In this paper, we study a class of Kirchhoff-type double phase equations with sublinear terms and two parameters in the framework of variable exponent function spaces. By employing a variant of the critical point theorem due to G. Bonanno, we establish the existence of at least three distinct solutions to the problem. Our result extends and complements recent contributions concerning double phase equations in the superlinear setting.

This paper investigates a class of Kirchhoff-type double phase problems distinguished by the inclusion of two nonlocal source terms and formulated in the framework of variable exponent Sobolev spaces. By applying a variant of Bonanno's critical point theorem [12], we establish the existence of at least three distinct solutions. These results not only enhance the current understanding of double phase equations exhibiting sublinear growth but also extend several recent developments by incorporating variable exponents and nonlocal effects into the analysis.

Abstract Let D ⊂ R N $\mathcal{D}\subset {\mathbb{R}}^{N}$ ( N ≥ 2) be a smooth bounded domain. Consider a class of fourth-order problems involving a p ( x )-Hardy potential of the form Δ a ( x , Δ u ) = μ θ ( x ) | u | p ( x ) − 2 u δ ( x ) 2 p ( x ) + λ f ( x , u ) , in D , u = Δ u = 0 , on ∂ D , $$\begin{cases}\left(a\left(x,{\Delta}u\right)\right)=\mu \theta \left(x\right)\frac{\vert u{\vert }^{p\left(x\right)-2}u}{\delta {\left(x\right)}^{2p\left(x\right)}}+\lambda f\left(x,u\right),\hfill & \quad \text{in} \mathcal{D},\hfill \\ u={\Delta}u=0,\hfill & \quad \text{on} \partial \mathcal{D},\hfill \end{cases}$$ here Δ a ( x , Δ u ) ${\Delta}\left(a\left(x,{\Delta}u\right)\right)$ represents the fourth-order Leray–Lions, δ ( x ) is the Euclidian distance from x to the boundary ∂ D $\partial \mathcal{D}$ , λ and μ are two parameters. Under some suitable conditions on the nonlinear term f , we study the existence and behavior of the solutions of the problem by using a version of Ricceri’s variational principle constructed by Bonanno–Bisci (“Infinitely many solutions for a boundary value problem with discontinuous nonlinearities,” Boundary Value Probl. , vol. 2009, 2009, Art. no. 670675).

This paper deals with a class of fourth-order elliptic equations of p(x)-Kirchhoff type of the form {Delta(2)(p(x))u-M(integral(Omega)1/p(x)divided by del u divided by p(x)dx)Delta(p)(x)u=lambda W(x)divided by divided by u divided by divided by(q(x)-2)u in Omega, u=Delta u=0 on partial derivative Omega, where Omega subset of & Ropf;(N) (N >= 2) is a smooth bounded domain with boundary partial derivative Omega,p:Omega -> R is a log-Holder continuous function, M(t) = a + bt(kappa) is a Kirchhoff function with a,kappa>0,b >= 0,Delta 2p(x)u= Delta(|Delta u|(p)((x)-2)Delta u) is the operator of fourth order called the p(x)-biharmonic operator, Delta(p)((x))u = div (|del u|(p)((x)-2)del u) is the p(x)-Laplacian, W : Omega -> & Ropf; is a weighted function and lambda is a positive parameter. Using variational techniques, we establish some multiplicity results to the problem in two cases when the function W is sign-changing or not and two examples are given to illustrate the main results

In this paper, we consider a class of fourth-order elliptic equations of Kirchhoff type with variable exponents {Delta(2)(p(x)) u-M(integral(Omega)1/p(x)vertical bar del u vertical bar(p(x)) dx) Delta(p(x))u +vertical bar u vertical bar(p(x)-2)u = lambda f(x,u) in Omega, u = Delta u = 0 on partial derivative Omega, where p(-) := inf(x is an element of(Omega) over bar)p(x) > max{1,N/2}, lambda is a positive parameter, Omega subset of R-N (N >= 1) is a smooth bounded domain, Delta(2)(p(x))u = Delta(vertical bar Delta u vertical bar(p(x)-2)Delta u) is the operator of fourth order called the p(x)-biharmonic operator, Delta(p(x)) u= div(vertical bar del u vertical bar(p(x)-2)del u) is the p(x)-Laplacian, p : (Omega) over bar -> R is a log-Holder continuous function, M:[0, +infinity)-> R is a continuous function and f: (Omega) over bar x R -> R is an L-1-Caratheodory function satisfying some certain conditions. Using variational methods and critical point theory, we prove some existence and multiplicity results for the problem in an appropriate space of functions. Furthermore, we provide two examples to illustrate our main conclusions.

In this paper, we consider a class of p(x)-Laplacian like problems with indefinite weight involving no flux boundary condition. Using variational techniques and the critical point theorem of Bonanno and Marano [4], we prove the existence of at least three weak solutions to the problem in Sobolev variable exponent spaces.

The purpose of the present paper is to study the existence of solutions for the following nonhomogeneous singular problem involving the fractional p(x,.)-Laplace operator {(-Delta)(p(x,.))(s) u + vertical bar u vertical bar(q(x)-2)u = g(x)u(epsilon-1-.gamma(x)) -/+ lambda f (x, u) in Omega, u = 0 on partial derivative Omega, where Omega is a smooth bounded domain in R-N (N >= 3), 0 < s, epsilon < 1, lambda is a positive parameter and gamma : (Omega) over bar -> (0, epsilon) is a continuous function, p (Omega) over bar x (Omega) over bar -> (1, infinity) is a bounded, continuous and symmetric function, q : (Omega) over bar (1, infinity) is a continuous function, g is an element of L p(s)*(x)-epsilon/p(s)* (x)+ gamma(x)-2 epsilon (Omega) and g(x) > 0 with p(s) * (x) = Np(x,x)/N-sp(x,x). Here, the nonlinearity f is in C1((Omega) over bar x R) and assumed to satisfy suitable assumptions. Using variational methods combined with monotonicity arguments, we obtain the existence of solutions to the problem in a fractional Sobolev space with variable exponent. To our best knowledge, this paper is the first attempt in the study of singular problems involving fractional p(x,.)-Laplace operators.

In this paper, we study the p-biharmonic equation of Kirchhoff type {delta(2)(p)u - ( a + b integral (N )(R)| & nabla; u | (p)dx ) delta(u )(p)+ V (x) | u |(p-2)u = K (x) f (u) + lambda g (x)|u|(q-2)u, x in R-N; u in W-2,W-p (R-N) & cap; W-0(1,p )(R-N). where N >= 5, 1 < q < p < (2)/(N), a > 0, b >= 0, lambda is a positive parameter, delta(p)u = div( |& nabla; u | (p-2)& nabla;u ) is the p-Laplacian operator and delta(2)(p)u = delta( |delta u|( p-2 )delta u) is the p-biharmonic operator, V, K, g are nonnegative functions, V is vanishing at infinity in the sense that lim (|x|->+infinity) V (x) = 0. When the nonlinear term f(u)f(u) satisfies some suitable conditions, we prove that the above problem has at least two nontrivial solutions using the mountain pass theorem combined with the Ekeland variational principle.

This paper deals with the following singular system: {(-Delta)(p)(s)u + (-Delta)(q)(s) u = lambda f(x) vertical bar u vertical bar(r-2)u + 1-alpha/2-alpha-beta h(x) vertical bar u vertical bar(-alpha) vertical bar v vertical bar(1-beta) in Omega, (-Delta)(p)(s)v + (-Delta)(q)(s) v = mu g(x) vertical bar-x vertical bar(r-2)v + 1-beta/2-alpha-beta h(x) vertical bar u vertical bar(1-alpha) vertical bar v vertical bar(1-beta) in Omega, u=v=0 in R-N \ Omega. where Omega subset of R-N is a bounded smooth domain, lambda, mu are positive parameters, s is an element of(0, 1), 1 < p < N/s, 0 < alpha, beta < 1, 2 - alpha - beta < q < p < r < p(s)* = Np /(N - sp), and (-Delta)sigma(s)u denotes the fractional sigma-Laplacian, sigma =p, q. Under appropriate conditions on the weight functions f,g, h which may change sign in Omega, we establish the existence of multiple solutions by using the Nehari manifold method. Our paper is one of the first attempts to study the existence of solutions for fractional singular systems involving sign-changing weight functions.

In this paper, we are interested in a class of bi-nonlocal problems with nonlinear Neumann boundary conditions and sublinear terms at infinity. Using $(S_+)$ mapping theory and variational methods, we establish the existence of at least two non-trivial weak solutions for the problem provied that the parameters are large enough. Our result complements and improves some previous ones for the superlinear case when the Ambrosetti-Rabinowitz type conditions are imposed on the nonlinearities.

This article considers p(x)-Kirchhoff type problems with Robin boundary conditions. Using the mountain pass theorem, the Ekeland's variational principle, and Krasnoselskii's genus theory, we prove that the problem has at least two nontrivial weak solutions or infinitely many nontrivial weak solutions under some suitable conditions on the nonlinearities. The main results improve and generalize the previous ones introduced in [2,7].

We establish a concentration-compactness principle for the Sobolev space W-2,W-p(.)(Omega) boolean AND W-0(1,p(.)) (Omega) that is a tool for overcoming the lack of compactness of the critical Sobolev imbedding. Using this result, we obtain several existence and multiplicity results for a class of Kirchhoff type problems involving p(.)-biharmonic operator and critical growth.

This paper deals with the following singular problem: \begin{align*} \begin{cases} (-\Delta)^s_p u+ \mu(-\Delta)^s_q u =\frac{a(x)}{ u^\gamma} +\lambda f(x,u) & in \Omega, u = 0,& in \mathbb{R}^N\setminus\Omega, \end{cases} \end{align*} where $\Omega\subset\mathbb{R}^N$ ($N\geq 3$) are a bounded smooth domain, $f\in C(\Omega\times \mathbb{R}, \mathbb{R})$ is positively homogeneous of degree $r-1$, $a\in L^\infty(\Omega)$, $a(x)>0$ for almost every $x\in \Omega$, $\lambda$, $\mu >0$, $s\in(0,1)$, $N> ps$, and $0<\gamma<1<q<p<r<p^*_s$. Under appropriate conditions on the function $f$, we establish the existence of multiple solutions by using the Nehari manifold method.

This paper is concerned with the existence and multiplicity of nontrivial solutions for a class of Kirchhoff-type systems in R-N involving the fractional pseudo-differential operators defined as the generalizations of the p(x)-Laplace operator. Our main tools come from a direct variational methods, the Mountain Pass Theorem, the symmetric Mountain Pass Theorem and the Fountain Theorem in critical point theory. The obtained results of this note significantly contribute to the study of Kirchhoff-type systems in the sense that our situation covers not only differential operators of fractional order but also nonhomogeneous differential operators in Sobolev spaces with variable exponent.

. This paper is concerned with the existence and multiplicity of nontrivial solutions for a class of Kirchhoff-type systems in R N involving the fractional pseudo-differential operators defined as the generalizations of the p ( x )-Laplace operator. Our main tools come from a direct variational methods, the Mountain Pass Theorem, the symmetric Mountain Pass Theorem and the Fountain Theorem in critical point theory. The obtained results of this note significantly contribute to the study of Kirchhoff-type systems in the sense that our situation covers not only differential operators of fractional order but also nonhomogeneous differential operators in Sobolev spaces with variable exponent.

In this paper, we study the effect of Hardy potential on the existence or non-existence of solutions to a fractional Laplacian problem involving a singular nonlinearity. Also, we mention a stability result.

Abstract In this paper, we prove the existence of multiple solutions for the following sixth-order p(x)-Kirchhoff-type problem −M∫Ω1p(x)|∇Δu|p(x)dxΔp(x)3u=λf(x)|u|q(x)−2u+g(x)|u|r(x)−2u+h(x)inΩ,u=Δu=Δ2u=0,on∂Ω, $$\begin{array}{} \displaystyle \begin{cases} -M\left( \int\limits_{\it\Omega} \frac{1}{p(x)}|\nabla {\it\Delta} u|^{p(x)}dx\right){\it\Delta}^3_{p(x)} u = \lambda f(x)|u|^{q(x)-2}u + g(x)|u|^{r(x)-2}u + h(x) &\mbox{in}\quad {\it\Omega}, \\[0.3em] u = {\it\Delta} u = {\it\Delta}^2 u = 0, \quad &\mbox{on}\quad \partial{\it\Omega}, \end{cases} \end{array}$$ where Ω ⊂ ℝN is a smooth bounded domain, N>3,Δp(x)3u:=div⁡(Δ(|∇Δu|p(x)−2∇Δu)) $\begin{array}{} N \,\,\gt\,\, 3, {\it\Delta}_{p(x)}^3u\,\, : =\,\, \operatorname{div}\Big({\it\Delta}(|\nabla {\it\Delta} u|^{p(x)-2}\nabla {\it\Delta} u)\Big) \end{array}$ is the p(x)-triharmonic operator, p, q, r ∈ C(Ω), 1 < p(x) < N3 $\begin{array}{} \displaystyle \frac N3 \end{array}$ for all x ∈ Ω, M(s) = a − bsγ, a, b,γ > 0, λ > 0, g : Ω × ℝ → ℝ is a nonnegative continuous function while f, h : Ω × ℝ → ℝ are sign-changing continuous functions in Ω. To the best of our knowledge, this paper is one of the first contributions to the study of the sixth-order p(x)-Kirchhoff type problems with sign changing Kirchhoff functions.

In this paper, we study the existence of a nontrival weak solution for a class of Kirchhoff type problems with singular potentials and critical exponents. The proofs are essentially based on an appropriated truncated argument, Caffarelli-Kohn-Nirenberg inequalities, combined with a variant of the concentration compactness principle. We also get a priori estimates of the obtained solution

In this paper, we investigate the existence and multiplicity of solutions for a class of fractional (p(1) (x, .), p(2) (x, .))-Kirchhoff type problems with Dirichlet boundary data of the following form (P-Mi(a)){Sigma M-2(i=)1(i) (integral(Q)1/p(i)(x, y) vertical bar u(x) - u(y)vertical bar(pi(x,y))/vertical bar x - y vertical bar(N+spi(x, y)) dxdy) (-Delta)(pi(x, .))(s) u(x) +Sigma(2)(i=1)vertical bar u vertical bar((p) over bari(x)-2) u = f(x, u) in Omega, u = 0 in R-N\Omega. More precisely, by means of mountain pass theorem with Cerami condition, we show that the above problem has at least one nontrivial solution. Moreover, using Fountain theorem, we prove that (P-Mi(s)) possesses infinitely many (pairs) of solutions with unbounded energy.