Symmetric Mountain Pass Theorem Research Articles

Kirchhoff problems with logarithmic double phase operator: Existence and multiplicity results

In this paper, we focus on Kirchhoff type problems driven by a logarithmic double phase operator with variable exponents. Under very general assumptions on the nonlinearity and using variational tools, like the mountain pass theorem, we establish the existence of at least one nontrivial weak solution for the problem under consideration. Then, under different hypotheses on the reaction term, we are also able to derive a multiplicity result of solutions for our problem. We stress that in order to produce such a multiplicity result a key role is played by a variant of the symmetric mountain pass theorem.

On class of bi-nonlocal problems involving fractional p(.,.)-Laplacian operators

This work aims to obtain the existence and multiplicity of solutions for a bi-nonlocal problem involving the p(.,.)-Laplacian operator. Under specific assumptions, the paper uses the Mountain Pass Theorem combined with the Ekeland variational approach to establish the existence of two nontrivial solutions for the problem. The existence of infinitely many solutions is proved using the Symmetric Mountain Pass Theorem and the Ljusternik-Schnirelman Theorem. Finally, we study a p(.,.)-Laplacian problem with non-variational form.

Infinitely of solutions for fractional κ(ξ)$$ \kappa \left(\xi \right) $$‐Kirchhoff equation in Hκ(ξ)ϖ,ν;μ(Λ)$$ {\mathcal{H}}_{\kappa \left(\xi \right)}^{\varpi, \nu; \mu}\left(\Lambda \right) $$

This work aims to develop the variational framework for some Kirchhoff problems involving the ‐Hilfer operator. Precisely, we use the symmetric mountain pass theorem to prove the existence of unfairly of nontrivial solutions. Further, we research the results from the theory of variable exponent Sobolev spaces and from the theory of ‐fractional space .

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.

Infinitely Many Solutions for Fractional Hamiltonian Systems with Locally Defined Potentials

In this paper, we are concerned with the existence of infinitely many solutions for the following fractional Hamiltonian where −∞ Dtα and t Dα∞ are left and right Liouville-Weyl fractional derivatives of order 1 < α < 1 on the whole axis respectively, 2 L ∈C(R,RN2) is a symmetric matrix valued function unnecessary coercive and W(t,x) ∈C1(R×RN,R). The novelty of this paper is that, assuming that L is bounded from below and unnecessarily coercive at infinity, and W is only locally defined near the origin with respect to the second variable, we show that (1) possesses infinitely many solutions via a variant Symmetric Mountain Pass Theorem.

On a m(x)$$ m(x) $$‐polyharmonic Kirchhoff problem without any growth near 0 and Ambrosetti–Rabinowitz conditions

In this paper, we study a higher order Kirchhoff problem with variable exponent of type where is a smooth bounded domain, for all ; is a Kirchhoff function, and it may be degenerate at zero; is a continuous function; and is the main ‐order differential operator. The main feature of our paper is the fact that the nonlinearity considered here satisfies some conditions which are much weaker than the classical Ambrosetti–Rabinowitz condition, the standard subcritical polynomial growth, and the strong ‐superlinear conditions required in [16]. In case of odd nonlinearity in and without requiring any control on near 0, we obtain the existence of infinitely many solutions of the above problem via the Symmetric mountain pass theorem. We improve and extend some recent results in the literature.

Ground state sign-changing solution for a logarithmic Kirchhoff-type equation in R 3

We investigate the following logarithmic Kirchhoff-type equation: ( a + b ∫ R 3 | ∇ u | 2 + V ( x ) u 2 d x ) [ − Δ u + V ( x ) u ] = | u | p − 2 u ln ⁡ | u | , x ∈ R 3 , where a , b > 0 are constants, 4 < p < 2 ∗ = 6 . Under some appropriate hypotheses on the potential function V , we prove the existence of a positive ground state solution, a ground state sign-changing solution and a sequence of solutions by using the constraint variational methods, topological degree theory, quantitative deformation lemma and symmetric mountain pass theorem. Our results complete those of Gao et al. [Appl. Math. Lett. 139(2023), 108539] with the case of 4 < p < 6 .

Infinitely many solutions for a critical $ p(x) $-Kirchhoff equation with Steklov boundary value

<p>In this paper, we aim to tackle the questions of existence and multiplicity of solutions of the $ p(x) $-Kirchhoff problem involving critical exponent and the Steklov boundary value. Further, we research the results from the theory of variable exponent Sobolev spaces, the concentration-compactness principle, and the symmetric mountain pass theorem.</p>

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 λ M 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;