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We prove the existence of infinitely many solutions to a fractional Choquard type equation (− Δ) p s u + V (x) | u | p − 2 u = (K ⁎ g (u)) g ′ (u) + ε W W (x) f ′ (u) in R N involving the fractional p -Laplacian and a general convolution term with critical growth. In order to obtain infinitely many solutions, we use a type of the symmetric mountain pass lemma which gives sequences of critical values converging to zero for even functionals. To assure the (P S) c conditions, we also use a nonlocal version of the concentration compactness lemma. [ABSTRACT FROM AUTHOR]

This paper deals with a pseudo-parabolic equation involving p(x)-Laplacian and variable nonlinear sources. Firstly, we use the Faedo-Galerkin method to give the existence and uniqueness of weak solution in the Sobolev space with variable exponents. Secondly, in the frame of variational methods, we classify the blow-up and the global existence of solutions completely by using the initial energy, characterized with the mountain pass level, and Nehari energy. In the supercritical case, we construct suitable auxiliary functions to determine the quantitative conditions on the initial data for the existence of blow-up or global solutions. The results in this paper are compatible with the problems with constant exponents. Moreover, we give broader ranges of exponents of source and diffusion terms in the discussion of blow-up or global solutions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence and multiplicity of solutions for a class of problems involving the Φ‐Laplacian operator with general assumptions on the nonlinearities, which include both semipositone cases and critical concave convex problems. The research is based on the subsupersolution technique combined with a truncation argument and an application of the Mountain Pass Theorem. The results in this paper improve and complement some recent contributions to this field. [ABSTRACT FROM AUTHOR]

This article concerns the existence and multiplicity of solutions for the singular p-biharmonic problem involving the Hardy potential and the critical Hardy-Sobolev exponent. To this end we use variational methods combined with the Mountain pass theorem and the Ekeland variational principle. We illustrate the usefulness of our results with and example. [ABSTRACT FROM AUTHOR]

This paper deals with a class of fractional Schrödinger‐Poisson system (−Δ)su+V(x)u−K(x)ϕ|u|2s∗−3u=a(x)f(u),x∈ℝ3(−Δ)sϕ=K(x)|u|2s∗−1,x∈ℝ3$$ \left\{\begin{array}{cc}{\left(-\Delta \right)}^su+V(x)u-K(x)\phi {\left|u\right|}^{2_s^{\ast }-3}u=a(x)f(u),\kern0.30em & x\in {\mathbb{R}}^3\\ {}{\left(-\Delta \right)}^s\phi =K(x){\left|u\right|}^{2_s^{\ast }-1},\kern0.30em & x\in {\mathbb{R}}^3\end{array}\right. $$with a critical nonlocal term and multiple competing potentials, which may decay and vanish at infinity, where s∈(34,1),2s∗=63−2s$$ s\in \left(\frac{3}{4},1\right),{2}_s^{\ast }=\frac{6}{3-2s} $$ is the fractional critical exponent. The problem is set on the whole space, and compactness issues have to be tackled. By employing the mountain pass theorem, concentration‐compactness principle, and approximation method, the existence of a positive ground state solution is obtained under appropriate assumptions imposed on V$$ V $$, K$$ K $$, a$$ a $$, and f$$ f $$. [ABSTRACT FROM AUTHOR]

Mountain Pass Theorem (MPT) is an important result in variational methods with multiple applications in partial differential equations involved in mathematical physics. Starting from a variant of MPT, a new result concerning the existence of the solution for certain mathematical physics problems involving p-Laplacian and p-pseudo-Laplacian has been obtained. Based on the main theorem, the existence, possibly the uniqueness, and characterization of solutions for models such as nonlinear elastic membrane, glacier sliding, and pseudo torsion problem have been obtained. The novelty of the work consists of the formulation of the central result under weaker conditions requested by the chosen variant of MPT, the proof of this statement, and its application in solving above mentioned problems. While the expressions of such Dirichlet and/or von Neumann problems were already completed, this proposed solving method suggests some specific numerical methods to construct the appropriate solution. A general goal of this paper is the extension of the applicative pallet of this way to construct the solutions encountered in modeling real processes developed within new emerging technologies. [ABSTRACT FROM AUTHOR]

In this paper, we study the Schrödinger equation involving the N/s -fractional Laplacian... where εis a positive parameter, N = ps, s ∈ (0, 1). The nonlinear function f has exponential growth and the potential function V is a continuous function satisfying some suitable conditions. Our problem lacks compactness. By using the Ljusternik-Schnirelmann theory, we obtain the existence, multiplicity and concentration of nontrivial nonnegative solutions for small values of the parameter. [ABSTRACT FROM AUTHOR]

In the present article deal with the existence and multiplicity of solutions to a class of p(x)-Kirchhoff type problem with Steklov boundary-value. By variational approach and theory of the variable exponent Sobolev spaces, under appropriate assumptions on f, we obtain existence of infinitely solutions and at least one nontrivial weak solution. [ABSTRACT FROM AUTHOR]

In this paper, a special class of boundary value problems, − ▵ u = λ u q + u r , in Ω , u > 0 , in Ω , n · ∇ u + g (u) u = 0 , on ∂ Ω , where 0 < q < 1 < r < N + 2 N − 2 and g : [ 0 , ∞) → (0 , ∞) is a nondecreasing C 1 function. Here, Ω ⊂ R N ( N ≥ 3 ) is a bounded domain with smooth boundary ∂ Ω and λ > 0 is a parameter. The existence of the solution is verified via sub- and super-solutions method. In addition, the influences of parameters on the minimum solution are also discussed. The second positive solution is obtained by using the variational method. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in existence and multiplicity of solutions to anisotropic elliptic equations of Kirchhoff-type given by L (u) = λ ∫ Ω F (x , u) r f (x , u) + | u | p ∗ − 2 u in Ω , u = 0 on ∂ Ω , (P) λ where L (u) := − ∑ i = 1 N M ∫ Ω ∂ u ∂ x i p i ∂ ∂ x i × ∂ u ∂ x i p i − 2 ∂ u ∂ x i , Ω is a smooth bounded domain of R N , N > 2 , λ > 0 and r ≥ 0 are parameters, 1 < p 1 ≤ ⋯. ≤ p N < p ∗ , where p ∗ = N p ¯ / (N − p ¯) is the critical exponent and p ¯ = N / ∑ i = 1 N 1 p i . The function M : [ 0 , + ∞) → (− ∞ , + ∞) is only continuous with M (0) > 0 and can change its sign, f : Ω × R → R is a function with subcritical growth and F (x , t) = ∫ 0 t f (x , s) d s. Under appropriate assumptions on f, applying an adequate truncation argument on M and the Concentration Compactness-Principle for the anisotropic operator [1], we obtain the existence of a nontrivial solution for (P) λ by the Mountain Pass Theorem [2] and multiplicity of solutions by Krasnoselskii's genus and the symmetric Mountain Pass Lemma [3]. As a model case for M and f, we can consider M (t) = a + b t or M (t) = a − b t with a > 0 , b ≥ 0 and f (x , t) = a (x) | t | q − 2 t with a ∈ L ∞ (Ω) and appropriate q ∈ (1 , p ∗). [ABSTRACT FROM AUTHOR]

In this paper, we study an initial boundary value problem for a nonlocal parabolic equation that includes a diffusion term and convex–concave nonlinearities. By establishing the L q -estimate and analyzing its energy, we prove the existence of global solutions and obtain some blow-up conditions. Given the variational structure of the problem, we utilize the Mountain-pass theorem to demonstrate the existence of nontrivial steady-state solutions. Additionally, we establish the dynamical behavior of global solutions whose trajectory is relatively compact in H 0 1 (Ω) , and which uniformly converge to some non-zero steady state after a long time, owing to the energy functional satisfying the P.S. condition. Finally, we derive an unstable steady states sequence by using another minimax theorem. [ABSTRACT FROM AUTHOR]

We study the fractional p-Kirchhoff equation ... where (-∆)s2 is the fractional p-Laplacian operator, a and b are strictly positive real numbers, s ∈ (0, 1), 1 < p < N/s, and p < q < p*s - 2 with p*s = Np/N-ps. By using the variational method, we prove the existence and uniqueness of global minimum or mountain pass type critical points on the Lp-normalized manifold S(c):= {u∈ 2 Ws,p(RN): RRN = cp. [ABSTRACT FROM AUTHOR]

In this article, we study the existence of nontrivial weak solutions for the following boundary value problem: − Δ γ u = f (x , u) in Ω , u = 0 on ∂ Ω , where Ω is a bounded domain with smooth boundary in ℝ N , Ω ∩ { x j = 0 } ≠ ∅ for some j, Δγ is a subelliptic linear operator of the type Δ γ : = ∑ j = 1 N ∂ x j (γ j 2 ∂ x j ) , ∂ x j : = ∂ ∂ x j , N ≥ 2 , where γ(x) = (γ1(x), γ2(x), ..., γN(x)) satisfies certain homogeneity conditions and degenerates at the coordinate hyperplanes and the nonlinearity f(x, ξ) is of subcritical growth and does not satisfy the Ambrosetti-Rabinowitz (AR) condition. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the existence of solutions to the anisotropic nonlocal problem { (P) δ } − ∑ i = 1 N ∂ ∂ x i ∂ u ∂ x i p i − 2 ∂ u ∂ x i = ∫ Ω F (x , u) r f (x , u) + δ | u | p ∗ − 2 u in Ω , u ≥ 0 in Ω , u = 0 on ∂ Ω , where Ω is a smooth bounded domain of R N , N ≥ 2 , δ = 0 or δ = 1 , p i and r ≥ 0 , 1 < p 1 ≤ ⋯ ≤ p N < p ∗ are parameters where p ∗ = N p ¯ / (N − p ¯) is a critical exponent, p ¯ = N / ∑ i = 1 N 1 / p i and p ¯ < N. The nonlinearity f : Ω × R → R can be discontinuous, has subcritical growth subcritical growth and F (x , t) = ∫ 0 t f (x , s) d s. Under appropriate assumptions on f, we obtain the existence of nontrivial solutions for (P) δ using the Ekeland's Variational Principle, Nonsmooth Mountain Pass Theorem and a Concentration Compactness-Principle. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the existence results of a sequence of infinitely many homoclinic orbits for the discrete p‐Laplacian with unbounded potentials without the Ambrosetti and Rabinowitz condition. The strategy of the proof for these results is to approach the problem using the mountain pass theorem, the fountain theorem, and dual fountain theorem. [ABSTRACT FROM AUTHOR]

We are concerned with the following p (x) -Laplacian equations in R N − △ p (x) u + | u | p (x) − 2 u = f (x , u) in R N. The nonlinearity is superlinear but does not satisfy the Ambrosetti-Rabinowitz type condition. Our main difficulty is that the weak limit of (PS) sequence is not always the weak solution of this problem. To overcome this difficulty, by adding potential term and using mountain pass theorem, we get the weak solution u λ of perturbation equations. First, we prove that u λ ⇀ u as λ → 0 . Second, by using vanishing lemma, we get that u is a nontrivial solution of the original problem. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence and multiplicity of solutions for the fractional Choquard‐type Schrödinger‐Kirchhoff equations with electromagnetic fields and critical nonlinearity: ε2sM([u]s,A2)(−Δ)Asu+V(x)u=(|x|−α∗F(|u|2))f(|u|2)u+|u|2s∗−2u,x∈RN,u(x)→0,as|x|→∞,where (−Δ)As is the fractional magnetic operator with 00 is a positive parameter. The electric potential V∈C(RN,R0+) satisfies V(x)=0 in some region of RN, which means that this is the critical frequency case. We first prove the (PS)c condition, by using the fractional version of the concentration compactness principle. Then, applying also the mountain pass theorem and the genus theory, we obtain the existence and multiplicity of semiclassical states for the above problem. The main feature of our problems is that the Kirchhoff term M can vanish at zero. [ABSTRACT FROM AUTHOR]

The aim of this paper is the study of existence of homoclinic solutions for a class of nonlinear differential equations involving p -Laplacian. Under suitable growth conditions we prove that the considered problem has at least one nontrivial homoclinic solution. The proof is based on the mountain-pass theorem with Cerami's condition, Brezis–Lieb lemma and variational method. [ABSTRACT FROM AUTHOR]

We consider a Brezis-Nirenberg problem on the hyperbolic space Hn. By using the stereographic projection, the problem becomes a singular problem on the boundary of the open ball B1(0) ⊂ Rn. Thanks to the Hardy inequality, in a version due to Brezis-Marcus, the difficulty involving singularities can be overcame. We use the mountain pass theorem due to Ambrosetti-Rabinowitz and Brezis-Nirenberg arguments to obtain a nontrivial solution. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of quasilinear elliptic equations with Φ-Laplacian operator and critical growth. Using the symmetric mountain pass theorem and the concentration-compactness principle, we demonstrate that there exists λi>0 such that our problem admits i pairs of nontrivial weak solutions provided λ∈(0,λi). [ABSTRACT FROM AUTHOR]

I study the existence of ground state for the following periodic Schrödinger equation under the circumstance that the nonlinear term satisfies odd conditions by using the Mountain Pass Lemma and variational method. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of homoclinic solutions for a class of fourth-order nonautonomous differential equations [ABSTRACT FROM AUTHOR]

In this article, we prove the existence of solutions for some discrete nonlinear difference equations subjected to a potential boundary type condition. We use a variational technique that relies on Szulkin's critical point theory, which ensures the existence of solutions by ground state and mountain pass methods. [ABSTRACT FROM AUTHOR]

Using variational arguments, we prove the existence of infinitely many solutions to a class of p-biharmonic equation in ℝN. The existence of nontrivial solution is established under a new set of hypotheses on the potential V (x) and the weight functions h1(x), h2(x). [ABSTRACT FROM AUTHOR]

In this paper, we study a transmission problem given by a system of two nonlinear equations of p(x)-Kirchhoff type with nonstandard growth conditions. Using a variational approach, we establish at least one nontrivial weak solution. [ABSTRACT FROM AUTHOR]

In this paper, we study the p -Laplace elliptic equations under the Dirichlet boundary condition. We give a general and weak sufficient condition for the existence of a sequence of solutions converging to zero. This result is proved by applying the symmetric mountain pass lemma obtained in our earlier paper. For some elliptic equations with parameters, we decide whether the zero solution is an accumulation point or an isolated point in the set of all solutions. [ABSTRACT FROM AUTHOR]

This article concerns the nonhomogeneous Klein-Gordon-Maxwell equation -Δu + u - (2w + Φ) Φu + |u|p-1u + h(x), in ℝ³; ΔΦ = (w + Φ)u², in ℝ³, where w > 0 is constant, p ∈ (1; 5). Under appropriate assumptions on h(x), the existence of at least two solutions is obtained by applying the Ekeland's variational principle and the Mountain Pass Theorem in critical point theory. [ABSTRACT FROM AUTHOR]

This article concerns the existence of solutions to boundary-value problems involving the p(x)-biharmonic operator. Our technical approach is the variational-hemivariational inequality on bounded domains by using the mountain pass theorem and the critical point theory for Motreanu-Panagiotopoulos type functionals. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of multiple positive solutions for a degenerate nonlocal problem on unbounded domain. Using the Ekeland's variational principle combined with the mountain pass theorem, we show that problem admits at least two positive solutions under several different conditions. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of infinitely many solutions to following nonlinear Kirchhoff–Schrödinger–Poisson system a + b ∫ R 3 | ∇ u | 2 + V ( x ) u 2 - Δ u + V ( x ) u + λ l ( x ) ϕ u = f ( x , u ) , x ∈ R 3 , - Δ ϕ = λ l ( x ) u 2 , x ∈ R 3 , where constants a > 0 , b ⩾ 0 and λ ⩾ 0 . When f has sublinear growth in u , we obtain infinitely many solutions under certain assumption that V do not have a positive lower bound. The technique we use in this paper is the symmetric mountain pass theorem established by Kajikiya (2005). [ABSTRACT FROM AUTHOR]

We study semilinear elliptic equations which are called MEMS model with fringing field. Using the variational method and dual approach, we show the multiple existence of positive solutions. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence and concentration of ground state solutions for the following nonlinear Schrödinger–Poisson-type system with doubly critical growth - ε 2 Δ u + V (x) u - ϕ | u | 3 u = | u | 4 u + f (u) , in R 3 , - ε 2 Δ ϕ = | u | 5 , in R 3 , where ε > 0 is a small parameter. By employing the concentration-compactness principle and mountain pass theorem, we prove the existence of positive ground state solutions v ε with exponential decay at infinity for ε sufficiently small under some suitable assumptions on the potential V and nonlinearity f. Moreover, as ε → 0 + , v ε concentrates around a global minimum point of V. [ABSTRACT FROM AUTHOR]

In this article, we consider the quasilinear Choquard equation with critical nonlinearity in R N : − ε 2 Δ u + V (x) u − ε 2 u Δ u 2 = ∫ R N | u | 2 2 μ ∗ | x − y | μ d y | u | 2 2 μ ∗ − 2 u + h (x , u) u , where 0 < μ < N , N ≥ 3 , 2 μ ∗ = (2 N − μ) ∕ (N − 2) is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality, ε is a real parameter. By using a change of variables, the quasilinear equations are reduced to a semilinear one, whose associated functionals are well defined in the usual Sobolev space and satisfy the geometric conditions of the mountain pass theorem. Under suitable assumptions on V and h , we investigate the existence and multiplicity of solutions for the above problem by using the mountain pass theorem and index theory. In order to overcome the lack of compactness, we apply the concentration-compactness principle. [ABSTRACT FROM AUTHOR]

We are concerned with the following p-biharmonic equations: Δ p 2 u + M (∫ R N Φ 0 (x , ∇ u) d x) div (φ (x , ∇ u)) + V (x) | u | p − 2 u = λ f (x , u) in R N , where 2 < 2 p < N , Δ p 2 u = Δ (| Δ u | p − 2 Δ u) , the function φ (x , v) is of type | v | p − 2 v , φ (x , v) = d d v Φ 0 (x , v) , the potential function V : R N → (0 , ∞) is continuous, and f : R N × R → R satisfies the Carathéodory condition. We study the existence of weak solutions for the problem above via mountain pass and fountain theorems. [ABSTRACT FROM AUTHOR]

In this paper, we first obtain the existence of solutions for a class of elliptic equations involving critical variable exponents and nonlinear boundary values by the mountain pass theorem and concentration compactness principle. Then, under suitable assumptions, we obtain a sequence of solutions with positive energies going towards infinity by Fountain Theorem. [ABSTRACT FROM AUTHOR]

In this paper, the authors investigate the following fractional Kirchhoff boundary value problem: {(a+b∫0T(0Dtαu)2dt)tDTα(0Dtαu)+λV(t)u=f(t,u),t∈[0,T],u(0)=u(1)=0, where the parameter λ>0 and constants a,b>0. By applying the mountain pass theorem and the linking theorem, some existence results on the above fractional boundary value problem are obtained. It should be pointed out that the potential V may be sign-changing. [ABSTRACT FROM AUTHOR]

We are concerned with the following quasilinear Choquard equation: −Δpu+V(x)|u|p−2u=λ(Iα∗F(u))f(u)in RN,F(t)=∫0tf(s)ds, where 1, Δpu=∇⋅(|∇u|p−2∇u) is the p-Laplacian operator, the potential function V:RN→(0,∞) is continuous and F∈C1(R,R). Here, Iα:RN→R is the Riesz potential of order α∈(0,p). We study the existence of weak solutions for the problem above via the mountain pass theorem and the fountain theorem. Furthermore, we address the behavior of weak solutions to the problem near the origin under suitable assumptions for the nonlinear term f. [ABSTRACT FROM AUTHOR]