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This article concerns the existence of solutions to the Schrödinger- Poisson system –Δpu +|u|p–2u + λΦu =|u|q–2u + h(x) in R³, –ΔΦ = u² in R³, where 4=3 < p < 12=5, p < q < p* = 3p=(3 – p), Δpu = div(|Δu|p–2Δu), λ > 0, and h ≠ 0. The multiplicity results are obtained by using Ekeland's variational principle and the mountain pass theorem. [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 article concerns the nonhomogeneous Klein-Gordon equation coupled with a Born-Infeld type equation, ... where ω is a positive constant. We obtain the existence of two solutions using the Mountain Pass Theorem, and the Ekeland's variational principle in critical point theory. [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 consider the existence of localized sign-changing solutions for the semiclassical Kirchhoff equation ... where 4 < p < 2* = 6, ε> 0 is a small parameter, V (x) is a positive function that has a local minimum point P. When ε → 0, by using a minimax characterization of higher dimensional symmetric linking structure via the symmetric mountain pass theorem, we obtain an infinite sequence of localized sign-changing solutions clustered at the point P. [ABSTRACT FROM AUTHOR]

We consider a Dirichlet problem having a double phase differential operator with unbalanced growth and reaction involving the combined effects of a concave (sublinear) and of a convex (superlinear) terms. We allow the coefficient E ∈ L∞(Ω) of the concave term to be sign changing. We show that when ||E||∞ is small the problem has at least two bounded positive solutions. [ABSTRACT FROM AUTHOR]

This work establishes the existence of positive solutions to a quasilinear singular elliptic equations involving the (p-q)-Laplacian operator with singularities and a vanishing potential. We adapt the penalization method developed by del Pino and Felmer and we consider an auxiliary problem whose corresponding functional satisfies the geometry of the mountain-pass theorem; then, we prove that the Palais-Smale sequences are bounded in a Sobolev space; after that, we show that the auxiliary problem has a solution. Finally, we use the Moser iteration scheme to obtain an appropriate estimate and we conclude that the solution to the auxiliary problem is also a solution to the original problem. [ABSTRACT FROM AUTHOR]

This article considers p(x)-Kirchhoff type problems with Robin boundary conditions. Using the mountain pass theorem, the Ekeland's variational principle, and Krasnoselskii's genus theory, we prove that the problem has at least two nontrivial weak solutions or infinitely many nontrivial weak solutions under some suitable conditions on the nonlinearities. The main results improve and generalize the previous ones introduced in [2, 7]. [ABSTRACT FROM AUTHOR]

This article concerns the ground state solution of the p-Laplacian equation with finitely many critical nonlinearities. By using the refined Sobolev inequality with Morrey norm and variational methods, we establish the existence of nonnegative ground state solution. [ABSTRACT FROM AUTHOR]

Let Ω⊂ℝN be a bounded domain with smooth boundary and 0∈Ω. For 00, c

In this article, we consider the multiplicity of solutions for nonhomogeneous Schrödinger-Poisson systems under the Berestycki-Lions type conditions. With the aid of Ekeland's variational principle, the mountain pass theorem and a Pohožaev type identity, we prove that the system has at least two positive solutions. [ABSTRACT FROM AUTHOR]

This article concerns the existence of positive solutions for the second order equation involving a nonlocal term -Δu = γ(-Δ)-1u + |u|p-1u, under Dirichlet boundary conditions. We prove the existence of positive solutions depending on the positive real parameter γ>0, and up to the critical value of the exponent p, i.e. when 1

In this article we establish the existence of solutions for elliptic problem involving a gradient term. To handle the so-called non-variational problem, we use a variational methods. We assume that the nonlinear term satisfies an asymptotically linear growth condition or a superlinear growth condition. We show the existence of at least one positive solution and one negative solution. [ABSTRACT FROM AUTHOR]

In this work, we establish the existence of solutions for the nonlinear nonlocal system of equations involving the fractional Laplacian, (–Δ)s u = au + bv + 2p/p+q ∫Ω | v(y) |q/|x – y|μ dy|u|p–2 u + 2ε1 ∫Ω | u(y) |2*μ/|x – y|μ dy|u|2*μ –2u in Ω, (–Δ)s v = bu + cv + 2q/p+q ∫Ω | u(y) |p/|x – y|μ dy|v|q–2 v + 2ε2 ∫Ω |v(y)|2*μ/|x – y|μ dy|v|2*μ –2v in Ω, u = v = 0 in RN \ Ω, where (–Δ)s is the fractional Laplacian operator, Ω is a smooth bounded domain in RN 0 < s < 1, N > 2s, 0 < μ < N, ε1, ε2 ≥ 0, 1 < p, q ≤ 2μ* and 2μ* = 2N – μ/N – 2s is the upper critical exponent in the Hardy-Littlewood-Sobolev inequality. The nonlinearities can interact with the spectrum of the fractional Laplacian. More specifically, the interval defined by the two eigenvalues of the real matrix from the linear part contains an eigenvalue of the spectrum of the fractional Laplacian. In this case, resonance phenomena can occur. [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]

We investigate the Henon type equation involving the critical Sobolev exponent with Dirichret boundary condition ..., in Ω included in a unit ball, under several conditions. Here, Ψ is a non-trivial given function with 0 ≤ Ψ ≤ 1 which may vanish on ∂Ω. Let λ1 be the first eigenvalue of the Dirichret eigenvalue problem -Δφ = λΨφ in Ω. We show that if the dimension N ≥ 4 and 0 < λ < λ1, there exists a positive solution for small α > 0. Our methods include the mountain pass theorem and the Talenti function. [ABSTRACT FROM AUTHOR]

In this article, we study the nonhomogeneous Klein-Gordon-Maxwell system -Δu + λV(x)u - K(x)(2w +φ)φu = f(x, u) + h(x), x ∈ ℝ³, Δφ = K(x)(w + φ)u², x ∈ ℝ³, where w > 0 is a constant and λ > 0 is a parameter. Using the Linking theorem and Ekeland's variational principle in critical point theory, we prove the existence of multiple solutions, under suitable assumptions that allow a sign-changing potential. [ABSTRACT FROM AUTHOR]

In this article we obtain weak radial solutions for the inhomogeneous elliptic system ∆2u + V1(|x|)|u| q−2u = Q(|x|)Fu(u, v) in R N , ∆2 v + V2(|x|)|v| q−2 v = Q(|x|)Fv(u, v) in R N , u, v ϵ D 2,2 0 (R N ), N ≥ 5, where ∆2 is the biharmonic operator, Vi, Q ચ C0 ((0, +∞), [0, +∞)), i = 1, 2, are radially symmetric potentials, 1 < q < N, q 6= 2, and F is a s-homogeneous function. Our approach relies on an application of the Symmetric Mountain Pass Theorem and a compact embedding result proved in [17]. [ABSTRACT FROM AUTHOR]

In this article, we study the nonlinear fractional Schrödinger equation (−∆) αu+ V (x) u= f (x, u) u∊ Hα (Rn, R), where (−∆) α (α∊(0, 1)) stands for the fractional Laplacian of order α, x∊ Rn, V∊ C (Rn, R) may change sign and f is only locally defined near the origin with respect to u. Under some new assumptions on V and f, we show that the above system has infinitely many solutions near the origin. Some examples are also given to illustrate our main theoretical result. [ABSTRACT FROM AUTHOR]

In this article we study two problems, a nonlinear eigenvalue problem involving the p(x)-Laplacian and a subcritical boundary value problem for the same operator. We work on the variable exponent Sobolev spaces and use one of the variants of the Mountain-Pass Lemma. [ABSTRACT FROM AUTHOR]

We consider a nonlinear elliptic equation driven by the Dirichlet p-Laplacian with a singular term and a (p - 1)-linear perturbation which is resonant at +∞ with respect to the principal eigenvalue. Using variational tools, together with suitable truncation and comparison techniques, we show the existence of at least two positive smooth solutions. [ABSTRACT FROM AUTHOR]

In this article, we study the quasilinear Schrödinger equation with the critical exponent and singular coefficients, ... where N ≤ 3, 2 < 22* (μ), 2*(s) = 2(N-s)/N-2, and λ, μ, v are parameters with λ, > 0, μ, v ∈ [0, 2). By applying the Mountain Pass Theorem and the Concentration Compactness Principle, we establish the existence of the ground state solutions to the above problem. [ABSTRACT FROM AUTHOR]

In this article, by using the Fountain theorem and Mountain pass theorem in critical point theory without Palais-Smale (PS) condition, we show the existence and multiplicity of solutions to the degenerate Kirchhoff type problem with the fractional p-Laplacian ... where (Δ)sp is the fractional p-Laplace operator with 0 < s < 1 < p < 1 < p <∞, Ω is a smooth bounded domain of RN, N > 2s, a; b > 0 are constants and f : Ω R → R is a continuous function. [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]

In this article, we prove the existence of multiple solutions for following fractional Schrödinger-Poisson system with sign-changing potential (-∆)su + V (x)u + λφu = ƒ(x, u), x ∈ R 3, (-∆)tø = u², x ∈ R³, where (-∆)α denotes the fractional Laplacian of order α ∈ (0, 1), and the potential V is allowed to be sign-changing. Under certain assumptions on ƒ, we obtain infinitely many solutions for this system. [ABSTRACT FROM AUTHOR]

By using the Mountain Pass Lemma, we establish sufficient conditions for the existence of solitons for the discrete nonlinear Schródinger equations. [ABSTRACT FROM AUTHOR]

In this article, we study the Schródinger-Poisson system... where h1(x); h2(x); h3(x) and V (x) are allowed to be sign-changing and μ > 0 is a parameter. Under some appropriate assumptions on V (x), we obtain the existence of two different solutions for the above system via variational methods. [ABSTRACT FROM AUTHOR]

In this article, we consider the Kirchhoff equation ... and under suitable assumptions on the main term f in the equation, some existence results are obtained by the variational methods and some analysis techniques. [ABSTRACT FROM AUTHOR]

This article concerns the existence of sign-changing solutions to nonlocal Kirchhoff type problems of the form -...∆u = f(x, u) in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in RN (N = 1, 2, 3) with smooth boundary, a > 0, b ≥ 0, and f : Ω × R → R is a continuous function. We first establish a new sign-changing version of the symmetric mountain pass theorem and then apply it to prove the existence of a sequence of sign-changing solutions with higher and higher energy. [ABSTRACT FROM AUTHOR]

This article concerns the existence of solutions for the fractional Hamiltonian system ... where α ∊ (1/2, 1), L ∊ C(R;Rn2) is a symmetric and positive definite matrix. The novelty of this article is that if ... and the nonlinearity W(t; u) involves a combination of superquadratic and subquadratic terms, the Hamiltonian system possesses at least two nontrivial solutions. [ABSTRACT FROM AUTHOR]

In this article, we study the quasilinear elliptic equation ... where the potential V (x) and the nonlinearity g(x, u) are allowed to be signchanging. Under some suitable assumptions on V and g, we obtain the multiplicity of solutions by using minimax methods. [ABSTRACT FROM AUTHOR]

By using variational methods we study the existence of positive solutions for a class of quasilinear elliptic problems with Robin boundary con- ditions. [ABSTRACT FROM AUTHOR]

We study the existence and multiplicity of homoclinic solutions for the second-order damped differential equation ü + ců - L(t)u(t,u) = 0 where L(t) and W(t, u) are neither autonomous nor periodic in t. Under certain assumptions on L and W, we obtain infinitely many homoclinic solutions when the nonlinearity W(t, u) is sub-quadratic or super-quadratic by using critical point theorems. Some recent results in the literature are generalized, and the open problem proposed by Zhang and Yuan is solved. In addition, with the help of the Nehari manifold, we consider the case where W(t, u) is indefinite and prove the existence of at least one nontrivial quasi-homoclinic solution. [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]

In this article, we prove the existence of solutions for Kirchhoff type equations with Dirichlet boundary-value condition. We use the Mountain Pass Theorem in critical point theory, without the (PS) condition [ABSTRACT FROM AUTHOR]

In this article, we consider the existence of multiple solutions to the elliptic problem -Δu = λuq + µu,p in Ω, u > 0 in Ω, u = 0 on ∂Ω, where Ω ... RN (N ≥ 3) is a bounded domain with smooth boundary ∂Ω, 0 < q < 1 < s < 2* - 1 ≤ p, 2* := 2N/N-2, λ and µ are nonnegative parameters. By using variational methods, truncation and Moser iteration techniques, we show that if the parameters λ and µ are small enough, then the problem has at least two positive solutions. [ABSTRACT FROM AUTHOR]

The aim of this article is to study the existence and multiplicity of solutions to operator equations involving duality mappings on Sobolev spaces with variable exponents. Our main tools are the well known Mountain Pass Theorem and its ℤ2-symmetric version. [ABSTRACT FROM AUTHOR]

In this article, we prove multiplicity of solutions for the sum of polyharmonic equation with critical Sobolev exponent. The proof is based upon the methods of weakly lower semi-continuous of the functionals and the Mountain Pass Lemma without (PS) conditions. [ABSTRACT FROM AUTHOR]

We obtain a positive solution for a superlinear p-Laplacian equations with the Dirichlet boundary-value conditions. Our main tool is a variation of the mountain pass theorem. [ABSTRACT FROM AUTHOR]

This article concerns the existence and the multiplicity of solutions for nonhomogeneous elliptic Kirchhoff problems involving the critical Sobolev exponent, defined on a regular bounded domain of ℝ³. Our approach is essentially based on Ekeland's Variational Principle and the Mountain Pass Lemma. [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]

We consider a nonlinear periodic problem driven by a nonhomogeneous differential operator plus an indefinite potential and a reaction having the competing effects of concave and convex terms. For the superlinear (concave) term we do not employ the usual in such cases Ambrosetti-Rabinowitz condition. Using variational methods together with truncation, perturbation and comparison techniques, we prove a bifurcation-type theorem describing the set of positive solutions as the parameter varies. [ABSTRACT FROM AUTHOR]