Symmetric Mountain Pass Theorem Research Articles

High Perturbations of a Fractional Kirchhoff Equation with Critical Nonlinearities

This paper concerns a fractional Kirchhoff equation with critical nonlinearities and a negative nonlocal term. In the case of high perturbations (large values of α, i.e., the parameter of a subcritical nonlinearity), existence results are obtained by the concentration compactness principle together with the mountain pass theorem and cut-off technique. The multiplicity of solutions are further considered with the help of the symmetric mountain pass theorem. Moreover, the nonexistence and asymptotic behavior of positive solutions are also investigated.

Existence and multiplicity of solutions for a class of nonlocal elliptic transmission systems

By using the approach based on variationnel methods and critical point theory, more precisely, the symmetric mountain pass theorem, we study the existence and multiplicity of weak solutions for a class of elliptic transmision system with nonlocal term.

Indefinite Schrödinger equation with nonlinearity sublinear at zero

We consider stationary Schrödinger equations with indefinite potential and nonlinearity sublinear at u = 0. Using linking theorem and symmetric mountain pass theorem, existence and multiplicity results are obtained.

Multiple weak solutions for pi(x)‐Kirchhoff‐type quasilinear elliptic systems

The existence of infinitely many weak solutions for a class of quasilinear elliptic ‐Kirchhoff system is investigated by the symmetric mountain pass theorem. Furthermore, for the case where , I introduce a positive quantity to find a mountain pass weak solution and to consider the sublinear case under large growth conditions at infinity and zero.

Existence and multiplicity of solutions for m-polyharmonic Kirchhoff problems without Ambrosetti–Rabinowitz conditions

In this paper, we prove the existence of infinitely many solutions for a class of quasilinear elliptic m-polyharmonic Kirchhoff equations where the nonlinear function has a quasicritical growth at infinity and without assuming the Ambrosetti and Rabinowitz type condition. The new aspect consists in employing the notion of a Schauder basis to verify the geometry of the symmetric mountain pass theorem. Furthermore, we introduce a positive quantity similar to the first eigenvalue of the m-polyharmonic operator to find a mountain pass solution, and also to discuss the sublinear case under large growth conditions at infinity and at zero. Our results are an improvement and generalization of the corresponding results obtained by Colasuonno-Pucci (Nonlinear Analysis: Theory, Methods and Applications, 2011) and Bae-Kim (Mathematical Methods in the Applied Sciences, 2020).

Divergent sequence of nontrivial solutions for superlinear double phase problems

We consider a double phase (unbalanced growth) Dirichlet problem with a Carathéodory reaction f ( z , x ) which is superlinear in x but without satisfying the AR-condition. Using the symmetric mountain pass theorem, we produce a whole sequence of distinct bounded solutions which diverge to infinity.

Existence and multiplicity of solutions for a class of Helmholtz systems

This paper establishes a dual variational framework for studying the existence and multiplicity of solutions to a class of Helmholtz systems. We first focus on a compact nonlinear potential and prove the existence and multiplicity of solutions via the symmetric Mountain Pass theorem. Next, we investigate the periodic nonlinear potential case and prove a nonvanishing theorem to recover the loss of compactness. We then demonstrate the existence of solutions using the Nehari manifold method and the multiplicity of solutions using the pseudoindex theory. Additionally, the regularity of solutions is given.

Quasilinear equation with critical exponential growth in the zero mass case

In this work we study the existence of solutions for the following class of quasilinear elliptic equations −divA(|x|)|∇u|N−2∇u=Q(|x|)f(u),inRN,where N≥2 and f has critical exponential growth. We establish conditions on the non-homogeneous weights A and Q to introduce a suitable function space where we are able to apply Variational Methods to obtain weak solutions. The main key is a Hardy type inequality for radial functions. Our approach is based on a new Trudinger–Moser type inequality, a version of the Symmetric Criticality Principle and Mountain Pass Theorem.

Existence and multiplicity of solutions for a quasilinear system with locally superlinear condition

Abstract We investigate the existence and multiplicity of weak solutions for a nonlinear Kirchhoff type quasilinear elliptic system on the whole space R N {{\mathbb{R}}}^{N} . We assume that the nonlinear term satisfies the locally super- ( m 1 , m 2 ) \left({m}_{1},{m}_{2}) condition, that is, lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 = + ∞ {\mathrm{lim}}_{| \left(u,v)| \to +\infty }\frac{F\left(x,u,v)}{| u{| }^{{m}_{1}}+| v{| }^{{m}_{2}}}=+\infty for a.e. x ∈ G x\in G , where G G is a domain in R N {{\mathbb{R}}}^{N} , which is weaker than the well-known Ambrosseti-Rabinowitz condition and the naturally global restriction, lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 = + ∞ {\mathrm{lim}}_{| \left(u,v)| \to +\infty }\frac{F\left(x,u,v)}{| u{| }^{{m}_{1}}+| v{| }^{{m}_{2}}}=+\infty for a.e. x ∈ R N x\in {{\mathbb{R}}}^{N} . We obtain that the system has at least one weak solution by using the classical mountain pass theorem. To a certain extent, our theorems extend the results of Tang et al. [Nontrivial solutions for Schrodinger equation with local super-quadratic conditions, J. Dynam. Differ. Equ. 31 (2019), no. 1, 369–383]. Moreover, under the aforementioned naturally global restriction, we obtain that the system has infinitely many weak solutions of high energy by using the symmetric mountain pass theorem, which is different from those results of Wang et al. [Existence and multiplicity of solutions for a class of quasilinear elliptic systems in Orlicz-Sobolev spaces, J. Nonlinear Sci. Appl. 10 (2017), no. 7, 3792–3814] even if we consider the system on the bounded domain with Dirichlet boundary condition.

Multiplicity results for $p$-Kirchhoff modified Schrödinger equations with Stein-Weiss type critical nonlinearity in $\mathbb R^N$

In this article, we consider the following modified quasilinear critical Kirchhoff-Schrödinger problem involving Stein-Weiss type nonlinearity: $$ \mathcal K(u)= \lambda f(x) |u(x)|^{q-2} u(x)+ \Big ( \int_{\mathbb R^N}\frac{|u(y)|^{2p_{\beta,\mu}^{*}}} {|x-y|^{\mu}|y|^{\beta}}dy \Big ) \frac {|u(x)|^{2p_{\beta,\mu}^{*}-2} u(x)}{|x|^\beta} \; \text{ in }\; \mathbb R^N, $$ where $\lambda > 0$ is a parameter, $N\geq 3$, $$ \mathcal K(u) = \Big ( a+b \int_{\mathbb R^N}|\nabla u|^{p}dx \Big ) \Delta_{p} u - a u \Delta_{p}(u^2) $$ with $a > 0$, $b\geq 0$, $\beta\geq0,$ $0 < \mu < N$, $0 < 2\beta+ \mu < N$, $2\leq q < 2 p^*$. Here, $p^*=\frac{Np}{N-p}$ is the Sobolev critical exponent and $ p_{\beta,\mu}^{*}:= \frac p2\frac{(2N-2\beta-\mu)}{N-p} $ is the critical exponent with respect to the doubly weighted Hardy-Littlewood-Sobolev inequality (also called Stein-Weiss type inequality). Then by establishing a concentration-compactness argument for our problem, we show the existence of infinitely many nontrivial solutions to the equations with respect to the parameter $\lambda$ by using Krasnoselskii's genus theory, symmetric mountain pass theorem and $\mathbb Z_2$- symmetric version of mountain pass theorem for different ranges of $q$. We further show that these solutions belong to $L^\infty(\mathbb R^N)$.

A variant of symmetric mountain pass theorem and its applications for generalized quasi-linear elliptic equations

By using a variant of the variational perturbation techniques introduced by Rabinowitz [Multiple critical points of perturbed symmetric functionals. Tran Amer Math Soc. 1982;272:753–769], we build a new non-smooth symmetric Mountain Pass Theorem without symmetric condition. This new abstract result can be applied to generalized quasi-linear elliptic equations with small perturbations. Our results extend the results concerning a semi-linear elliptic equation made by Li-Liu [Perturbations from symmetric elliptic boundary value problems. J Differ Equ. 2002;185:271–280].

Solutions for quasilinear Schrödinger equation on involving indefinite potentials

This paper is concerned with the quasilinear Schrödinger equation where , V is a given potential allowed to be indefinite, or equivalently, the Schrödinger operator can be indefinite. We consider the case that V is coercive so that the working space can be compactly embedded into Lebesgue spaces. Using the local linking theorem and Morse theory, we obtain a nontrivial solution for the above problem. Moreover, by the symmetric mountain pass theorem, we get an unbounded sequence of solutions.

Existence and multiplicity of solutions involving the $ p(x) $-Laplacian equations: On the effect of two nonlocal terms

&lt;p style='text-indent:20px;'&gt;We study a class of &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ p(x) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-Kirchhoff problems which is seldom studied because the nonlinearity has nonstandard growth and contains a bi-nonlocal term. Based on variational methods, especially the Mountain pass theorem and Ekeland's variational principle, we obtain the existence of two nontrivial solutions for the problem under certain assumptions. We also apply the Symmetric mountain pass theorem and Clarke's theorem to establish the existence of infinitely many solutions. Our results generalize and extend several existing results.&lt;/p&gt;

Existence and multiplicity of nontrivial solutions for a class of Kirchhoff-Schrödinger-Poisson systems

We consider a class of Kirchhoff-Schrödinger-Poisson systems. By using the Mountain Pass Theorem and the Symmetric Mountain Pass Theorem of Rabinowitz, we prove the existence and multiplicity of nontrivial solutions. Our conditions weaken the Ambrosetti Rabinowitz type condition.

Infinitely Many Solutions for the Fractional p&amp;q-Laplacian Problems in RN

In this paper, we consider the following class of the fractional p&amp;q-Laplacian problem: (−Δ)psu+(−Δ)qsu+V(x)(|u|p−2u+|u|q−2u)+g(x)|u|r−2u=K(x)f(x,u)+h(u),x∈RN,V:RN→R+ is a potential function, and h:R→R is a perturbation term. We studied two cases: if f(x,u) is sublinear, by means of Clark’s theorem, which considers the symmetric condition about the functional, we get infinitely many solutions; if f(x,u) is superlinear, using the symmetric mountain-pass theorem, infinitely many solutions can be obtained.

Multiplicity of nontrivial solutions for Kirchhoff type equations with zero mass and a critical term

In this paper, a class of Kirchhoff type equations in ℝN (N ⩾ 3) with zero mass and a critical term is studied. Under some appropriate conditions, the existence of multiple solutions is obtained by using variational methods and a variant of Symmetric Mountain Pass theorem. The Second Concentration Compactness lemma is used to overcome the lack of compactness in critical problem. Compared to the usual Kirchhoff-type problems, we only require the nonlinearity to satisfy the classical superquadratic condition (Ambrosetti-Rabinowitz condition).

Existence and multiplicity of nontrivial solutions for poly-Laplacian systems on finite graphs

In this paper, we investigate the existence and multiplicity of nontrivial solutions for poly-Laplacian system on a finite graph G=(V, E), which is a generalization of the Yamabe equation on a finite graph. When the nonlinear term F satisfies the super-(p, q)-linear growth condition, by using the mountain pass theorem we obtain that the system has at least one nontrivial solution, and by using the symmetric mountain pass theorem, we obtain that the system has at least dim W nontrivial solutions, where W is the working space of the poly-Laplacian system. We also obtain the corresponding result for the poly-Laplacian equation. In some sense, our results improve some results in (Grigor’yan et al. in J. Differ. Equ. 261(9):4924–4943, 2016).

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.

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.