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In this paper, we study the existence of weak solution to $ (p,q) $ (p , q) -fractional Choquard equation in $ {\mathbb {R}}^N $ R N as follows $$\begin{align*} {\mathcal{L}}_{p}^{s}u+{\mathcal{L}}_{q}^{s}u+ V(x)(|u|^{p-2}u+|u|^{q-2}u)=\left(\frac{1}{|x|^{\mu}}*F(u)\right)f(u), \end{align*} $$ L p s u + L q s u + V (x) (| u | p − 2 u + | u | q − 2 u) = (1 | x | μ ∗ F (u)) f (u) , where $ 2\le \frac {N}{s}=p 2 ≤ N s = p < q , $ 0\le \mu 0 ≤ μ < N , f has exponential growth. The function f and V are continuous which satisfy some suitable assumptions. In our best knowledge, sofar there is not any work about existence of weak solution to $ (p,q) $ (p , q) -fractional Choquard equation with exponential growth. Our results are even new in the case $ (N,q) $ (N , q) -Laplace equation which are complement the results of Fiscella and Pucci [ $ (p,N) $ (p , N) equations with critical exponential nonlinearities in $ {\mathbb {R}}^N $ R N . J Math Anal Appl. 2021;501(1), Article 123379]. [ABSTRACT FROM AUTHOR]

We investigate the following nonlinear bosonic equation on Euclidean space arising in string theory and cosmology: P − Δ e − c Δ u + m u = f (x , u) , x ∈ R n , where n ≥ 3 , m > 0 , c > 0 and f (x , u) u tends to a positive function h (x) independent of u as u → + ∞ , e − c Δ is given by a power series with Δ is the Euclidean Laplace operator. Here, the nonlinear term f (x , u) does not satisfy the usual condition: AR 0 ≤ F (x , u) : = ∫ 0 u f (x , t) d t ≤ 1 2 + θ f (x , u) u , for θ > 0 and | u | is large, which is important in using the mountain pass theorem, see Alves et al. (J. Differ. Equ. 323:229-252, 2022) and Corrêa et al. (J. Differ. Equ. 363:491-517, 2023). This paper is devoted to discuss how to use the mountain pass theorem to obtain the existence of nontrivial solution to problem (P) without the (AR) condition. [ABSTRACT FROM AUTHOR]

The article is about an elliptic problem defined on a stratified Lie group. Both sub and superlinear cases are considered whose solutions are guaranteed to exist in light of the interplay between the nonlinearities and the weak L 1 datum. The existence of infinitely many solutions is proved for suitable values of λ , p , q by using the Symmetric Mountain Pass Theorem. [ABSTRACT FROM AUTHOR]

Using Mountain Pass Theorem, we consider the existence of weak solutions of weighted Robin problem involving p. -biharmonic operator a x Δ p x 2 u = λ b (x) u q (x) - 2 u , i n Ω a (x) Δ u p (x) - 2 ∂ u ∂ υ + β (x) u p (x) - 2 u = 0 , o n ∂ Ω under some conditions in the space W a , b 2 , p (.) Ω. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following nonhomogeneous quasilinear Schrödinger–Poisson system with exponential and logarithmic nonlinearities - Δ u + ϕ u = | u | p - 2 u log | u | 2 + λ f (u) + h (x) , in Ω , - Δ ϕ - ε 4 Δ 4 ϕ = u 2 , in Ω , u = ϕ = 0 , on ∂ Ω , where 4 < p < + ∞ , ε , λ > 0 are parameters, Δ 4 ϕ = div (| ∇ ϕ | 2 ∇ ϕ) , Ω ⊂ R 2 is a bounded domain, and f has exponential critical growth. First, using reduction argument, truncation technique, Ekeland’s variational principle, and the Mountain Pass theorem, we obtain that the above system admits at least two solutions with different energy for λ large enough and ε fixed. Finally, we research the asymptotic behavior of solutions with respect to the parameters ε . [ABSTRACT FROM AUTHOR]

In this present paper, we concern investigating nonlinear Kirchhoff-type problems subject to Dirichlet boundary conditions, incorporating nonlocal terms and logarithmic nonlinearity in the ϕ -Hilfer fractional spaces with the η (·) -Laplacian operator by means of the do Mountain Pass Theorem, Fountain Theorem and Dual Fountain Theorem. [ABSTRACT FROM AUTHOR]

This article deals with the following fractional N s -Laplace Kichhoff equation involving exponential growth of the form: ε N K [ u ] s , N s N s (- Δ) N / s s u + Z (x) | u | N s - 2 u = f (u) in R N , where ε > 0 is a parameter, s ∈ (0 , 1) and (- Δ) p s is the fractional p-Laplace operator with p = N s ≥ 2 , K is a Kirchhoff function, f is a continuous function with exponential growth and Z is a potential function possessing a local minimum. Under some suitable conditions, we obtain the existence, multiplicity and concentration of solutions to the above problem via penalization methods and Lyusternik-Schnirelmann theory. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a new nonlocal Q-Laplacian equation on the Heisenberg group in the smooth bounded domain. The nonlinear term f (ξ , u) has exponential growth behaving like exp (β | t | Q Q - 1 ) as | t | → ∞ with some β > 0 . Existence of solutions are obtained by an application of the mountain pass theorem and Ekeland's variational principle. Moreover, our results are new even on the Euclidean case. [ABSTRACT FROM AUTHOR]

This article deals with the fractional Choquard equation involving exponential growth as follows ε N (- Δ) p s u + Z (x) | u | p - 2 u = ε μ - N 1 | x | μ ∗ H (u) h (u) in R N , where ε a positive parameter, s ∈ (0 , 1) , μ ∈ (0 , N) and (- Δ) p s is a fractional p-Laplace operator with p = N s ≥ 2. The function h is only continuous and has exponential growth. In addition, the potential function Z satisfies some appropriate conditions. We use the Trudinger–Moser inequality to deal with the function h involving exponential growth. Together with the Ljusternik–Schnirelmann category theory and variational method, the multiplicity and concentration behavior of positive solutions are obtained for the above problem. As far as we know, our results seem to be new for the fractional N s -Laplacian Choquard equation. [ABSTRACT FROM AUTHOR]

In this article, we consider the following quasilinear Schrödinger system: −εΔu+u+k2ε[Δ∣u∣2]u=2αα+β∣u∣α−2u∣v∣β,x∈RN,−εΔv+v+k2ε[Δ∣v∣2]v=2βα+β∣u∣α∣v∣β−2v,x∈RN,\left\{\begin{array}{ll}-\varepsilon \Delta u+u+\frac{k}{2}\varepsilon \left[\Delta \hspace{-0.25em}{| u| }^{2}]u=\frac{2\alpha }{\alpha +\beta }{| u| }^{\alpha -2}u{| v| }^{\beta },& x\in {{\mathbb{R}}}^{N},\\ -\varepsilon \Delta v+v+\frac{k}{2}\varepsilon \left[\Delta \hspace{-0.25em}{| v| }^{2}]v=\frac{2\beta }{\alpha +\beta }{| u| }^{\alpha }{| v| }^{\beta -2}v,& x\in {{\mathbb{R}}}^{N},\end{array}\right. where ε>0,k

We investigate some nonlinear elliptic problems of the form \Delta^2v + \sigma(x) v= h(x,v)\quadin \Omega,\quad v=\Delta v=0 \quadon \partial\Omega, \eqno({\rm P}) where $\Omega$ is a regular bounded domain in $\mathbb{R}^N$, $N\geq2$, $\sigma(x)$ a positive function in $L^{\infty}(\Omega)$, and the nonlinearity $h(x,t)$ is indefinite. We prove the existence of solutions to the problem (P) when the function $h(x,t)$ is asymptotically linear at infinity by using variational method but without the Ambrosetti-Rabinowitz condition. Also, we consider the case when the nonlinearities are superlinear and subcritical.

In this paper, we study the following Schrödinger–Hardy system \begin{equation*} \begin{cases} -\Delta_{\mathbb{G}}u-\mu\frac{\psi^2}{r(\xi)^2}u=F_u(\xi,u,v)\ &{\rm in}\ \Omega, \\ -\Delta_{\mathbb{G}}v-\nu\frac{\psi^2 }{r(\xi)^2}v=F_v(\xi,u,v)\ &{\rm in}\ \Omega, \\ u=v=0 \ \ & {\rm on}\ \partial\Omega, \end{cases} \end{equation*} where $\Omega $ is a smooth bounded domain on Carnot groups $\mathbb{G}$, whose homogeneous dimension is $Q\geq 3$, $\Delta_{\mathbb{G}}$ denotes the sub-Laplacian operator on $\mathbb{G}$, $\mu$ and $\nu$ are real parameters, $r(\xi)$ is the natural gauge associated with fundamental solution of $-\Delta_{\mathbb{G}}$ on $\mathbb{G}$, $\psi$ is the geometrical function defined as $\psi=|\nabla_{\mathbb{G}}r|$, and $\nabla_{\mathbb{G}}$ is the horizontal gradient associated with $\Delta_{\mathbb{G}}$. The difficulty is not only the nonlinearities $F_u$ and $F_v$ without Ambrosetti–Rabinowitz condition, but also the Hardy terms and the structure on Carnot groups. We obtain the existence of nonnegative solution for this system by mountain pass theorem in a new framework.

In this paper, we consider the existence and nonexistence of positive solution for the following critical Neumann problem (0.1) { − Δ u − 1 2 (x ⋅ ∇ u) = λ u + a | u | 2 ⁎ − 2 u in R + N , ∂ u ∂ n = μ | u | q − 2 u + | u | 2 ⁎ − 2 u on ∂ R + N , where R + N = { (x ′ , x N) : x ′ ∈ R N − 1 , x N > 0 } is the upper half-space, N ≥ 3 , λ , μ ∈ R are parameters, a ∈ { 0 , 1 } , n is the outward normal vector at the boundary ∂ R + N , 2 ≤ q < 2 ⁎ , 2 ⁎ = 2 N N − 2 is the usual critical exponent for the Sobolev embedding D 1 , 2 (R + N) ↪ L 2 ⁎ (R + N) and 2 ⁎ = 2 (N − 1) N − 2 is the critical exponent for the Sobolev trace embedding D 1 , 2 (R + N) ↪ L 2 ⁎ (∂ R + N). By applying the Mountain Pass Theorem without (PS) condition and the delicate estimates for Mountain Pass level, we obtain the existence of a positive solution under different assumptions on λ , μ and q. Meanwhile, some nonexistence results for problem (0.1) are also obtained by an improved Pohozaev identity and Hardy inequality according to the value of the parameters λ , μ and q. Particularly, for N ≥ 3 , λ ≥ 0 , μ > 0 and q ∈ (2 , 2 ⁎) , we obtain that problem (0.1) has a positive solution if and only if λ ∈ [ 0 , N 2) ; On the other hand, for N ≥ 3 and μ = 0 , we find a lower bound Λ ⁎ ∈ [ N 4 , N 2) depending on N such that problem (0.1) has a positive solution if λ ∈ (Λ ⁎ , N 2) and has no positive solution if λ ∈ (− ∞ , N 4) ∪ [ N 2 , + ∞). Moreover, we estimate that Λ ⁎ ∈ (N 4 , N − 2 2) if N ≥ 5 and Λ ⁎ = N 4 = 1 if N = 4 which gives the best lower bound of λ for the existence of a positive solution of problem (0.1) if N = 4. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence of standing wave solutions for a class of nonlinear fractional Schrödinger-Poisson system involving nonlinearity with subcritical and critical growth. We suppose that the potential V satisfies either Palais-Smale type condition or there exists a bounded domain Ω such that V has no critical point in ∂ Ω . To overcome the "lack of compactness" of the problem, we combine Del Pino-Felmer's penalization technique with Moser's iteration method and some ideas from Alves [1]. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study the p-Kirchhoff type equations with logarithmic nonlinear terms and critical growth: − M ∫ Ω ∇ u p d x Δ p u = u p ∗ − 2 u + λ u p − 2 u − u p − 2 u ln u 2           x ∈ Ω ,                                                                         u = 0                                                                                   x ∈ ∂ Ω ,                                   where Ω ⊂ ℝ N is a bounded domain with a smooth boundary, 2 < p < p ∗ < N , and both p and N are positive integers. By using the Nehari manifold and the Mountain Pass Theorem without the Palais-Smale compactness condition, it was proved that the equation had at least one nontrivial solution under appropriate conditions. It addresses the challenges posed by the critical term, the Kirchhoff nonlocal term and the logarithmic nonlinear term. Additionally, it extends partial results of the Brézis–Nirenberg problem with logarithmic perturbation from p = 2 to more general p-Kirchhoff type problems. [ABSTRACT FROM AUTHOR]

In the present paper, by the variational method coupled with mountain pass type theorems, we study nontrivial homoclinic solutions of nonlinear $ (p, q) $-Laplacian partial difference equations with a parameter $ \lambda>0 $:$ \begin{equation*} \begin{split} &\Delta^2_1(\phi_{p_2}(\Delta^2_1u(k-2, l)))+\Delta^2_2(\phi_{p_2}(\Delta^2_2u(k, l-2)))-a[\Delta_1(\phi_{p_1}(\Delta_1u(k-1, l)))\\ &+\Delta_2(\phi_{p_1}(\Delta_2u(k, l-1)))]+V(k, l)\phi_q(u(k, l))\\& = \lambda f((k, l), u(k, l)). \end{split} \end{equation*} $We prove that the above equation admits two nontrivial homoclinic solutions if $ \lambda $ is sufficiently large and one nontrivial homoclinic solutions if $ \lambda>0 $. Our obtained results generalize and improve some existing ones. [ABSTRACT FROM AUTHOR]

In this paper, we study a higher order Kirchhoff problem with variable exponent of type M∫Ω|Dru|m(x)m(x)dxΔm(x)ru=f(x,u)inΩ,Dαu=0,on∂Ω,for eachα∈ℝNwith|α|≤r−1,$$ \left\{\begin{array}{ll}M\left({\int}_{\Omega}\frac{{\left&#x0007C;{\mathcal{D}}_ru\right&#x0007C;}&#x0005E;{m(x)}}{m(x)} dx\right){\Delta}_{m(x)}&#x0005E;ru&#x0003D;f\left(x,u\right)& \mathrm{in}\kern0.30em \Omega, \\ {}{D}&#x0005E;{\alpha }u&#x0003D;0,\kern0.30em & \mathrm{on}\kern0.30em \mathrm{\partial \Omega },\kern0.30em \mathrm{for}\ \mathrm{each}\kern0.4em \alpha \in {\mathrm{\mathbb{R}}}&#x0005E;N\kern0.4em \mathrm{with}\kern0.4em \mid \alpha \mid \le r-1,\end{array}\right. $$where Ω⊂ℝN$$ \Omega \subset {\mathrm{\mathbb{R}}}&#x0005E;N $$ is a smooth bounded domain, r∈ℕ∗,m∈C(Ω‾),1

This paper deals with the existence and multiplicity results for a bi‐nonlocal problem without the Ambrosetti–Rabinwitz condition, in the context of variable exponents of Sobolev spaces on compact Riemannian manifolds. Using the mountain pass theorem, we obtain that our problem admits at least nontrivial weak solutions. Also, using the fountain and dual fountain theorems, we obtain the existence of infinitely many nontrivial high or small energy solutions. [ABSTRACT FROM AUTHOR]

Our focus in this study revolves around investigating a Kirchhoff problem involving the fractional p(x)-Laplacian operator. The purpose is to study the existence and multiplicity of weak solutions for the above problem under appropriate hypotheses on functions f and m. By using the Mountain Pass Theorem with Cerami condition, we show the existence of non-trivial weak solution for the problem without assuming the Ambrosetti-Rabinowitz condition. Furthermore, our second purpose is to determine the precise positive interval of λ for which the above problem admits at least two nontrivial weak solutions. It should be noted that the existence of infinitely many weak solutions is proved by employing the Fountain Theorem. [ABSTRACT FROM AUTHOR]

In this paper, we study the fractional critical Schrödinger–Poisson system (- Δ) s u + λ ϕ u = α u + μ | u | q - 2 u + | u | 2 s ∗ - 2 u , in R 3 , (- Δ) t ϕ = u 2 , in R 3 , having prescribed mass ∫ R 3 | u | 2 d x = a 2 , where s , t ∈ (0 , 1) satisfy 2 s + 2 t > 3 , q ∈ (2 , 2 s ∗) , a > 0 and λ , μ > 0 parameters and α ∈ R is an undetermined parameter. For this problem, under the L 2 -subcritical perturbation μ | u | q - 2 u , q ∈ (2 , 2 + 4 s 3) , we derive the existence of multiple normalized solutions by means of the truncation technique, concentration-compactness principle and the genus theory. In the L 2 -supercritical perturbation μ | u | q - 2 u , q ∈ (2 + 4 s 3 , 2 s ∗) , we prove two different results of normalized solutions when parameters λ , μ satisfy different assumptions, by applying the constrained variational methods and the mountain pass theorem. Our results extend and improve some previous ones of Zhang et al. (Adv Nonlinear Stud 16:15–30, 2016); and of Teng (J Differ Equ 261:3061–3106, 2016), since we are concerned with normalized solutions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the quasilinear Schrödinger equation \[ -\Delta u+V(x)u-\Delta[(1+u^2)^{1/2}]\frac{u}{2(1+u^2)^{1/2}}=h(x,u), \quad {\rm in}\ \mathbb{R}^{N}, \] − Δ u + V (x) u − Δ [ (1 + u 2) 1 / 2 ] u 2 (1 + u 2) 1 / 2 = h (x , u) , in R N , where $ N\geq 3 $ N ≥ 3 , V is a given potential allowed to be indefinite, or equivalently, the Schrödinger operator $ -\Delta +V $ − Δ + V 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. [ABSTRACT FROM AUTHOR]

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]. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the existence of at least two standing waves (and homoclinic solutions) for a class of time-dependent (and time-independent) discrete nonlinear Schrödinger systems or equations. The novelties of the paper are as follows. (1) Our nonlinearities are composed of three mixed growth terms, i.e., the nonlinearities are composed of sub-linear, asymptotically-linear and super-linear terms. (2) Our nonlinearities may be sign-changing. (3) Our results can also be applied to the cases of concave-convex nonlinear terms. (4) Our results can be applied to a wide range of mathematical models. [ABSTRACT FROM AUTHOR]

We present a variational framework for studying the existence and regularity of solutions to elliptic free boundary problems that do not necessarily minimize energy. As applications, we obtain mountain pass solutions of critical and subcritical superlinear free boundary problems, and establish full regularity of the free boundary in dimension N = 2 and partial regularity in higher dimensions. [ABSTRACT FROM AUTHOR]

We consider the following class of elliptic Kirchhoff-Boussinesq type problems given by Δ 2 u ± Δ p u = f (u) + β | u | 2 ∗ ∗ - 2 u in Ω and Δ u = u = 0 on ∂ Ω , where Ω ⊂ R N is a bounded and smooth domain, 2 < p ≤ 2 N N - 2 for N ≥ 3 , 2 ∗ ∗ = 2 N N - 4 if N ≥ 5 , 2 ∗ ∗ = ∞ if 3 ≤ N < 5 and f is a continuous function. We show existence and multiplicity of nontrivial solutions using minimization technique on the Nehari manifold, Mountain Pass Theorem and Genus theory. In this paper we consider the subcritical case β = 0 and the critical case β = 1 . [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of fourth-order dispersive wave equations with exponential source term. Firstly, by applying the contraction mapping principle, we establish the local existence and uniqueness of the solution. In the spirit of the variational principle and mountain pass theorem, a natural phase space is precisely divided into three different energy levels. Then we introduce a family of potential wells to derive a threshold of the existence of global solutions and blow up in finite time of solution in both cases with sub-critical and critical initial energy. These results can be used to extend the previous result obtained by Alves and Cavalcanti (Calc. Var. Partial Differ. Equ. 34 (2009) 377–411). Moreover, an explicit sufficient condition for initial data leading to blow up result is established at an arbitrarily positive initial energy level. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider a ( ϕ 1 , ϕ 2 ) $(\phi _{1},\phi _{2})$ -Laplacian system as follows: { Δ ϕ 1 ( Δ u ( t − 1 ) ) + ∇ u F ( t , u ( t ) , v ( t ) ) = 0 , Δ ϕ 2 ( Δ v ( t − 1 ) ) + ∇ v F ( t , u ( t ) , v ( t ) ) = 0 , $$\begin{aligned} \textstyle\begin{cases} \Delta \phi _{1} (\Delta u(t-1) )+\nabla _{u} F(t,u(t),v(t))=0, \\ \Delta \phi _{2} (\Delta v(t-1) )+\nabla _{v} F(t,u(t),v(t))=0, \end{cases}\displaystyle \end{aligned}$$ where F ( t , u ( t ) , v ( t ) ) = − K ( t , u ( t ) , v ( t ) ) + W ( t , u ( t ) , v ( t ) ) $F(t,u(t),v(t))=-K(t,u(t),v(t))+W(t,u(t),v(t))$ is T-periodic in t. By using the mountain pass theorem, we obtain that the ( ϕ 1 , ϕ 2 ) $(\phi _{1},\phi _{2})$ -Laplacian system has at least one periodic solution if W is asymptotically ( p , q ) $(p,q)$ -linear at infinity. Our results improve and extend some known works.

In this work, we focus our attention on the existence of nontrivial solutions to the following supercritical Schrödinger-Poisson type system with $ (p, q) $-Laplacian: $ \begin{equation*} \begin{cases} -\Delta_{p}u-\Delta_{q}u+\phi|u|^{q-2} u = f\left(x, u\right)+\mu|u|^{s-2} u & \text { in } \Omega, \\ -\Delta \phi = |u|^q & \text { in } \Omega, \\ u = \phi = 0 & \text { on } \partial \Omega, \end{cases} \end{equation*} $ where $ \Omega \subset \mathbb{R}^N $ is a bounded smooth domain, $ \mu > 0, N > 1 $, and $ -\Delta_{{\wp}}\varphi = div(|\nabla\varphi|^{{\wp}-2} \nabla\varphi) $, with $ {\wp}\in \{p, q\} $, is the homogeneous $ {\wp} $-Laplacian. $ 1 < p < q < \frac{q^*}{2} $, $ q^*: = \frac{Nq}{N-q} < s $, and $ q^* $ is the critical exponent to $ q $. The proof is accomplished by the Moser iterative method, the mountain pass theorem, and the truncation technique. Furthermore, the $ (p, q) $-Laplacian and the supercritical term appear simultaneously, which is the main innovation and difficulty of this paper.

In this paper, we analyze the existence of solutions to a double-phase fractional equation of the Kirchhoff type in Musielak-Orlicz Sobolev space with variable exponents. Our approach is mainly based on the sub-supersolution method and the mountain pass theorem. [ABSTRACT FROM AUTHOR]

By means of variational methods, we study the existence and multiplicity of nontrivial weak solutions for a nonlocal τ(m)-Laplacian-like problem, arising from a capillarity phenomenon, with logarithmic nonlinearity and without the Ambrosetti-Rabinwitz condition, in the setting of variable exponents of Sobolev spaces in compact Riemannian manifolds. [ABSTRACT FROM AUTHOR]

This article explores the concept of a ground-state solution for a quasilinear Schrödinger system. The authors employ the critical point theorem and Pohozaev identity to establish the existence of a non-trivial solution with the minimum energy. They also offer a brief overview of previous research on the quasilinear Schrödinger system and its practical applications. The primary focus of the paper is to prove the existence of a ground-state solution for the given system. The article is structured into sections that outline the problem, provide foundational information, and present the proof of the main theorem. Additionally, the authors acknowledge the funding sources for their research and conclude the article with publication details and contact information. [Extracted from the article]

In this paper, we consider a (ϕ 1 , ϕ 2) -Laplacian system as follows: { Δ ϕ 1 (Δ u (t − 1)) + ∇ u F (t , u (t) , v (t)) = 0 , Δ ϕ 2 (Δ v (t − 1)) + ∇ v F (t , u (t) , v (t)) = 0 , where F (t , u (t) , v (t)) = − K (t , u (t) , v (t)) + W (t , u (t) , v (t)) is T-periodic in t. By using the mountain pass theorem, we obtain that the (ϕ 1 , ϕ 2) -Laplacian system has at least one periodic solution if W is asymptotically (p , q) -linear at infinity. Our results improve and extend some known works. [ABSTRACT FROM AUTHOR]

In this paper, for dimension N≥2$N\ge 2$ and prescribed mass m>0$m>0$, we consider the following nonlinear scalar field equation with L2$L^2$ constraint: −Δu+V(x)u+λu=g(u)inRN,∫RNu2=m,$$\begin{equation*} \left\{ \def\eqcellsep{&}\begin{array}{l} -\Delta u+V(x)u+\lambda u=g(u)\qquad \hbox{in} \; \mathbb {R}^N, \\ \int _{\mathbb {R}^N} u^2=m, \end{array} \right. \end{equation*}$$where λ∈R$\lambda \in \mathbb {R}$ is a Lagrange multiplier, V(x)∈C1(RN,R)$V(x)\in C^1 (\mathbb {R}^N,\mathbb {R})$. In particular, g(x)∈C(R,R)$g(x)\in C(\mathbb {R},\mathbb {R})$ satisfies mass supercritical and Sobolev subcritical growth. We prove the existence results of the normalized solution and infinitely many normalized solutions to the above system under some proper assumptions on the functions V(x),g(x)$V(x), g(x)$ by the mountain pass argument. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we study the following non-local problem in fractional Orlicz–Sobolev spaces: (− Δ Φ) s u + V (x) a (| u |) u = f (x , u) ,   x ∈ R N , where (− Δ Φ) s (s ∈ (0 , 1)) denotes the non-local and maybe non-homogeneous operator, the so-called fractional Φ -Laplacian. Without assuming the Ambrosetti–Rabinowitz type and the Nehari type conditions on the non-linearity f, we obtain the existence of ground state solutions for the above problem with periodic potential function V (x) . The proof is based on a variant version of the mountain pass theorem and a Lions' type result in fractional Orlicz–Sobolev spaces. [ABSTRACT FROM AUTHOR]

This paper is concerned with the quasilinear Schrödinger–Poisson system − Δ p u − l (x) ϕ | u | p − 2 u = | u | p * − 2 u + μ h (x) | u | q − 2 u in R 3 and −Δϕ = l(x)|u|p in R 3 , where μ > 0, p * = 3 p 3 − p and Δpu = div(|∇u|p−2∇u). By using the Ekeland's variational principle and the mountain pass theorem, we prove that the system admits two positive solutions for 1 ⩽ q < p and 1 < p < 3, and the system admits one positive solution for p ⩽ q < p* and 3 2 < p < 3. [ABSTRACT FROM AUTHOR]

In this work, we focus our attention on the existence of nontrivial solutions to the following supercritical Schrödinger-Poisson type system with (p, q)-Laplacian:... [ABSTRACT FROM AUTHOR]

In this paper, two problems related to the following class of elliptic Kirchhoff–Boussinesq-type models are analyzed in the subcritical ( β = 0 ) and critical ( β = 1 ) cases: Δ 2 u - Δ p u = τ | u | q - 2 u ln | u | + β | u | 2 ∗ ∗ - 2 u in Ω and Δ u = u = 0 on ∂ Ω , where τ > 0 , 2 < p < 2 ∗ = 2 N N - 2 for N ≥ 3 and 2 ∗ ∗ = ∞ for N = 3 , N = 4 , 2 ∗ ∗ = 2 N N - 4 for N ≥ 5 . The first one is concerned with the existence of a nontrivial ground-state solution via variational methods. As for the second problem, we prove the multiplicity of such a solution using the Mountain Pass Theorem. [ABSTRACT FROM AUTHOR]

We exploit the existence and non-existence of positive solutions to the eigenvalue problem driven by the nonhomogeneous fractional $ p\& q $ p &q Laplacian operator with indefinite weights \[ \left(-\Delta_p\right)^{\alpha}u + \left(-\Delta_q\right)^{\beta}u = \lambda\left[a \left|u\right|^{p-2}u + b \left|u\right|^{q-2}u \right]\quad{\rm in}\ \Omega, \] (− Δ p) α u + (− Δ q) β u = λ [ a | u | p − 2 u + b | u | q − 2 u ] in Ω , where $ \Omega \subseteq \mathbb {R}^N $ Ω ⊆ R N is a smooth bounded domain that has been extended by zero. We further show the existence of a continuous family of eigenvalues in the case $ \Omega =\mathbb {R}^N $ Ω = R N and $ b\equiv 0 $ b ≡ 0 a.e. Our approach relies strongly on variational Analysis, in which the Mountain pass theorem plays the key role. Due to the lack of spatial compactness and the embedding $ \mathcal {W}^{\alpha, p}\left (\mathbb {R}^N\right) \hookrightarrow \mathcal {W}^{\beta, q}\left (\mathbb {R}^N\right) $ W α , p (R N) ↪ W β , q (R N) in $ \mathbb {R}^N $ R N , we employ the concentration-compactness principle of P.L. Lions [The concentration-compactness principle in the calculus of variations. The limit case. II, Rev Mat Iberoamericana. 1985;1(2):45–121]. to overcome the difficulty. Our paper can be considered as a counterpart to the important works [Alves et al. Existence, multiplicity and concentration for a class of fractional $ p\& q $ p &q Laplacian problems in $ \Bbb R^N $ R N , Commun Pure Appl Anal, 2019;18(4):2009–2045], [Benci et al. An eigenvalue problem for a quasilinear elliptic field equation. J Differ Equ, 2002;184(2):299–320], [Bobkov et al. On positive solutions for $ (p,q) $ (p , q) -Laplace equations with two parameters, Calc Var Partial Differ Equ, 2015;54(3):3277–3301], [Colasuonno and Squassina. Eigenvalues for double phase variational integrals, Ann Mat Pura Appl (4), 2016;195(6):1917–1956], [Papageorgiou et al. Positive solutions for nonlinear Neumann problems with singular terms and convection, J Math Pures Appl (9), 2020;136:1–21], [Papageorgiou et al. Ground state and nodal solutions for a class of double phase problems, Z Angew Math Phys, 2020;71:1–15], and may have further applications to deal with other problems. [ABSTRACT FROM AUTHOR]

This work deals with the existence and multiplicity of solutions for a class of variable-exponent equations involving the Kirchhoff term in variable-exponent Sobolev spaces according to some conditions, where we used the sub-supersolutions method combined with the mountain pass theory. [ABSTRACT FROM AUTHOR]

Let Ω be a finite connected subset of a locally finite graph G = (V, E) with the vertex set V and the edge set E. We investigate the logarithmic Schrödinger equation on Ω with the nonlinear term |u|p−2u log u2. For p > 2, through two different approaches which are the Brouwer degree theory and mountain-pass theorem, we obtain the existence of ground state solutions. We also apply the Brouwer degree theory together with the constraint variational method to prove that the equation admits a sign-changing solution which implies the multiplicity of solutions to the equation. Finally, we illustrate that as p → 2, up to a subsequence, the solutions for p > 2 shall converge to a non-trivial solution of the equation with p = 2. [ABSTRACT FROM AUTHOR]

We are devoted to the study of the following sub-Laplacian system with Hardy-type potentials and critical nonlinearities { −ΔGu−μ1 ψ2u d(z)2 =λ1 ψα|u|2∗(α)−2u d(z)α +βp1f(z) ψγ|u|p1−2u|v|p2 d(z)γ in G, −ΔGv−μ2 ψ2v d(z)2 =λ2 ψα|v|2∗(α)−2v d(z)α +βp2f(z) ψγ|u|p1|v|p2−2v d(z)γ in G, where −ΔG is the sub-Laplacian on Carnot group G, μ1, μ2∈[0,μG), α,γ∈(0,2), λ1, λ2, β, p1, p2>0 with 1

In this paper we study the following nonlinear fractional Hartree (or Choquard-Pekar) equation (− Δ) s u + μ u = ( I α * F (u)) F ′ (u) in R N , (*) where μ > 0, s ∈ (0, 1), N ≥ 2, α ∈ (0, N), I α ∼ 1 | x | N − α is the Riesz potential, and F is a general subcritical nonlinearity. The goal is to prove existence of multiple (radially symmetric) solutions u ∈ H s ( R N ) , by assuming F odd or even: we consider both the case μ > 0 fixed and the case ∫ R N u 2 = m > 0 prescribed. Here we also simplify some arguments developed for s = 1 (S. Cingolani, M. Gallo, and K. Tanaka, "Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities," Calc. Var. Partial Differ. Equ., vol. 61, no. 68, p. 34, 2022). A key point in the proof is given by the research of suitable multidimensional odd paths, which was done in the local case by Berestycki and Lions (H. Berestycki and P.-L. Lions, "Nonlinear scalar field equations II: existence of infinitely many solutions," Arch. Ration. Mech. Anal., vol. 82, no. 4, pp. 347–375, 1983); for (*) the nonlocalities play indeed a special role. In particular, some properties of these paths are needed in the asymptotic study (as μ varies) of the mountain pass values of the unconstrained problem, then exploited to describe the geometry of the constrained problem and detect infinitely many normalized solutions for any m > 0. The found solutions satisfy in addition a Pohozaev identity: in this paper we further investigate the validity of this identity for solutions of doubly nonlocal equations under a C1-regularity. [ABSTRACT FROM AUTHOR]

In this paper we aim to combine tools from variational calculus with modern techniques from quaternionic analysis that involve Dirac type operators and related hypercomplex integral operators. The aim is to develop new methods for showing geometry independent explicit global existence and uniqueness criteria as well as new computational methods with special focus to the stationary incompressible viscous magnetohydrodynamic equations. We first show how to specifically apply variational calculus in the quaternionic setting. To this end we explain how the mountain pass theorem can be successfully applied to guarantee the existence of (weak) solutions. To achieve this, the quaternionic integral operator calculus serves as a key ingredient allowing us to apply Schauder’s fixed point theorem. The advantage of the approach using Schauder’s fixed point theorem is that it is also applicable to large data since it does not require any kind of contraction property. These considerations will allow us to provide explicit iterative algorithms for its numerical solution. Finally to obtain more precise a-priori estimates one can use in the situations dealing with small data the Banach fixed point theorem which then also grants the uniqueness. [ABSTRACT FROM AUTHOR]

We study the following singular Kirchhoff type problem ... where Ω ⊂ R² is a bounded domain with smooth boundary and 0 ∈, β ∈ (0,2), α > 0 and m is a continuous function on R+. Here, h is a suitable preturbation of ᵉau2 as u → ∞. In this paper, we prove the existence of solutions of (P). Our tools are Trudinger-Moser inequality with a singular weight and the mountain pass theorem. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following p(x)-biharmonic problem with Hardy nonlinearity:... ... where Ω ⊂ RN (N≥ 3),Δp (x)isp (x)-laplacian andΔ²p(x) isp (x)- biharmonic operator. More precisely, to prove the existence and multiplicity of solutions, variational methods are combined with the theory of generalized Lebesgue and Sobolev spaces to prove the existence and the multiplicity of solutions. [ABSTRACT FROM AUTHOR]

We investigate the existence of infinitely many periodic solutions for the p(t)- Laplacian Hamiltonian systems. By virtue of several auxiliary functions, we obtain a series of new super-p + growth and asymptotic-p + growth conditions. Using the minimax methods in critical point theory, some multiplicity theorems are established, which unify and generalize some known results in the literature. Meanwhile, we also present an example to illustrate our main results are new even in the case p(t) ≡ p = 2. [ABSTRACT FROM AUTHOR]

We investigate some nonlinear elliptic problems of the form (P) ∆2 v + σ(x)v = h(x, v) in Ω, v = ∆v = 0 on ∂Ω, where Ω is a regular bounded domain in RN, N > 2, σ(x) a positive function in L∞(Ω), and the nonlinearity h(x, t) is indefinite. We prove the existence of solutions to the problem (P) when the function h(x, t) is asymptotically linear at infinity by using variational method but without the Ambrosetti-Rabinowitz condition. Also, we consider the case when the nonlinearities are superlinear and subcritical. [ABSTRACT FROM AUTHOR]

In this paper we consider quasilinear elliptic equations driven by the variable exponent double phase operator with superlinear right-hand sides. Under very general assumptions on the nonlinearity, we prove a multiplicity result for such problems whereby we show the existence of a positive solution, a negative one and a solution with changing sign. The sign-changing solution is obtained via the Nehari manifold approach and, in addition, we can also give information on its nodal domains. [ABSTRACT FROM AUTHOR]