Using the Nehari manifold, we establish the existence of two non-negative weak solutions for a fractional type problem driven by a non-local operator of the elliptic type in fractional Orlicz-Sobolev spaces. We show how the existence of solutions depends on the properties of the Nehari manifold. Moreover, under some suitable assumptions, continuous and compact embeddings results are established.

In this paper, using the three critical points theorem we obtain the existence of three weak solutions for a Kirchhoff type problem driven by a nonlocal operator of the elliptic type in a fractional Orlicz-Sobolev space, with homogeneous Dirichlet boundary conditions.

In this paper, we develop some properties of the ax,y(.)-Neumann derivative for the fractional ax,y(.)-Laplacian operator. Therefore we prove the basic proprieties of the correspondent function spaces. In the second part of this paper, by means of Ekeland’s variational principal and direct variational approach, we prove the existence of weak solutions for a nonlocal problem with nonhomogeneous Neumann and Robin boundary condition.

This paper is concerned with a class of fractional p -Laplace type problems with Dirichlet boundary data of the following form (P_s) {[ M(||u||^p)( (-Δ )^s_p u +|u|^p-2u) = λ f(x,u) in Ω; ; u = 0 in ℝ^N∖Ω . ]. By means of Ekeland’s variational principle and a direct variational approach, we investigate the existence of nontrivial weak solution for the above problem.

In this paper, we are concerned with some qualitative properties of the new fractional Musielak-Sobolev spaces (WL)-L-s phi(x, y) such that the generalized Poincare type inequality and some continuous and compact embedding results. Moreover, we prove that any function in (WL)-L-s phi(x, y) (Omega) may be extended to a function in (WL)-L-s phi(x, y) (R-N), with Omega subset of R-N is a bounded domain of class C-0,C-1. In addition, we establish a result that relates to the complemented subspace in (WL)-L-s phi(x, y) (R-N). As an application, using the mountain pass theorem and some variational methods, we investigate the existence of a nontrivial weak solution for a class of nonlocal fractional type problems with Dirichlet boundary data.

In this research we prove an existence result of renormalized solutions in Musielak-Orlicz spaces for a class of nonlinear elliptic equations with two lower order terms and L^1 -data. We assume that the Musielak function satisfies only the log-Hölder continuity condition, therefore we work with Musielak-Orlicz spaces which are not necessarily reflexive.

Let Omega be a bounded open subset of R-N, N >= 2. In this paper we give an existence result of bounded solution, in Musielak spaces, for unilateral problems associated to the nonlinear elliptic equation -div a(x, u, del u) + g (x, u, del u) = f in Omega, where the nonlinearity g does not satisfy the well known sign condition and f is an integrable source.

We study the existence of solutions for some nonlinear elliptic problems of the type $$-{\text {div}}(b(x, u, \nabla u)+F(x,u))=\nu$$ in $$\Omega ,$$ in the setting of Musielak–Orlicz spaces. The lower order term F verifies the natural growth condition, no $$\Delta _{2}$$ -condition is assumed on the Musielak function, and the datum $$\nu$$ is assumed to belong to $$L^{1}(\Omega )+W^{-1} E_{\psi }(\Omega )$$ .

We are interested in the multiplicity of weak solutions for a binonlocal fractional p (x, .)-Kirchhoff type problems. Our technical approach is based on the general three critical points theorem obtained by B. Ricceri.

In the present paper, we study the existence and regularity of positive solutions for the following boundary value problem : $\mathrm{-div}\> \big( \lvert\nabla u\rvert^{p-2}\nabla u ) + u^{s} = \dfrac{f}{u^{\alpha}}\mbox{ in } \Omega \mbox{ and } u=0\mbox{ on } \partial\Omega,$ where $ \Omega $ is an open and bounded subset of $ \mathbb{R}^{N} $ $ (N> p>1) $, $ 0<\alpha\leq 1 $, $ s\geq 1 $ and $f$ is a nonnegative function that belongs to some Lebesgue space.

In this paper, we prove the existence result of solutions for the nonlinear unilateral problem associated to the parabolic equation partial differential u partial differential t - div a(x, t, u, Vu) - div phi(x, t, u) = mu in QT = omega x (0, T), where the lower order term phi satisfies a generalized natural growth condi-tion described by the appropriate Orlicz function psi, and the data mu is an integrable source term. No growth restrictions are assumed either on psi or on its complementary psi. Therefore the solution is natural in this context.

ABSTRACT In this paper, we investigate the existence theorem of entropy solutions for nonlinear elliptic problem of the type in Ω, in the setting of Musielak-Orlicz spaces. The lower order term Φ verifies the natural growth condition, the nonlinearity g has a natural growth with respect to its third argument and with sign condition, no -condition is assumed on the Musielak function, and the datum μ is assumed to belong to .

In this paper, we first introduce the new fractional Musielak-Sobolev spaces, and we establish some qualitative properties of these spaces. Then, using the direct variational approach, we investigate the existence of a nontrivial weak solution for a class of fractional type problems with Dirichlet boundary data of the following form (P-a){(-Delta)(a(x,.))(s)u+(a) over cap (x)(vertical bar u vertical bar)u=f(x,u) in omega, u=0 in R-N\Omega, where s is an element of(0,1), omega is an open bounded subset in R-N with Lipschitz boundary delta Omega, and f: Omega x R -> R is a Caratheodory function with suitable growth conditions.

In the present paper, we discuss the existence of bounded weak solutions for the degenerate nonlinear elliptic problem $$\begin{aligned} \text{ div }(\Gamma (x,u,\nabla u))+\kappa (x,u,\nabla u)=f, \end{aligned}$$ where $$\text{ div }(\Gamma (x,u,\nabla u))$$ is a degenerate Leray−Lions operator and defined on non-reflexive Musielak space $$D(A)\subset W_{0}^{1}L_{\varphi }(\Omega )$$ , such that $$\varphi$$ is a Musielak function. The lower term $$\kappa$$ satisfies only the growth condition and no sign condition is assumed on it. The source data f are in $$L^{N}(\Omega )$$ .

With only the log-Hölder continuity on the modular function, we prove in non-reflexive Musielak spaces an existence result of solutions for a strongly nonlinear obstacle problem associated to the elliptic equation -div𝒜(x,u,∇ u)-divΦ (x,u)= f in Ω, where the lower order term Φ is a non-coercive Carathéodory function satisfying a generalized natural growth condition described by the appropriate Musielak function φ and f is an integrable datum. We do not assume any growth restrictions neither on φ nor on its complementary φ .

In this paper, we establish the existence of two positive constants $$\lambda _0$$ and $$\lambda _1$$ with $$\lambda _0\leqslant \lambda _1$$ , such that any $$\lambda \in [\lambda _1, \infty )$$ is an eigenvalue, while any $$\lambda \in (0,\lambda _0)$$ is not an eigenvalue, for a Kirchhoff type problem driven by nonlocal operators of elliptic type in a fractional Orlicz-Sobolev space, with Dirichlet boundary conditions.

This paper is concerned with a class of fractional -Kirchhoff type problems with Dirichlet boundary data of the following form By means of mountain pass theorem of Ambrosetti and Rabinowitz, direct variational approach and Ekeland's variational principle, we investigate the existence of nontrivial weak solutions for the above problem in different cases of the competition between the growth rates of functions p and r involved in problem , this fact is essential in describing the set of eigenvalues of this problem.

In this article, we prove an existence theorem of renormalized solutions for nonlinear elliptic problem of the type -div𝒜(x,u,∇ u)-div (x,u)+ℋ(x,u,∇ u)= f in , where the first lower-order term satisfies only a generalized natural growth condition without any supplementary assumptions. The approach does not require any particular type of growth condition on .

In this paper , we investigate an existence result of solutions for the nonlinear elliptic unilateral problem $$\begin{aligned} \left\{ \begin{array}{llll} u\ge \psi \text { a.e. in }\Omega , T_k(u)\in W_{0}^{1}L_{M}(\Omega ),\\ \displaystyle \int _{\Omega }{\mathcal {A}}(x,u,\nabla u)\nabla T_k(u-\varphi )\>\mathrm{d}x+\displaystyle \int _{\Omega }\Phi (x,u)\nabla T_k(u-\varphi )\>\mathrm{d}x\\ \le \displaystyle \int _{\Omega }fT_k(u-\varphi )\>\mathrm{d}x,\\ \quad \forall \varphi \in L^{\infty }(\Omega )\cap \Big ({\mathbf {K}}_{\psi }= \Big \{u\in W_{0}^{1}L_{M}(\Omega ):u\ge \psi \text{ a.e. } \text{ in } \Omega \Big \}\Big ), \end{array}\right. \end{aligned}$$ where the lower-order term $$\Phi $$ verifies a generalized natural growth condition described by a suitable N-function M and the data f is an element of $$L^1(\Omega )$$ . No restriction is assumed neither on M nor on its conjugate $${\overline{M}}$$ .

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.