Existence Of Nontrivial Weak Solutions Research Articles

Singular Hamiltonian elliptic systems involving double exponential growth in dimension two

In this research, we are interested to investigate the existence of nontrivial weak solutions to the following Hamiltonian elliptic system −div(ω(x)∇u)=g(v)|x|a,x∈B1(0),−div(ω(x)∇v)=f(u)|x|b,x∈B1(0),with Dirichlet boundary conditions, where a,b∈[0,2), the weight ω(x) is of logarithmic type and the nonlinearities f and g possess double exponential growth. To establish the existence of solutions, our approach involves utilizing the linking theorem and a finite-dimensional approximation.

On a class of fractional p( x, y)−Kirchhoff type problems with indefinite weight

This paper is concerned with a class of fractional \(p(x,y)-\)Kirchhoff type problems with Dirichlet boundary data along with indefinite weight of the following form \begin{equation*} \left\lbrace\begin{array}{ll} M\left(\int_{Q}\frac{1}{p(x,y)}\frac{|u(x)-u(y)|^{p(x,y)}}{|x-y|^{N+sp(x,y)}}\,dx\,dy\right)\\ (-\triangle_{p(x)})^s+|u(x)|^{q(x)-2}u(x) & \\ =\lambda V(x)|u(x)|^{r(x)-2}u(x)& \text{in }\Omega,\\ u=0, & \text{in }\mathbb{R}^N\Omega. \end{array}\right. \end{equation*} By means of direct variational approach and Ekeland’s variational principle, we investigate the existence of nontrivial weak solutions for the above problem in case of the competition between the growth rates of functions \(p\) and \(r\) involved in above problem, this fact is essential in describing the set of eigenvalues of this problem.

Existence of two non-zero weak solutions for a $p(\cdot)$-biharmonic problem with Navier boundary conditions

In this paper, the existence of non-trivial weak solutions for some problems with Navier boundary conditions driven by the p(\cdot) -biharmonic operator is investigated. The proofs combine variational methods with topological arguments.

A Critical p(x)-Laplacian Steklov Type Problem with Weights

ABSTRACT This paper is concerned with the existence of nontrivial weak solutions for a class of p(x)-Laplacian problem under Steklov boundary condition. The variable exponent theory of generalized Lebesgue-Sobolev spaces and the concentration-compactness principle for weighted variable exponent spaces are used for this purpose.

Nontrivial weak solutions for a (p,q)-Laplacian equation involving discontinuities with critical exponent

We review the existence of nontrivial weak solutions for the following problem{−Δpu−Δqu=f(u)+λH(u−β)uq⁎−1inΩ,u=0 on ∂Ω where Ω⊂RN is a smooth bounded domain, q⁎ is the critical Sobolev exponent, f:R⟶R is a discontinuous function that does not satisfy necessarily the Ambrosetti-Rabinowitz condition and H is the Heaviside function. Applying convenable critical point theory for nondifferentiable functionals, we prove that such problem has at least one nontrivial solution for any parameters λ>0,β>0. We also establish other results if the nonlinearity satisfies the Ambrosetti-Rabinowitz condition.

Existence of Nontrivial Weak Solutions for Some Discrete p(.)-Laplacian Boundary Value Problems in n-Dimensional

Existence of Nontrivial Weak Solutions for Some Discrete p(.)-Laplacian Boundary Value Problems in n-Dimensional

Existence of nontrivial weak solutions for nonuniformly elliptic equation with mixed boundary condition in a variable exponent Sobolev space

In In this paper, we consider a mixed boundary value problem for nonuniformly elliptic equation in a variable exponent Sobolev space containing p ( ⋅ ) -Laplacian and mean curvature operator. More precisely, we are concerned with the problem with the Dirichlet condition on a part of the boundary and the Steklov boundary condition on an another part of the boundary. We show the existence of a nontrivial weak solution and at least two nontrivial weak solutions according to some hypotheses on given functions.

Nonlinear Liouville-type theorems for generalized Baouendi-Grushin operator on Riemannian manifolds

We are concerned with differential inequalities of the form $$ -[d_{M_2}(y_0,y)]^{2\rho_2}\Delta_{M_1} u-[d_{M_1}(x_0,x)]^{2\rho_1}\Delta_{M_2}u\geq V |u|^p\mbox { in } M_1\times M_2, $$ where $M_i$ ($i=1,2$) are complete noncompact Riemannian manifolds, $(x_0,y_0)\in M_1\times M_2$ is fixed, $d_{M_1}(x_0,\cdot)$ is the distance function on $M_1$, $d_{M_2}(y_0,\cdot)$ is the distance function on $M_2$, $\Delta_{M_i}$ is the Laplace-Beltrami operator on $M_i$, $V=V(x,y) > 0$ is a measurable function, and $p>1$. Namely, we establish necessary conditions for existence of nontrivial weak solutions to the considered problem. The obtained conditions depend on the parameters of the problem as well as the geometry of the manifolds $M_i$. Next, we discuss some special cases of potential functions $V$. The proof of our main result is based on the nonlinear capacity method and a result due to Bianchi and Setti (2018) about the construction of cut-off functions with controlled gradient and Laplacian, under certain assumptions on the Ricci curvatures of the manifolds.

Existence of mountain-pass solutions for p(・)-biharmonic equations with Rellich-type term

This manuscript discusses the existence of nontrivial weak solution for the following nonlinear eigenvalue problem driven by the p(?)-biharmonic operator with Rellich-type term {?(|?u|p(x)?2?u) = ?|u|q(x)?2u/?(x)2q(x), for x ? ?, u = ?u = 0, for x ? ??. Considering the case 1 < min x?? p(x) ? max x?? p(x) < min x?? q(x) ? max x?? q(x) < min (N 2, Np(x) N ? 2p(x) ), we extend the corresponding result of the reference [8], for the case 1 < min x?? q(x) ? max x?? q(x) < min x?? p(x) ? max x?? .p(x) < N 2 . The proofs of the main results are based on the mountain pass theorem

The Existence of Solutions for Local Dirichlet (r(u),s(u))-Problems

In this paper, we consider local Dirichlet problems driven by the (r(u),s(u))-Laplacian operator in the principal part. We prove the existence of nontrivial weak solutions in the case where the variable exponents r,s are real continuous functions and we have dependence on the solution u. The main contributions of this article are obtained in respect of: (i) Carathéodory nonlinearity satisfying standard regularity and polynomial growth assumptions, where in this case, we use geometrical and compactness conditions to establish the existence of the solution to a regularized problem via variational methods and the critical point theory; and (ii) Sobolev nonlinearity, somehow related to the space structure. In this case, we use a priori estimates and asymptotic analysis of regularized auxiliary problems to establish the existence and uniqueness theorems via a fixed-point argument.

Existence of Solutions for Superlinear Second-Order System with Noninstantaneous Impulses

Variational methods are used in order to establish the existence of nontrivial weak solution for superlinear second-order system with noninstantaneous impulses. The main result is obtained when a kind of definition of the weak solution for this system is introduced. Meanwhile, an example is presented to illustrate the main result.

A mountain pass algorithm for quasilinear boundary value problem with [formula omitted]-Laplacian

In this paper, we deal with a specific type of quasilinear boundary value problem with Dirichlet boundary conditions and with p-Laplacian. We show two ways of proving the existence of nontrivial weak solutions. The first one uses the mountain pass theorem, the other one is based on our new minimax theorem. This method is novel even for p=2. In the paper, we also present a numerical algorithm based on the introduced approach. The suggested algorithm is illustrated on numerical examples and compared with a current approach to demonstrate its efficiency.

Biharmonic equations with totally characteristic degeneracy

In this paper, we study the biharmonic operator on a manifold with conical singularities and consider the existence of non-trivial weak solution for the corresponding semilinear degenerate elliptic equation with critical cone Sobolev exponents by the cone Sobolev inequality and Poincaré inequality. Furthermore, we show an interesting integral identity, which can be used to discuss the non-existence results for some biharmonic equations under Navier boundary conditions.

Nonlocal eigenvalue type problem in fractional Orlicz-Sobolev space

This paper is concerned with a class of fractional a-Laplace type problem with Dirichlet boundary data of the following form $$\begin{aligned} (P_a) \left\{ \begin{array}{clclc} (-\varDelta )^s_a u +a(|u|)u = \lambda f(x,u) &{} \text { in }&{} \varOmega , \\ u = 0 &{} \text { in } &{} {\mathbb {R}} ^N\setminus \varOmega . \end{array} \right. \end{aligned}$$ By means of Ekeland’s variational principal and direct variational approach, we investigate the existence of nontrivial weak solutions for the above problem.

A class of elliptic equations involving nonlocal integrodifferential operators with sign-changing weight functions

In this paper, we investigate the existence of nontrivial weak solutions to a class of elliptic equations involving a general nonlocal integrodifferential operator LAK, two real parameters, and two weight functions, which can be sign-changing. Considering different situations concerning the growth of the nonlinearities involved in the problem (P), we prove the existence of two nontrivial distinct solutions and the existence of a continuous family of eigenvalues. The proofs of the main results are based on ground state solutions using the Nehari method, Ekeland’s variational principle, and the direct method of the calculus of variations. The difficulties arise from the fact that the operator LAK is nonhomogeneous and the nonlinear term is undefined.

A problem of capillarity under Neumann condition

We formulate a Neumann variational problem, which characterizes the capillary phenomena, and disLaplacian‐like operator defined bycuss the existence of nontrivial weak solutions. We get the results using sufficient hypotheses on the reaction term.

Combined effects for non-autonomous singular biharmonic problems

We study the existence of nontrivial weak solutions for a class of generalized $p(x)$-biharmonic equations with singular nonlinearity and Navier boundary condition. The proofs combine variational and topological arguments. The approach developed in this paper allows for the treatment of several classes of singular biharmonic problems with variable growth arising in applied sciences, including the capillarity equation and the mean curvature problem.

EXISTENCE OF SOLUTIONS FOR A FRACTIONAL ADVECTION-DISPERSION EQUATION WITH IMPULSIVE EFFECTS VIA VARIATIONAL APPROACH

In this paper, based on the variational approach and iterative technique, the existence of nontrivial weak solutions is derived for a fractional advection-dispersion equation with impulsive effects, and the nonlinear term of fractional advection-dispersion equation contain the fractional order derivative. In addition, an example is presented as an application of the main result.

On a class of fractional p(x) -Kirchhoff type problems

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.

On a Nonlocal Fractional p(., .)-Laplacian Problem with Competing Nonlinearities

The aim of this paper is to study the existence of nontrivial weak solutions for the problem $$\begin{aligned} \left\{ \begin{array}{ll} M\left( \int _{\Omega \times \Omega }\frac{|u(x)-u(y)|^{p(x,y)}}{p(x,y)|x-y|^{N+p(x,y)s}}dxdy\right) (\Delta )^s_{p(x,.)}u(x)\\ \quad = \lambda f(x,u) - |u(x)|^{q(x)-2}u(x)\quad \hbox {in}~\Omega , \\ u = 0\quad \hbox {in}~\partial \Omega , \end{array} \right. \end{aligned}$$ where $$\Omega \subset \mathbb R^N$$ , $$N\ge 2$$ is a bounded smooth domain, M and f are two continuous functions and $$(\Delta )^s_{p(.,.)}$$ is the fractional p(., .)-Laplacian while $$\lambda $$ is a positive parameter and $$0<s<1$$ . Using variational techniques combined with the theory of the generalized Lebesgue Sobolev spaces, we prove some existence and multiplicity results for the problem in an appropriate space of functions.