (p 1 (x) , ... , p n (x)) -Kirchhoff-type potential systems type with critical exponent. [ABSTRACT FROM AUTHOR]

Using monotonicity methods and some variational argument we consider nonlinear problems which involve monotone potential mappings satisfying condition (S) and their strongly continuous perturbations. We investigate when functional whose minimum is obtained by a direct method of the calculus of variations satisfies the Palais-Smale condition, relate minimizing sequence and Galerkin approximaitons when both exist, then provide structure conditions on the derivative of the action functional under which bounded Palais-Smale sequences are convergent. Finally, we make some comment concerning the convergence of Palais-Smale sequence obtained in the mountain pass theorem due to Rabier.

Abstract We consider the existence of multiple solutions of the following singular nonlocal elliptic problem: { − M ( ∫ R N | x | − a p | ∇ u | p ) div ( | x | − a p | ∇ u | p − 2 ∇ u ) = h ( x ) | u | r − 2 u + H ( x ) | u | q − 2 u , u ( x ) → 0 as | x | → ∞ , $$\begin{aligned} \textstyle\begin{cases} -M(\int _{\mathbb{R} ^{N}}{ \vert x \vert ^{-ap} \vert \nabla u \vert ^{p}})\operatorname{div}( \vert x \vert ^{-ap} \vert \nabla u \vert ^{p-2}\nabla u)= h(x) \vert u \vert ^{r-2}u+H(x) \vert u \vert ^{q-2}u, \\ u(x)\rightarrow 0 \quad \text{as } \vert x \vert \rightarrow \infty , \end{cases}\displaystyle \end{aligned}$$ where x ∈ R N $x\in \mathbb{R} ^{N}$ , and M ( t ) = α + β t $M(t)=\alpha +\beta t$ . By the variational method we prove that the problem has infinitely many solutions when some conditions are fulfilled.

We discuss the existence and multiplicity of positive solutions for a class of second order quasilinear equations. To obtain our results we will use the Ekeland variational principle and the Mountain Pass Theorem.

In this paper we study the asymptotic behaviour of a family of elliptic systems, as far as the existence of solutions is concerned. We give a special attention to the asymptotic behaviour of W and V as ε goes to zero in the system − ε 2 Δ u + W (x) u = Q u (u , v) in R N , − ε 2 Δ v + V (x) v = Q v (u , v) in R N , u , v ∈ H 1 (R N) , u (x) , v (x) > 0 for each x ∈ R N , where ε > 0 , W and V are positive potentials of C 2 class and Q is a p-homogeneous function with subcritical growth. We establish the existence of a positive solution by considering two classes of potentials W and V. Our arguments are based on penalization techniques, variational methods and the Moser iteration scheme. [ABSTRACT FROM AUTHOR]

Introduction Fourth order equations have attracted many author's interest in different area of applied mathematics and physics. The interest in studying such problems arise in many applications such as: Micro Electro Mechanical systems, thin film theory, surface diffusion on solids, interface dynamics and flow in Hele-Shaw cells. The importance of considering such equations is due to many physical examples using mathematical modeling, which is mostly seen in the field of Newtonian fluids and elastic mechanics, in particular, the electro-rheological fluids. Consider the following p-biharmonic elliptic equation: {Δp² u + Δpu + λV(x)|u|p-2u = f(x, u) x ∈ RN, as λ → ∞ (2) where λ > 0, p > 1, N > 2p and V ∈ C(RN, R+) satisfying some certain conditions. Giri Choudhuri and Pradhan proved the existence and concentration phenomena of solutions on the set V-1(0). In 2017, the existence at the least one non-decreasing sequence of positive eigenvalues has been studied by several authors for following problem (Δp(x)² u - Δp(x)u = λ|u|p(x)-2u in Ω u ∈ W2,p(x) Ո W01,p(x) (3) Main Result The purpose of this paper is to investigate the existence of solution to the following problem involving both p(x)-Laplacian and p(x)-biharmonic. (Δp(x)² u + Δp(x)u = λ|u|p(x)-2u + μf(x, u) in Ω u ∈ W2,p(x) Ո W01,p(x) (1) where Ω is a bounded smooth domain RN. Parameters λ, μ are positive and p, q: Ω → R are functions that satisfying in the following conditions 1 < p- = minx∈Ω p(x) ≤ p(x) ≤ p+ = maxx∈Ω p(x) < ∞ Δ²p(x) u: = Δ(|Δu|p(x)-2Δu). is the p(x)-biharmonic operator which is a natural generalization of the p-biharmonic. In this paper, using the Mountain Pass Theorem, we prove that the problem (1) has at least one nontrivitial solution. Therefore we can prove our main result. Theorem1 (Mountain Pass Theorem): Let (X, II. II) be a real Banach space and let I: X → R be a continuously Gateaux differentiable function, such that I(0) = 0 and satisfying the (PS) condition. Suppose that (a) there exist constants p, α > 0 such that I(u) ≥ α if ||u|| = p, (b) there exists e ∈ X with ||e|| > p such that I(e) ≤ 0. Then I possesses a critical value C ≥ α, which can be characterized as C = infγ∈Гmaxu∈γ([0,1])I(u), where Г = {γ ∈ C([0,1], X) ; γ(0) = 0, γ(1) = e}. We prove main result using a positive constant that is introduced as follows and Mountain Pass Theorem. λ* = infu∈X ||un||>1 ∫Ω 1/p(x) (|Δu|p(x) + |∇u|p(x) )dx/∫Ω|u|p-dx Theorem2: Let p- < p(x) ≤ p2 * (x) and F: Ω × R → R with F(x, 0) = 0. Assume that following assumptions hold: (a) for all t ∈ R, x ↦ F(X, t) is measurable. (b) for almost every where x ∈ Ω ،x ↦ F(x, t) is locally lipschitz. (c) for all λ > 0, there exists cλ > 0 such that λ|u|p(x) < up- + cλ ∀u ∈ Ω (d) for all x ∈ Ω and t ∈ R, there exists c > 0 such that |f(x, t)| ≤ c(1 + |t|p(x)-1) (e) for λ < p-λ*/p+ and uniformly for x ∈ Ω, we have lim t→0 f(x, t) t|t|p(x)-2 ≤ λ (f) there exists t* > 0 such that for θ > p+, have 0 <θF(x, ξ) ≤ f(x, ξ)ξ ; ∀ x ∈ Ω, |ξ| ≥ t*. Then there exists μ* > 0 such that for μ ∈ (0, μ*), the problem (1) has at least one nontrivial weak solution in X. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and multiplicity of solutions for a class of quasi-linear elliptic problems driven by a nonlocal integro-differential operator with homogeneous Dirichlet boundary conditions. As a particular case, we study the following problem: { (−Δ)s pu = f(x, u) in Ω, u = 0 in RN \ Ω, where (−Δ)s p is the fractional p-Laplacian operator, Ω is an open bounded subset of RN with Lipschitz boundary and f : Ω×R → R is a generic Carathéodory function satisfying either a p−sublinear or a p−superlinear growth condition. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of solutions of the following Kirchhoff-type problem: { − (a + b ∫ R 3 | ∇ u | 2 d x) Δ u + V (x) u = f (x , u) , in R 3 , u ∈ H 1 (R 3) , where a , b > 0 are constants, and the potential V (x) is indefinite in sign. Under some suitable assumptions on f, the existence of solutions is obtained by Morse theory. [ABSTRACT FROM AUTHOR]

In this work we consider second order Dirichelet boundary value problem with nonlinearity depending on the derivative. Using a global diffeomorphism theorem we propose a new variational approach leading to the existence and uniqueness result for such problems.

In this article, we derive a sensitivity result for a nonlinear fractional ordinary elliptic system on a bounded interval with Dirichlet boundary conditions. More precisely, using a global implicit function theorem, we show that for each functional parameter there exists a unique solution, and that its dependence on the functional parameters is continuously differentiable. [ABSTRACT FROM AUTHOR]

Using monotonicity methods and some variational argument we consider nonlinear problems which involve monotone potential mappings satisfying condition (S) and their strongly continuous perturbations. We investigate when functional whose minimum is obtained by a direct method of the calculus of variations satisfies the Palais-Smale condition, relate minimizing sequence and Galerkin approximaitons when both exist, then provide structure conditions on the derivative of the action functional under which bounded Palais-Smale sequences are convergent. Finally, we make some comment concerning the convergence of Palais-Smale sequence obtained in the mountain pass theorem due to Rabier. [ABSTRACT FROM AUTHOR]

This note is concerned with a weighted Dirichlet-Steklov problem driven by the p-biharmonic operator ... Our approach is based on variational method and Ljusternick-Schnirelmann principle, we establish that the above problem admits a non-decreasing sequence of non-negative eigenvalues. [ABSTRACT FROM AUTHOR]

The paper focuses on a nonlocal Dirichlet problem with asymmetric nonlinearities. The equation is driven by the fractional Laplacian (− Δ) s for s ∈ (0 , 1) and exhibits a sublinear term containing a parameter λ , a linear term interfering with the spectrum of (− Δ) s and a superlinear term with fractional critical growth. The corresponding local problem governed by the standard Laplacian operator was investigated by F. O. de Paiva and A. E. Presoto. It can be recovered by letting s ↑ 1. The statement given here in the nonlocal setting is also related to extensively studied topics for local elliptic operators as the Brezis-Nirenberg problem and asymmetric nonlinearities. We go beyond the case of the standard Laplacian taking advantage of recent contributions on nonlocal fractional equations. Our main result establishes the existence of at least three nontrivial solutions, with one nonnegative and one nonpositive, provided the parameter λ > 0 is sufficiently small. In order to overcome the difficulties in the nonlocal setting we develop new arguments that are substantially different from those used in previous works. [ABSTRACT FROM AUTHOR]