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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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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 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

In this paper, we deal with the existence of nontrivial nonnegative solutions for a ( p , N ) {(p,N)} -Laplacian Schrödinger–Kirchhoff problem in ℝ N {\mathbb{R}^{N}} with singular exponential nonlinearity. The main features of the paper are the ( p , N ) {(p,N)} growth of the elliptic operators, the double lack of compactness, and the fact that the Kirchhoff function is of degenerate type. To establish the existence results, we use the mountain pass theorem, the Ekeland variational principle, the singular Trudinger–Moser inequality, and a completely new Brézis–Lieb-type lemma for singular exponential nonlinearity. [ABSTRACT FROM AUTHOR]

We investigate the existence of nontrivial solutions for a class of poly‐Laplacian nonlinear system with mixed nonlinearity term on a locally finite graph or a finite graph. By using the mountain pass theorem, we obtain some sufficient conditions about existence of a nontrivial solution. Our results are different from some known results even for the scalar equation on graph. [ABSTRACT FROM AUTHOR]

In this article, we consider a class of nonlocal p(x)-Laplace equations with nonlinear boundary conditions. When the nonlinear boundary involves critical exponents, using the concentration compactness principle, mountain pass lemma, and fountain theorem, we can prove the existence and multiplicity of solutions. [ABSTRACT FROM AUTHOR]

In this thesis, we research quasilinear Schrödinger system as follows in which 3 < N ∈ R , 2 < p < N , 2 < q < N , V 1 (x) , V 2 (x) are continuous functions, k , ι are parameters with k , ι > 0 , and nonlinear terms f , h ∈ C (R N × R 2 , R) . We find a nontrivial solution (u , v) for all ι > ι 1 (k) by means of the mountain-pass theorem and change of variable theorem. Our main novelty of the thesis is that we extend Δ to Δ p and Δ q to find the existence of a nontrivial solution. [ABSTRACT FROM AUTHOR]

We prove the existence of infinitely many solutions to a fractional Choquard type equation (− Δ) p s u + V (x) | u | p − 2 u = (K ⁎ g (u)) g ′ (u) + ε W W (x) f ′ (u) in R N involving the fractional p -Laplacian and a general convolution term with critical growth. In order to obtain infinitely many solutions, we use a type of the symmetric mountain pass lemma which gives sequences of critical values converging to zero for even functionals. To assure the (P S) c conditions, we also use a nonlocal version of the concentration compactness lemma. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and multiplicity of solutions for the Schrödinger–Bopp–Podolsky system { - Δ ⁢ u + V ⁢ (x) ⁢ u + ϕ ⁢ u = f ⁢ (u) + λ ⁢ | u | 4 ⁢ u in ⁢ ℝ 3 , - Δ ⁢ ϕ + a 2 ⁢ Δ 2 ⁢ ϕ = 4 ⁢ π ⁢ u 2 in ⁢ ℝ 3 , where x ∈ ℝ 3 , a > 0 , V ⁢ (x) ∈ 풞 ⁢ (ℝ 3 , ℝ) . Using variational methods and the symmetric mountain pass theorem, we establish the existence of multiple solutions for this system. [ABSTRACT FROM AUTHOR]

In this paper, we study the following nonautonomous Kirchhoff problem: −1+b∫ℝN|∇u|2dxΔu+V(x)u=a(x)|u|p−2u+λ|u|q−2u,x∈ℝN,u∈H1ℝN,$$ \left\{\begin{array}{l}-\left(1&#x0002B;b{\int}_{{\mathrm{\mathbb{R}}}&#x0005E;N}{\left&#x0007C;\nabla u\right&#x0007C;}&#x0005E;2\mathrm{d}x\right)\Delta u&#x0002B;V(x)u&#x0003D;a(x){\left&#x0007C;u\right&#x0007C;}&#x0005E;{p-2}u&#x0002B;\lambda {\left&#x0007C;u\right&#x0007C;}&#x0005E;{q-2}u,x\in {\mathrm{\mathbb{R}}}&#x0005E;N,\\ {}u\in {H}&#x0005E;1\left({\mathrm{\mathbb{R}}}&#x0005E;N\right),\end{array}\right. $$where N≥2$$ N\ge 2 $$, p∈(1,2∗)$$ p\in \left(1,{2}&#x0005E;{\ast}\right) $$, q∈[2,2∗)$$ q\in \left[2,{2}&#x0005E;{\ast}\right) $$, b$$ b $$ is a positive constant, λ≥0$$ \lambda \ge 0 $$ is a parameter, and the potential functions a,V$$ a,V $$ belong to C(ℝN,(0,+∞))$$ C\left({\mathrm{\mathbb{R}}}&#x0005E;N,\left(0,&#x0002B;\infty \right)\right) $$. The existence of radially symmetric and positive solution to the above problem is first established for all λ≥0$$ \lambda \ge 0 $$ when a,V$$ a,V $$ are radially symmetric and p,q∈(4,2∗)$$ p,q\in \left(4,{2}&#x0005E;{\ast}\right) $$, and the range of q$$ q $$ can be extended to q∈(2,2∗)$$ q\in \left(2,{2}&#x0005E;{\ast}\right) $$ with the aid of a coercive type assumption on V$$ V $$. Moreover, we show the existence of infinitely many solutions with high energies via the fountain theorem under more general assumption on V$$ V $$ which allows it to be sign‐changing. When p∈(1,2)$$ p\in \left(1,2\right) $$ and q∈(2,2∗)$$ q\in \left(2,{2}&#x0005E;{\ast}\right) $$, we show that the above problem possesses infinitely many solutions with negative critical values for small λ$$ \lambda $$ provided that the function V−1$$ {V}&#x0005E;{-1} $$ belongs to a suitable space. In particular, by imposing a hypothesis on the potential V$$ V $$ controlling its growth at infinity, we obtain a nonradial solution via the mountain pass theorem and the principle of symmetric criticality. [ABSTRACT FROM AUTHOR]

The existence of infinitely many weak solutions for a class of quasilinear elliptic pi(x)$$ {p}_i(x) $$‐Kirchhoff system is investigated by the symmetric mountain pass theorem. Furthermore, for the case pi(x)≡Const$$ {p}_i(x)\equiv Const $$ where i=1,2,...,n$$ i&amp;#x0003D;1,2,\dots, n $$, I introduce a positive quantity λM$$ {\lambda}_M $$ to find a mountain pass weak solution and to consider the sublinear case under large growth conditions at infinity and zero. [ABSTRACT FROM AUTHOR]

This paper is concerned with the following first-order Hamiltonian system -z = JHz(t, z), where the Hamiltonian function H(t, z) = 12 Lz · z + A(?t)G(|z|) and - > 0 is a small parameter. Under some natural conditions, we obtain a new existence result for ground state homoclinic orbits by applying variational methods. Moreover, the concentration behavior and exponential decay of these ground state homoclinic orbits are also investigated. [ABSTRACT FROM AUTHOR]

We are concerned with the following fractional Kirchhoff variational inequality: .... In this paper, by applying penalization techniques from Bensoussan and Lions (1978) combined with mountain pass theorem, we show the existence and concentration behavior of positive solution to the cited variational inequality. This result extend some results established by Alves, Barros and Torres [J. Math. Anal. Appl. 494 (2021)] to the fractional case. [ABSTRACT FROM AUTHOR]

In this paper, by using the mountain pass theorem and the concentration compactness principle, we prove the existence of a positive solution for a p -Kirchhoff-type problem with critical Sobolev exponent. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The aim of this paper is the study a class of double phase problems with variable exponent. Using the Clark's theorem and the symmetric mountain pass lemma, we prove the existence of infinitely many small solutions and infinitely many large solutions, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of solutions for a class of integrodifferential Kirchhoff equations. These equations involve a nonlocal operator with a measurable kernel that satisfies "structural properties" that are more general than the standard kernel of the fractional Laplacian operator. Additionally, the potential can be periodic or asymptotically periodic, and the nonlinear term exhibits critical exponential growth in the sense of Trudinger–Moser inequality. To guarantee the existence of solutions, we employ variational methods, specifically the mountain-pass theorem. In this context, it is important to emphasize that we have additional difficulties due to the lack of compactness in our problem, because we deal with critical growth nonlinearities in unbounded domains. Moreover, the Kirchhoff term adds complexity to the problem, as it requires suitable calculations for control the estimate the minimax level, representing the main challenge in this work. Finally, we consider two different approaches to estimate the minimax level. The first approach is based on a hypothesis proposed by D. M. Cao, while the second one involves a slightly weaker assumption addressed by Adimurthi and Miyagaki. [ABSTRACT FROM AUTHOR]

We investigate the uniqueness, multiplicity, and qualitative properties of the positive solutions for a Brezis–Oswald problem perturbed with an indefinite λ -Kirchhoff operator - 1 + λ ‖ u ‖ 2 Δ u = f (u) in Ω , u = 0 on ∂ Ω , where Ω ⊂ R N is a smooth bounded domain, ‖ u ‖ 2 = ∫ Ω | ∇ u | 2 , λ ∈ R is a real parameter and f is a suitable sublinear nonlinearity. The main difficulties in proving the main results lie in the presence of the sublinearity f and that the parameter λ might be negative. We overcome the difficulties by showing the existence of the first solution by minimizing on a suitable compact set involving the well-known properties of the energy functional associated with the Brezis–Oswald problem ( λ = 0 ). The second solution is obtained via mountain pass theorem by controlling the contribution of the Kirchhoff part in the energy functional. We also prove a comparison principle (it is important in itself) for this class of problems that permit us to take advantage of it to prove the uniqueness of solutions for λ ≥ 0 and some qualitative properties of the solutions. [ABSTRACT FROM AUTHOR]

We consider the coupled propagation of an optical field and its second harmonic in a quadratic nonlinear medium governed by a coupled system of Schrodinger equations. We prove the existence of ring‐profiled optical vortex solitons appearing as solutions to a constrained minimization problem and as solutions to a min–max problem. In the case of the constrained minimization problem, solutions are shown to be positive with undetermined wave propagation constants, but in the min–max approach, the wave propagation constants can be prescribed. The quadratic nonlinearity introduces some interesting properties not commonly observed in other coupled systems in the context of nonlinear optics, such as the system not accepting any semi‐trivial solutions, meaning that optical solitons cannot be observed when, say, one of the beams is off. Additionally, the second harmonic always remains positive. [ABSTRACT FROM AUTHOR]

We prove the existence of periodic traveling wave solutions for general discrete nonlinear Klein‐Gordon systems, considering both cases of hard and soft on‐site potentials. In the case of hard on‐site potentials, we implement a fixed‐point theory approach, combining Schauder's fixed‐point theorem and the contraction mapping principle. This approach enables us to identify a ring in the energy space for nontrivial solutions to exist, energy (norm) thresholds for their existence, and upper bounds on their velocity. In the case of soft on‐site potentials, the proof of existence of periodic traveling wave solutions is facilitated by a variational approach based on the mountain pass theorem. Thresholds on the averaged kinetic energy for these solutions to exist are also derived. [ABSTRACT FROM AUTHOR]

The paper deals with the existence and multiplicity of nontrivial solutions for the doubly elliptic problem { Δ u = 0 in Ω , u = 0 on Γ 0 , − Δ Γ u + ∂ ν u = | u | p − 2 u on Γ 1 , where Ω is a bounded open subset of R N ( N ≥ 2 ) with C 1 boundary ∂ Ω = Γ 0 ∪ Γ 1 , Γ 0 ∩ Γ 1 = ∅ , Γ 1 being nonempty and relatively open on Γ , H N − 1 (Γ 0) > 0 and p > 2 being subcritical with respect to Sobolev embedding on ∂ Ω . We prove that the problem admits nontrivial solutions at the potential-well depth energy level, which is the minimal energy level for nontrivial solutions. We also prove that the problem has infinitely many solutions at higher energy levels. The paper deals with the existence and multiplicity of nontrivial solutions for the doubly elliptic problem where is a bounded open subset of ( ) with boundary , , being nonempty and relatively open on , and being subcritical with respect to Sobolev embedding on . We prove that the problem admits nontrivial solutions at the potential-well depth energy level, which is the minimal energy level for nontrivial solutions. We also prove that the problem has infinitely many solutions at higher energy levels. [ABSTRACT FROM AUTHOR]

This paper is devoted to a class of general nonlocal fourth-order elliptic equation Δ 2 u − (a + b ∫ R N | ∇ u | 2 d x) Δ u + V (x) u = f (x , u) , x ∈ R N , where Δ 2 = Δ (Δ) is the biharmonic operator, a > 0 , b ≥ 0 are constants and the potential V is allowed to be sign-changing. Under certain assumptions on f, infinitely many high energy solutions are established. Our results enrich that obtained in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in a class of critical nonlocal problems with variable exponents of the form: { M (T p (⋅ , ⋅) (u)) [ (− Δ) p (x , y) s u + | u | p ~ (x) − 2 u ] = λ f (x , u) + | u | q (x) − 2 u , in R N , u ∈ W s , p (⋅ , ⋅) (R N) , p ~ (x) = p (x , x) , where T p (⋅ , ⋅) (u) := ∬ R 2 N | u (x) − u (y) | p (x , y) p (x , y) | x − y | N + s p (x , y) d x d y + ∫ R N 1 p ~ (x) | u | p ~ (x) d x , M is the Kirchhoff function, λ is a real parameter and f is a continuous function. We also assume that { x ∈ R N : q (x) = p s ∗ (x) } ≠ ∅ , where p s ∗ (x) = N p ~ (x) / (N − s p ~ (x)) is the critical Sobolev exponent for variable exponents. The strategy of the proof for these results is to approach the problem variationally by using the mountain pass theorem and the concentration-compactness principles for fractional Sobolev spaces with variable exponents. In addition, we obtain the existence and multiplicity of nontrivial solutions for the above problem in non-degenerate and degenerate cases. [ABSTRACT FROM AUTHOR]

We consider the integro-differential system (P m) : - a k + b k ∫ R N | ∇ u k | p d x p - 1 Δ p u k + V (x) | u k | p - 2 u k = ∂ k F (u 1 , … , u m) , where x ∈ R N , a k > 0 , b k ≥ 0 , N ≥ 2 and 1 < p < N , u k ∈ W 1 , p (R N) , for k = 1 , … , m . By ∂ k F (u 1 , … , u m) , it is denoted the k-th partial generalized gradient in the sense of Clarke. The potential V ∈ C R N verifies inf (V) > 0 and a coercivity property introduced by Bartsch et al. The coupling function F : R m ⟶ R is locally Lipschitz and verifies conditions introduced by Duan and Huang. By applying tools from the non-smooth critical point theory, we prove the existence of a non-trivial mountain pass solution of (P m) . [ABSTRACT FROM AUTHOR]

This paper deals with a pseudo-parabolic equation involving p(x)-Laplacian and variable nonlinear sources. Firstly, we use the Faedo-Galerkin method to give the existence and uniqueness of weak solution in the Sobolev space with variable exponents. Secondly, in the frame of variational methods, we classify the blow-up and the global existence of solutions completely by using the initial energy, characterized with the mountain pass level, and Nehari energy. In the supercritical case, we construct suitable auxiliary functions to determine the quantitative conditions on the initial data for the existence of blow-up or global solutions. The results in this paper are compatible with the problems with constant exponents. Moreover, we give broader ranges of exponents of source and diffusion terms in the discussion of blow-up or global solutions. [ABSTRACT FROM AUTHOR]

The goal of the paper is to investigate a Kirchhoff-type elliptic problem driven by a generalized nonlocal fractional p-Laplacian whose nonlocal term vanishes at finitely many points. Multiple nontrivial solutions are obtained by applying a variational method combined with truncation techniques. [ABSTRACT FROM AUTHOR]

In this paper, the existence and nonexistence of positive solutions for a relativistic nonlinear Schrödinger equation with vanish potential and general nonlinear term were obtained. First, we reduce this equation to a semilinear elliptic equation by a change. Then, using mountain pass theorem and Strauss compactness lemma, we get the existence of positive radial ground state solutions for this equation. Moreover, the nonexistence of nontrivial solutions for this equation was gotten by Pohozaev identity. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The aim of this article is twofold: firstly, we deal with the existence and multiplicity of weak solutions to the Kirchhoff problem: { − a (∫ Ω | ∇ u | N d x) Δ N u = f (x , u) | x | b + λ h (x) in Ω , u = 0 on ∂ Ω , where Ω is a smooth bounded domain in R N (N ≥ 2) and 0 ≤ b < N. Secondly, we deal with the existence and multiplicity of weak solutions to the Kirchhoff problem: − a (∫ R N | ∇ u | N + V (x) | u | N d x) (Δ N u + V (x) | u | N − 2 u) = g (x , u) | x | b + λ h (x) in R N , where N ≥ 2 and 0 ≤ b < N. We assume that f and g have critical exponential growth at infinity. To establish our existence results, we use the mountain pass theorem, Ekeland variational principle and Moser–Trudinger inequality. [ABSTRACT FROM AUTHOR]

We present a novel approach to solving a specific type of quasilinear boundary value problem with p-Laplacian that can be considered an alternative to the classic approach based on the mountain pass theorem. We introduce a new way of proving the existence of nontrivial weak solutions. We show that the nontrivial solutions of the problem are related to critical points of a certain functional different from the energy functional, and some solutions correspond to its minimum. This idea is new even for p = 2. We present an algorithm based on the introduced theory and apply it to the given problem. The algorithm is illustrated by numerical experiments and compared with the classic approach. [ABSTRACT FROM AUTHOR]

Here, the solvability of the following inclusion elliptic problem − Δ H n , p u ∈ F (ξ , u) in Ω , u = 0 on ∂ Ω , is proved, via variational technique, where Ω is a Korányi ball in the Heisenberg Lie group ℍn and 퓕: Ω × ℝ → 풫(ℝ) is a real set-valued mapping. [ABSTRACT FROM AUTHOR]

In this work, we study a quasilinear elliptic problem in the whole space ℝN$$ {\mathrm{\mathbb{R}}}^N $$ involving the 1‐biharmonic operator with potentials that can vanish at infinity. We consider two different geometrical assumptions in the nonlinearity and use variational methods to obtain nontrivial bounded variation solutions. [ABSTRACT FROM AUTHOR]

In this paper we use variational methods to establish the existence of heteroclinic solution for the prescribed mean curvature equation of the form − ( q ′ 1 + (q ′) 2 ) ′ + a (t) V ′ (q) = 0 in R , where V is a double-well potential with minima at t = ± α and a ∈ L ∞ (R) is an even non-negative function with 0 < a 0 : = inf t ≥ M ⁡ a (t) for some M > 0. [ABSTRACT FROM AUTHOR]

In this paper, we devote to studying the existence of normalized solutions for the following Schrödinger equation with Sobolev critical nonlinearities. { − Δ u = λ u + μ | u | q − 2 u + | u | p − 2 u in R N , ∫ R N | u | 2 d x = a 2 , where N ⩾ 3 , 2 < q < 2 + 4 N , p = 2 ∗ = 2 N N − 2 , a , μ > 0 and λ ∈ R is a Lagrange multiplier. Since the existence result for 2 + 4 N < p < 2 ∗ has been proved, using an approximation method, that is let p → 2 ∗ , we obtain that there exists a mountain-pass type solution for p = 2 ∗ . [ABSTRACT FROM AUTHOR]

In the present manuscript, we focus on a novel tri-nonlocal Kirchhoff problem, which involves the p (x) -fractional Laplacian equations of variable order. The problem is stated as follows: { M (σ p (x , y) (u)) (− Δ) p (⋅) s (⋅) u (x) = λ | u | q (x) − 2 u (∫ Ω 1 q (x) | u | q (x) d x) k 1 + β | u | r (x) − 2 u (∫ Ω 1 r (x) | u | r (x) d x) k 2 in Ω , u = 0 on ∂ Ω , where the nonlocal term is defined as σ p (x , y) (u) = ∫ Ω × Ω 1 p (x , y) | u (x) − u (y) | p (x , y) | x − y | N + s (x , y) p (x , y) d x d y. Here, Ω ⊂ R N represents a bounded smooth domain with at least N ≥ 2 . The function M (s) is given by M (s) = a − b s γ , where a ≥ 0 , b > 0 , and γ > 0 . The parameters k 1 , k 2 , λ and β are real parameters, while the variables p (x) , s (⋅) , q (x) , and r (x) are continuous and can change with respect to x . To tackle this problem, we employ some new methods and variational approaches along with two specific methods, namely the Fountain theorem and the symmetric Mountain Pass theorem. By utilizing these techniques, we establish the existence and multiplicity of solutions for this problem separately in two distinct cases: when a > 0 and when a = 0 . To the best of our knowledge, these results are the first contributions to research on the variable-order p (x) -fractional Laplacian operator. In the present manuscript, we focus on a novel tri-nonlocal Kirchhoff problem, which involves the -fractional Laplacian equations of variable order. The problem is stated as follows: where the nonlocal term is defined as Here, represents a bounded smooth domain with at least . The function is given by , where , , and . The parameters , , and are real parameters, while the variables , , , and are continuous and can change with respect to . To tackle this problem, we employ some new methods and variational approaches along with two specific methods, namely the Fountain theorem and the symmetric Mountain Pass theorem. By utilizing these techniques, we establish the existence and multiplicity of solutions for this problem separately in two distinct cases: when and when . To the best of our knowledge, these results are the first contributions to research on the variable-order -fractional Laplacian operator. [ABSTRACT FROM AUTHOR]