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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. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and multiplicity solutions for the following Klein-Gordon-Maxwell system ∫-Δu + V(x)u - (2ω + ϕ)ϕu = f (x, u), x ∈ R³, Δϕ = (ω + ϕ)u², x ∊ R³, where ω > 0 is a constant and the nonlinearity f (x, u) is either asymptotically linear in u at infinity or the primitive of f (x, u) is of 4-superlinear growth in u at infinity. Under some suitable assumptions, the existence and multiplicity of solutions are proved by using the Mountain Pass theorem and the fountain theorem, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of positive solutions for the following generalized quasilinear Schrödinger equation - div(gp(u)|∇u|p-2∇u) + gp-1(u)g′(u)|∇u|p + V(x)|u|p-2u = K(x) f (u) + Q(x)g(u)|G(u)|p*-2G(u), x ∈ RN, where N ≥ 3, 1 < p ≤ N, p* = Np/N-p, g ∈ C¹(R,R+), V(x) and K(x) are positive continuous functions and G(u) =∫u0g(t)dt. By using a change of variable, we obtain the existence of positive solutions for this problem by using the Mountain Pass Theorem. Our results generalize some existing results. [ABSTRACT FROM AUTHOR]

In this article, we study the following degenerated Schrödinger equations: {-Δγu+λV(x)u= f(x,u)u∈Eλ, in RN, where λ>0 is a parameter, Δγ is a degenerate elliptic operator, the potential V(x) has a potential well with bottom and the nonlinearity f(x,u) is either super-linear or sub-linear at infinity in u. The existence of ground state solution be obtained by using the variational methods. [ABSTRACT FROM AUTHOR]

In the present paper we deal with a quasilinear problem involving a singular term and a parametric superlinear perturbation. We are interested in the existence, nonexistence and multiplicity of positive solutions as the parameter λ > 0 varies. In our first result, the superlinear perturbation has an arbitrary growth and we obtain the existence of a solution for the problem by using the sub-supersolution method. For the second result, the superlinear perturbation has subcritical growth and we employ the Mountain Pass Theorem to show the existence of a second solution. [ABSTRACT FROM AUTHOR]

In this paper, we study the following generalized quasilinear Schrödinger equation... where λ > 0, N ≥ 3, g ∈ C¹ (R, R+). By using a change of variable, we obtain the existence of positive solutions for this problem with concave and convex nonlinearities via the Mountain Pass Theorem. Our results generalize some existing results. [ABSTRACT FROM AUTHOR]

In this paper we show the existence and multiplicity of positive solutions using the sub-supersolution method and Mountain Pass Theorem in a general singular system which the operator is not homogeneous neither linear. [ABSTRACT FROM AUTHOR]

In this article, we study the following quasilinear Schrödinger equation ... where V(x) is a given positive potential and the nonlinearity f (x, u) is allowed to be sign-changing. Under some suitable assumptions, we obtain the existence of infinitely many nontrivial solutions by a change of variable and Symmetric Mountain Pass Theorem. [ABSTRACT FROM AUTHOR]

We study the following class of double-phase nonlinear eigenvalue problems − div [φ(x, ∣∇u∣)∇u + ψ(x, ∣∇u∣)∇u] = λ f(x, u) in Ω, u = 0 on ∂Ω, where Ω is a bounded domain from RN with smooth boundary and the potential functions φ and ψ have (p1(x); p2(x)) variable growth. The main results of this paper are to prove the existence of a continuous spectrum consisting in a bounded interval in the near proximity of the origin, the fact that the multiplicity of every eigenvalue located in this interval is at least two and to establish the existence of infinitely many solutions for our problem. The proofs rely on variational arguments based on the Ekeland's variational principle, the mountain pass theorem, the fountain theorem and energy estimates. [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]

In this paper, we consider the following nonhomogeneous Klein-Gordon-Maxwell system ... where ω>0 is a constant, the primitive of the nonlinearity f is of 2-superlinear growth at infinity. The nonlinearity considered here is weaker than the local (AR) condition and the (Je) condition of Jeanjean. The existence of two solutions is proved by the Mountain Pass Theorem and Ekeland's variational principle. [ABSTRACT FROM AUTHOR]

This paper is concerned with the following quasilinear Schrödinger equations with critical exponent: -Δpu+V(x)|u|p-2u-Δp(|u|2ω)|u|2ω-2u=ak(x)|u|q-2u+b|u|2ωp*-2u,x∈RN. Here Δpu=div(|∇u|p-2∇u) is the p-Laplacian operator with 1

We study the existence of weak solutions for a degenerate p(x)-Laplace equation. The main tool used is the variational method, more precisely, the Mountain Pass Theorem. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of multiple solutions for the following quasilinear Schrödinger equation -Δu - uΔ(|u|²) + V(|x|)u = f (|x|, u), x ∈ ℝN. Under some generalized assumptions on f, we obtain infinitely many radial solutions for N ≥ 2, many non-radial solutions for N = 4 and N ≥ 6, and a non radial solution for N = 5. Our results generalize and extend some existing results. [ABSTRACT FROM AUTHOR]

In this paper, we study the following Schrödinger-Poisson system {-Δu + V(x)u + φu = f (x, u) + g(x), x ∈ ℝ³, -Δφ = u², x ∈ ℝ³. Under appropriate assumptions on V, f and g, using the Mountain Pass Theorem and the Ekeland's variational principle, we establish two existence theorems to ensure that the above system has at least two different solutions. Recent results from the literature are extended and improved. [ABSTRACT FROM AUTHOR]

This paper is concerned with the following quasilinear Schrödinger system in RN: {-ε2∆u + V1(x)u - ε2∆(u2)u = K1(x)∣u∣22*-2u + h1(x, u, v)u,-ε2∆v + V2(x)v - ε2∆(v2)v = K2(x)∣v∣22*-2v + h2(x, u, v)v, where N ≥ 3, Vi(x) is a nonnegative potential, Ki(x) is a bounded positive function, i = 1, 2. h1(x, u, v)u and h2(x, u, v)v are superlinear but subcritical functions. Under some proper conditions, minimax methods are employed to establish the existence of standing wave solutions for this system provided that ε is small enough, more precisely, for any m ∊ N, it has m pairs of solutions if ε is small enough. And these solutions (uε, vε) → (0,0) in some Sobolev space as ε → 0. Moreover, we establish the existence of positive solutions when ε = 1. The system studied here can model some interaction phenomena in plasma physics. [ABSTRACT FROM AUTHOR]

In this paper we study an existence result of the quasilinear problem -div[ϕ'(∣▿ u∣2)▿ u] + a(x)∣u∣α-2u = ∣u∣γ-2u + ∣u∣β-2u in RN(N ≥ 3), where ϕ(t) behaves like tq/2 for small t and tq/2 for large t, a is a positive potential, 1 < p < q < N, 1 < α ≤ p*q'/p' and max {α, q} < γ < β < p* = pN/(N - p), with p' and q' the conjugate exponents of p, respectively q. Our main result is the proof of the existence of a weak solution, based on the mountain pass theorem. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the existence of solutions to the following quasilinear Schrödinger type system driven by the fractional p-Laplacian (-Δ)psu + a(x)|u|p-2u = Hu(x, u, v) in RN, (-Δ)qsv + b(x)|v|q-2v = Hv(x, u, v) in RN, where 1 < q ≤ p, sp < N, (-Δ)ms is the fractional m-Laplacian, the coefficients a, b are two continuous and positive functions, and Hu, Hv denote the partial derivatives of H with respect to the second variable and the third variable. By using the mountain pass theorem, we obtain the e×istence of nontrivial and nonnegative solutions for the above system. The main feature of this paper is that the nonlinearities do not necessarily satisfy the Ambrosetti-Rabinowitz condition. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following ordinary p-Laplacian system d/dt(∣∣u(t)∣p-2u(t) ) - ∇K(t, u(t)) + ∇W(t, u(t)) = f (t), (HS) where t ∈ R and p > 1. Using the Mountain Pass Theorem, we establish the existence of a nontrivial homoclinic solution for (HS) under new assumptions on the growth of the potential which allow W(t, x) to be either super p-linear or asymptotically plinear at infinity. Also, contrary to previous works, W(t, x) will be neither periodic nor bounded with respect to the variable t. Recent results in the literature are generalized even if p = 2. [ABSTRACT FROM AUTHOR]

In this paper, a fourth-order boundary value problem on the half-line is considered and existence of solutions is proved using a minimization principle and the mountain pass theorem. [ABSTRACT FROM AUTHOR]

In this paper we study a class of fractional Laplace equations with asymptotically linear right-hand side. The existence results of three nontrivial solutions under the resonance and non-resonance conditions are established by using the minimax method and Morse theory. [ABSTRACT FROM AUTHOR]