Mountain Pass Theorem Research Articles

Existence of solutions for critical (p,q)-Laplacian equations in ℝN

In this paper, we are mainly interested in existence properties for a class of nonlinear PDEs driven by the ([Formula: see text])-Laplace operator where the reaction combines a power-type nonlinearity at critical level with a subcritical term. In addition, nonnegative nontrivial weights and a positive parameter [Formula: see text] are included in the nonlinearity. An important role in the analysis developed is played by the two potentials. Precisely, under suitable conditions on the exponents of the nonlinearity, first a detailed proof of the tight convergence of a sequence of measures is given, then the existence of a nontrivial weak solution is obtained provided that the parameter [Formula: see text] is far from [Formula: see text]. Our proofs use concentration compactness principles by Lions and Mountain Pass Theorem by Ambrosetti and Rabinowitz.

Multiplicity Results of Solutions to Non-Local Magnetic Schrödinger–Kirchhoff Type Equations in RN

In this paper, we establish the existence of a nontrivial weak solution to Schrödinger-kirchhoff type equations with the fractional magnetic field without Ambrosetti and Rabinowitz condition using mountain pass theorem under a suitable assumption of the external force. Furthermore, we prove the existence of infinitely many large- or small-energy solutions to this problem with Ambrosetti and Rabinowitz condition. The strategy of the proof for these results is to approach the problem by applying the variational methods, that is, the fountain and the dual fountain theorem with Cerami condition.

Infinitely many solutions for the discrete Schr\xf6dinger equations with a nonlocal term

In the present paper, we consider the following discrete Schrödinger equations −(a+b∑k∈Z|Δuk−1|2)Δ2uk−1+Vkuk=fk(uk)k∈Z,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ - \\biggl(a+b\\sum_{k\\in \\mathbf{Z}} \\vert \\Delta u_{k-1} \\vert ^{2} \\biggr) \\Delta ^{2} u_{k-1}+ V_{k}u_{k}=f_{k}(u_{k}) \\quad k\\in \\mathbf{Z}, $$\\end{document} where a, b are two positive constants and V={V_{k}} is a positive potential. Delta u_{k-1}=u_{k}-u_{k-1} and Delta ^{2}=Delta (Delta ) is the one-dimensional discrete Laplacian operator. Infinitely many high-energy solutions are obtained by the Symmetric Mountain Pass Theorem when the nonlinearities {f_{k}} satisfy 4-superlinear growth conditions. Moreover, if the nonlinearities are sublinear at infinity, we obtain infinitely many small solutions by the new version of the Symmetric Mountain Pass Theorem of Kajikiya.

Existence of solutions to a class of damped random impulsive differential equations under Dirichlet boundary value conditions

<abstract><p>In this paper, we study sufficient conditions for the existence of solutions to a class of damped random impulsive differential equations under Dirichlet boundary value conditions. By using variational method we first obtain the corresponding energy functional. Then the existence of critical points are obtained by using Mountain pass lemma and Minimax principle. Finally we assert the critical point of enery functional is the mild solution of damped random impulsive differential equations.</p></abstract>

Variational approach to <i>p</i>-Laplacian fractional differential equations with instantaneous and non-instantaneous impulses

<abstract><p>In this paper, we examine the existence of solutions of <italic>p</italic>-Laplacian fractional differential equations with instantaneous and non-instantaneous impulses. New criteria guaranteeing the existence of infinitely many solutions are established for the considered problem. The problem is reduced to an equivalent form such that the weak solutions of the problem are defined as the critical points of an energy functional. The main result of the present work is established by using a variational approach and a mountain pass lemma. Finally, an example is given to illustrate our main result.</p></abstract>

Existence of solutions for first-order Hamiltonian random impulsive differential equations with Dirichlet boundary conditions

<p style='text-indent:20px;'>In this paper, we study the sufficient conditions for the existence of solutions of first-order Hamiltonian random impulsive differential equations under Dirichlet boundary value conditions. By using the variational method, we first obtain the corresponding energy functional. And by using Legendre transformation, we obtain the conjugation of the functional. Then the existence of critical point is obtained by mountain pass lemma. Finally, we assert that the critical point of the energy functional is the mild solution of the first order Hamiltonian random impulsive differential equation. Finally, an example is presented to illustrate the feasibility and effectiveness of our results.</p>

STANDING WAVE SOLUTIONS FOR THE GENERALIZED MODIFIED CHERN-SIMONS-SCHRÖDINGER SYSTEM

In this paper, we study the modified gauged Schrödinger equation under some assumptions on the functions <i>V</i> and <i>f</i>. By using dual approach, Jeanjean’s monotone trick and Mountain Pass Theorem, we obtain the standing wave solutions for the generalized modified Chern-Simons-Schrödinger system.

Quasilinear double phase problems in the whole space via perturbation methods

We are concerned with the following double phase problems in the whole space $$ \begin{aligned} -{\rm div}(|\nabla u|^{p-2}\nabla u + & \mu(x)|\nabla u|^{q-2}\nabla u) \\ & +|u|^{p-2}u+\mu(x)|u|^{q-2}u= f(x,u)\;{\rm in}\; \mathbb{R}^N. \end{aligned}$$ The nonlinearity is super-linear but does not satisfy the Ambrosetti-Rabinowitz type condition. The main difficulty is that weak limits of $(PS)$ sequences are not always weak solutions of the problem. To overcome this difficulty, we add {a} potential term and, using the mountain pass theorem, we get weak solutions $u_\lambda$ of the perturbed equations. First, we prove that $u_\lambda\rightharpoonup u$ as $\lambda\rightarrow 0$. Then, via a vanishing lemma, we get that $u$ is a nontrivial solution of the original problem.

Existence results of nontrivial solutions for a new $ p(x) $-biharmonic problem with weight function

&lt;abstract&gt;&lt;p&gt;In this paper, we study a class of $ p(x) $-biharmonic problems with negative nonlocal terms and weight function. Using the mountain pass theorem and the Ekeland's variational principle, at least three solutions are obtained. We also give some comments on the existence of infinite many solutions for our problem when the nonlinear term is a general function.&lt;/p&gt;&lt;/abstract&gt;

Variational approach for a Steklov problem involving nonstandard growth conditions

&lt;abstract&gt;&lt;p&gt;The aim of this paper is to study the multiplicity of solutions for a nonlocal $ p(x) $-Kirchhoff type problem with Steklov boundary value in variable exponent Sobolev spaces. We prove the existence of at least three solutions and a nontrivial weak solution of the problem, using the Ricceri's three critical points theorem together with Mountain Pass theorem.&lt;/p&gt;&lt;/abstract&gt;

MULTIPLE SOLUTIONS FOR A NONHOMOGENEOUS SCHRÖDINGER-POISSON SYSTEM WITH CRITICAL EXPONENT

<abstract> In this paper, a nonhomogeneous Schrödinger-Poisson system with critical exponent was considered. By using the Mountain Pass Theorem and variational method, two positive solutions were obtained for the system which generalize and improve some recent results in the literature. </abstract>

Existence of the Solutions for a Class of Quasilinear Schr&amp;amp;#246;dinger Equations with Nonlocal Term

In this paper, we deal with the existence of solution for a class of quasilinear Schrödinger equations with a nonlocal term Where μ ∈ (0,3), the function K,V ∈ C(R3,R+) and V(x) may be vanish at infinity, g is a C1 even function with g’(t) ≤ 0 for all t ≥ 0, g(0) = 1, , 0 a F is the primitive function of f which is superlinear but subcritical at infinity in the sense of Hardy-littlewood-Sobolev inequality. By the mountain pass theorem, we prove that the above equation has a nontrivial solution.

Existence results for $(p_1,...,p_n)$-biharmonic systems under Navier boundary conditions

The authors study the existence of weak solutions for a $(p_1,...,p_n)$-biharmonic system via mountain pass theorem and establish semitrivial principal and strictly principal eigenvalues, positivity and simplicity results.

Existence and multiplicity of solutions for p(.)-Kirchhoff-type equations

his paper is concerned with the existence and multiplicity of solutions of a Dirichlet problem for $p(.)$-Kirchhoff-type equation% \begin{equation*} \left\{ \begin{array}{c} M\left( \int_{\Omega }\frac{\left\vert \nabla u\right\vert ^{p(x)}}{p(x)}% dx\right) \left( -\Delta _{p(x)}u\right) =f(x,u),\text{ in }\Omega , \\ u=0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{on }\partial \Omega .% \end{array}% \right. \end{equation*}% Using the mountain pass theorem, fountain theorem, dual fountain theorem and the theory of the variable exponent Sobolev spaces, under appropriate assumptions on $f$ and $M$, we obtain results on existence and multiplicity of solutions.

Image restoration via Picard's and Mountain-pass Theorems

&lt;abstract&gt;&lt;p&gt;In this work, we present existence results for some problems which arise in image processing namely image restoration. Our essential tools are Picard's fixed point theorem for a strict contraction and Mountain-pass Theorem for critical point.&lt;/p&gt;&lt;/abstract&gt;

Schrödinger equation with asymptotically linear nonlinearities

LRES, Faculty of Sciences of Tunis, University of Tunis El Manar, Tunisia EM makkia.dammak@gmail.com KW Asymptotically linear % variational method % Schr?dinger equations % Cerami sequence KR nema In this paper, we investigate a quasilinear Schr?dinger problem under Dirichlet boundary condition in a regular domain with asymptotically linear nonlinearities. We use Cerami version of the mountain pass theorem to prove the existence of solution without using the Ambrosetti-Rabionovitz condition or any of its refinements. Then, we prove that the same techniques work when the nonlinearity is superlinear and subcritical at infinity.

Existence, nonexistence and multiplicity of positive solutions for singular quasilinear problems

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 parameterλ&gt;0varies. 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.

Multiple Solutions for a Class of Variable-Order Fractional Laplacian Equations with Concave-Convex Nonlinearity

This paper is concerned with the following variable-order fractional Laplacian equations , where N ≥ 1 and N > 2s(x,y) for (x,y) ∈ Ω × Ω, Ω is a bounded domain in RN, s(⋅) ∈ C (RN × RN, (0,1)), (-Δ)s(⋅) is the variable-order fractional Laplacian operator, λ, μ > 0 are two parameters, V: Ω → [0, ∞) is a continuous function, f ∈ C(Ω × R) and q ∈ C(Ω). Under some suitable conditions on f, we obtain two solutions for this problem by employing the mountain pass theorem and Ekeland’s variational principle. Our result generalizes the related ones in the literature.

Multiple solutions for a quasilinear Choquard equation with critical nonlinearity

AbstractIn the present work, we are concerned with the multiple solutions for quasilinear Choquard equation with critical nonlinearity inRN{{\mathbb{R}}}^{N}. We show multiplicity results for this problem, which are characterized, respectively, by the new version of symmetric mountain-pass theorem and the mountain-pass theorem for even functionals. The novelty of our work is the appearance of the convolution terms as well as critical nonlinearities.

Multiple solutions for critical Kirchhoff–Poisson systems in the Heisenberg group

In this paper, we consider a class of the critical Kirchhoff–Poisson systems in the Heisenberg group. Under suitable assumptions on the Kirchhoff function and on the nonlinear terms, the existence of multiple solutions is obtained by using the symmetric mountain pass theorem. Moreover, our results are new even in the Euclidean case.