Mountain Pass Theorem Research Articles

Multiple positive solutions for nonhomogeneous Schrodinger-Poisson systems with Berestycki-Lions type conditions

In this article, we consider the multiplicity of solutions for nonhomogeneous Schrodinger-Poisson systems under the Berestycki-Lions type conditions. With the aid of Ekeland's variational principle, the mountain pass theorem and a Pohozaev type identity, we prove that the system has at least two positive solutions.
 For more information see https://ejde.math.txstate.edu/Volumes/2021/01/abstr.html

Existence and multiplicity of positive solutions for a singular system via sub-supersolution method and Mountain Pass Theorem

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.

Existence and concentration of nontrivial solutions for an elastic beam equation with local nonlinearity

<abstract><p>In this paper, by using the mountain pass lemma and the skill of truncation function, we investigate the existence and concentration phenomenon of nontrivial weak solutions for a class of elastic beam differential equation with two parameters $ \lambda $ and $ \mu $ when the nonlinear term satisfies some growth conditions only near the origin. In particular, we obtain a concrete lower bound of the parameter $ \lambda $, and analyze the relationship between $ \lambda $ and $ \mu $. In the end, we investigate the concentration phenomenon of solutions when $ \mu\to 0 $, and obtain a specific lower bound of the parameter $ \lambda $ which is independent of $ \mu $.</p></abstract>

Existence of Solutions for Klein-Gordon-Maxwell Equations Involving Hardy-Sobolev Critical Exponents

We investigate the existence of solutions for Klein-Gordon-Maxwell equations involving Hardy-Sobolev critical exponents. By means of the Ekeland’s variational principle and the Mountain Pass Theorem, we obtain that there is at least a nontrivial solution for the subcritical system. Then we prove that there are at least two different solutions for the critical system.

Existence and uniqueness of weak solution of p(x)- Laplacian in Sobolev spaces with variable exponents in complete manifolds

The paper deals with the existence and uniqueness of a non-trivial solution to non-homogeneous p(x)- Laplacian equations, managed by non polynomial growth operator in the framework of variable exponent Sobolev spaces on Riemannian manifolds. The mountain pass Theorem is used.

Positive solutions for a class of generalized quasilinear Schrödinger equation involving concave and convex nonlinearities in Orilicz space

In this paper, we study the following generalized quasilinear Schrödinger equation − div ( g 2 ( u ) ∇ u ) + g ( u ) g ′ ( u ) | ∇ u | 2 + V ( x ) u = λ f ( x , u ) + h ( x , u ) , x ∈ R N , where λ > 0 , N ≥ 3 , g ∈ C 1 ( 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.

A VARIATIONAL APPROACH FOR A PROBLEM INVOLVING A <i>ψ</i>-HILFER FRACTIONAL OPERATOR

Boundary value problems driven by fractional operators has drawn the attention of several researchers in the last decades due to its applicability in several areas of Science and Technology. The suitable definition of the fractional derivative and its associated spaces is a natural problem that arise on the study of this kind of problem. A manner to avoid of such problem is to consider a general definition of fractional derivative. The purpose of this manuscript is to contribute, in the mentioned sense, by presenting the $\psi-$fractional spaces $\mathbb{H}_{p}^{\alpha, \beta; \psi}([0, T], \mathbb{R})$. As an application we study a problem, by using the Mountain Pass Theorem, which includes an wide class of equations.

Positive and Negative Solutions of a Class of Fractional Schrödinger Equation

In this paper, we study the existence of positive and negative solutions for a class of fractional Schrodinger equations. Firstly, we give the definition of fractional Laplace operator and the conditions satisfied by nonlinear terms. This paper introduces the previous progress in this field, and gives the definitions of space and energy functional and the positive and negative parts of function. Then we introduce the main results of this paper. Next, we give the embedding relationship between workspace and <i>L^p</i> space and give the definition of inner product and norm of space. In order to obtain the existence of positive and negative solutions of the equation, we give the definitions of functions <i>u</i>⁺, <i>u</i>⁻ and functional weak solutions. This paper mainly uses mountain pass lemma to prove. Firstly, according to the embedding relationship of workspace and the condition of nonlinear term <i>f</i>, it is proved that functional <I>I</I> satisfies mountain road structure. Secondly, we need to prove that functional <I>I</I> satisfies the (<i>C_c</i>) condition, we first prove that the sequence <i>u_n</i> is bounded, then prove that UN has convergent subsequence by the definition of inner product and holder inequality. Therefore, we prove that functional <I>I</I> satisfies the (<i>C_c</i>) condition. Then, we define functional <I>I</I>^± and its inner product form to verify that functional <I>I</I>^± also has mountain path structure and satisfies (<i>C_c</i>) condition. Finally, taking <i>u</i>⁺ and <i>u</i>⁻ as experimental functions respectively, it is verified that they are the solutions of functional <I>I</I>. It is obtained that both <i>u</i>⁺ and <i>u</i>⁻ are the solutions of functional <I>I</I>. Therefore, we get the conclusion.

Periodic wave solutions of non-Newtonian filtration equations with nonlinear sources and singularities

This paper is concerned with a kind of non-Newtonian filtration equation with nonlinear sources and singularities. Based on the mountain pass theorem and variational methods, a sufficient criterion for the new results on the periodic wave solutions has been provided. Here not only the structure is more general and practical than the existing works but the conditions imposed are concise. Consequently, compared with the previous results on the singular equations and non-Newtonian filtration equations, the results we established are more generalized and some previous ones can been complemented and improved. Finally, the effectiveness of the established results are validated via two numerical examples and simulations.

EXISTENCE RESULTS FOR ANISOTROPIC FRACTIONAL (&lt;i&gt;p&lt;/i&gt;&lt;sub&gt;1&lt;/sub&gt;(&lt;i&gt;x&lt;/i&gt;, .), &lt;i&gt;p&lt;/i&gt;&lt;sub&gt;2&lt;/sub&gt;(&lt;i&gt;x&lt;/i&gt;, .))-KIRCHHOFF TYPE PROBLEMS

In this paper, we investigate the existence and multiplicity of solutions for a class of fractional $ (p_1(x,.),p_2(x,.)) $-Kirchhoff type problems with Dirichlet boundary data of the following form $ \left( \mathcal{P}_{M_i}^s\right) \; \; \left\{ \begin{array}{lllll} \sum\limits_{i = 1}^{2} M_i \left(\int_{Q}\frac{1}{p_i(x,y)}\frac{|u(x)-u(y)|^{p_i(x,y)}}{|x-y|^{N+sp_i(x,y)}}\; dxdy\right) \left( -\Delta\right)_{p_i(x,.)} ^{s}u(x)\\+\sum\limits_{i = 1}^{2}|u|^{\bar{p}_i(x)-2}u = f(x,u)\quad\rm{in}\; \; \Omega, u = 0 \quad \quad\rm{in}\; \; \; \mathbb{R}^{N}\setminus\Omega. \end{array}\right. $ More precisely, by means of mountain pass theorem with Cerami condition, we show that the above problem has at least one nontrivial solution. Moreover, using Fountain theorem, we prove that $ \left( \mathcal{P}_{M_i}^s\right) $ possesses infinitely many (pairs) of solutions with unbounded energy.

Existence of solutions for a p-Laplacian Kirchhoff type problem with nonlinear term of superlinear and subcritical growth

"This paper is concerned by the study of the existence of nonnegative and nonpositive solutions for a nonlocal quasilinear Kirchhoff problem by using the Mountain Pass lemma technique."

Existence and multiplicity results for critical anisotropic Kirchhoff-type problems with nonlocal nonlinearities

In this paper, we are interested in existence and multiplicity of solutions to anisotropic elliptic equations of Kirchhoff-type given by where Ω is a smooth bounded domain of and are parameters, where is the critical exponent and . The function is only continuous with and can change its sign, is a function with subcritical growth and 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 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 or with and with and appropriate

Multiple nodal solutions having shared componentwise nodal numbers for coupled Schrödinger equations

We investigate the structure of nodal solutions for coupled nonlinear Schrödinger equations in the repulsive coupling regime. Among other results, for the following coupled system of N equations, we prove the existence of infinitely many nodal solutions which share the same componentwise-prescribed nodal numbers(0.1){−Δuj+λuj=μuj3+∑i≠jβujui2inΩ,uj∈H0,r1(Ω),j=1,…,N, where Ω is a radial domain in Rn for n=2,3 and a bounded interval for n=1, λ>0, μ>0, and β<0. More precisely, let p be a prime factor of N and write N=pB. Suppose β≤−μp−1. Then for any given non-negative integers P1,P2,…,PB, (0.1) has infinitely many solutions (u1,…,uN) such that each of these solutions satisfies the same property: for b=1,...,B, upb−p+i changes sign precisely Pb times for i=1,...,p. The result reveals the complex nature of the solution structure in the repulsive coupling regime due to componentwise segregation of solutions. Our method is to combine a heat flow approach as deformation with a minimax construction of the symmetric mountain pass theorem using a Zp group action index. Our method is robust, also allowing to give the existence of one solution without assuming any symmetry of the coupling.

Existence of nontrivial solutions for Schrödinger-Kirchhoff type equations involving the fractional &lt;i&gt;p&lt;/i&gt;-Laplacian and local nonlinearity

In this paper, we deal with the existence of nontrivial solutions for the following Kirchhoff-type equation $ M\left(\,\,\displaystyle{\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\text{d}x\text{d}y}\right)(-\Delta)_{p}^{s} u+V(x)|u|^{p-2}u = \lambda f(x,u),\,\, \text{in}\,\,\mathbb{R}^N, $ where $0 0$ is a real parameter, $(-\Delta)_{p}^{s}$ is the fractional $p$-Laplacian operator, $V:\mathbb{R}^N\rightarrow\mathbb{R}^N$ is a potential function, $M$ is a Kirchhoff function, the nonlinearity $f:\mathbb{R}^N\times\mathbb{R}\rightarrow\mathbb{R}$ is a continuous function and just super-linear in a neighborhood of $u = 0$. By using an appropriate truncation argument and the mountain pass theorem, we prove the existence of nontrivial solutions for the above equation, provided that $\lambda$ is sufficiently large. Our results extend and improve the previous ones in the literature.

On Critical Schrödinger–Kirchhoff-Type Problems Involving the Fractional p-Laplacian with Potential Vanishing at Infinity

The aim of this paper is to study the existence of solutions for critical Schrodinger–Kirchhoff-type problems involving a nonlocal integro-differential operator with potential vanishing at infinity. As a particular case, we consider the following fractional problem: $$\begin{aligned}&M\left( \iint _{{\mathbb {R}}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\mathrm{dxdy}+\int _{{\mathbb {R}}^N}V(x)|u(x)|^{p}\mathrm{{d}}x\right) ((-\Delta )_p^{s}u(x)+V(x)|u|^{p-2}u)\\&\quad =K(x)(\lambda f(x,u)+|u|^{p_s^{*}-2}u), \end{aligned}$$ where $$M:[0, \infty )\rightarrow [0, \infty )$$ is a continuous function, $$(-\Delta )_p^{s}$$ is the fractional p-Laplacian, $$0<s<1<p<\infty $$ with $$sp<N,$$ $$p_s^{*}=Np/(N-ps),$$ K, V are nonnegative continuous functions satisfying some conditions, and f is a continuous function on $${\mathbb {R}}^N\times {\mathbb {R}}$$ satisfying the Ambrosetti–Rabinowitz-type condition, $$\lambda >0$$ is a real parameter. Using the mountain pass theorem, we obtain the existence of the above problem in suitable space W. For this, we first study the properties of the embedding from W into $$L_{K}^{\alpha }({\mathbb {R}}^N), \alpha \in [p, p_s^{*}].$$ Then, we obtain the differentiability of energy functional with some suitable conditions on f. To the best of our knowledge, this is the first existence results for degenerate Kirchhoff-type problems involving the fractional p-Laplacian with potential vanishing at infinity. Finally, we fill some gaps of papers of do O et al. (Commun Contemp Math 18: 150063, 2016) and Li et al. (Mediterr J Math 14: 80, 2017).

Variational and Topological Methods for a Class of Nonlinear Equations which Involves a Duality Mapping

The purpose of this paper is to show the existence results for the following abstract equation Jpu = Nfu,where Jp is the duality application on a real reflexive and smooth X Banach space, that corresponds to the gauge function φ(t) = tp-1, 1 &lt; p &lt; ∞. We assume that X is compactly imbedded in Lq(Ω), where Ω is a bounded domain in RN, N ≥ 2, 1 &lt; q &lt; p∗, p∗ is the Sobolev conjugate exponent.Nf : Lq(Ω) → Lq′(Ω), 1/q + 1/q′ = 1, is the Nemytskii operator that Caratheodory function generated by a f : Ω × R → R which satisfies some growth conditions. We use topological methods (via Leray-Schauder degree), critical points methods (the Mountain Pass theorem) and a direct variational method to prove the existence of the solutions for the equation Jpu = Nfu.

Existence and Concentration of Solutions for Choquard Equations with Steep Potential Well and Doubly Critical Exponents

Abstract In this paper, we investigate the non-autonomous Choquard equation - Δ ⁢ u + λ ⁢ V ⁢ ( x ) ⁢ u = ( I α ∗ F ⁢ ( u ) ) ⁢ F ′ ⁢ ( u ) in ⁢ R N , -\Delta u+\lambda V(x)u=(I_{\alpha}\ast F(u))F^{\prime}(u)\quad\text{in}\ \mathbb{R}^{N}, where N ≥ 4 N\geq 4 , λ &gt; 0 \lambda&gt;0 , V ∈ C ⁢ ( R N , R ) V\in C(\mathbb{R}^{N},\mathbb{R}) is bounded from below and has a potential well, I α I_{\alpha} is the Riesz potential of order α ∈ ( 0 , N ) \alpha\in(0,N) and F ⁢ ( u ) = 1 2 α * ⁢ | u | 2 α * + 1 2 * α ⁢ | u | 2 * α F(u)=\frac{1}{2_{\alpha}^{*}}\lvert u\rvert^{2_{\alpha}^{*}}+\frac{1}{2_{*}^{\alpha}}\lvert u\rvert^{2_{*}^{\alpha}} , in which 2 α * = N + α N - 2 2_{\alpha}^{*}=\frac{N+\alpha}{N-2} and 2 * α = N + α N 2_{*}^{\alpha}=\frac{N+\alpha}{N} are upper and lower critical exponents due to the Hardy–Littlewood–Sobolev inequality, respectively. Based on the variational methods, by combining the mountain pass theorem and Nehari manifold, we obtain the existence and concentration of positive ground state solutions for 𝜆 large enough if 𝑉 is nonnegative in R N \mathbb{R}^{N} ; further, by the linking theorem, we prove the existence of nontrivial solutions for 𝜆 large enough if 𝑉 changes sign in R N \mathbb{R}^{N} .

Existence of multiple solutions to semilinear Dirichlet problem for subelliptic operator

We study the existence of multiple solutions to the Dirichlet problem for a class of semilinear subelliptic equations associated with the Hormander type operator. By using mountain pass theorem and Ekeland’s variational principle respectively, we obtain two different nontrivial weak solutions for the problem.

Existence of Solutions for the Fractional (p, q)-Laplacian Problems Involving a Critical Sobolev Exponent

In this article, we study the following fractional (p, q)-Laplacian equations involving the critical Sobolev exponent: $$({P_{\mu ,\lambda }})\left\{ {\begin{array}{*{20}{l}} {( - \Delta )_p^{{s_1}}u + ( - \Delta )_q^{{s_2}}u = \mu |u{|^{q - 2}}u + \lambda |u{|^{p - 2}}u + |u{|^{p_{{s_1}}^* - 2}}u,}&{\text{in}\;\Omega ,} \\ {u = 0,}&{\text{in}\;{\mathbb{R}^N}\backslash \Omega ,} \end{array}} \right.$$ where Ω ⊂ ℝN is a smooth and bounded domain, λ, μ > 0, 0 < S2 < s1 < 1, $$1 < q < p < {\textstyle{N \over {{s_1}}}}.$$ We establish the existence of a non-negative nontrivial weak solution to (Pμ,λ) by using the Mountain Pass Theorem. The lack of compactness associated with problems involving critical Sobolev exponents is overcome by working with certain asymptotic estimates for minimizers.

A variable‐order fractional p(·)‐Kirchhoff type problem in ℝN

This paper is concerned with the existence and multiplicity of solutions for the following variable s(·)‐order fractional p(·)‐Kirchhoff type problem where N &gt; p(x, y)s(x, y) for any , is a variable s(·)‐order p(·)‐fractional Laplace operator with and , for , and M is a continuous Kirchhoff‐type function, g(x, v) is a Carathéodory function, and μ &gt; 0 is a parameter. Under the weaker conditions, we obtain that there are at least two distinct solutions for the above problem by applying the generalized abstract critical point theorem. Moreover, we also show the existence of one solution and infinitely many solutions by using the mountain pass lemma and fountain theorem, respectively. In particular, the new compact embedding result of the space into will be used to overcome the lack of compactness in . The main feature and difficulty of this paper is the presence of a double non‐local term involving two variable parameters.