Nontrivial Weak Solutions Research Articles

Analysis of a nonlinear elliptic system with variable coefficients and Hardy potential with Carathéodory nonlinearity: Existence

Abstract This study investigates existence result of a nontrivial weak solutions for a system of nonlinear elliptic equations involving Hardy Potentials. The problem arises in the mathematical modeling of complex physical phenomena.

Kirchhoff problems with logarithmic double phase operator: Existence and multiplicity results

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.

A critical Kirchhoff equation with a logarithmic type perturbation in dimension 4

In this paper, a critical Kirchhoff type elliptic equation involving a logarithmic type perturbation is considered in a bounded domain in . Three main features of the problem bring essential difficulties when proving the existence of weak solutions. The first one is the nonlocal term which makes the structure of the corresponding energy functional more complicated, the second one is that the problem is critical in the sense that the energy functional lacks compactness, and the third one is the appearance of the logarithmic term which satisfies neither the standard monotonicity condition nor the Ambrosetti–Rabinowitz condition. Moreover, the boundedness of the sequence is hard to obtain for Kirchhoff problems in dimension 4. By combining a result by Jeanjean (Proc. Roy. Soc. Edinburgh Sect. A, 1999, 129: 787–809) and a recent estimate by Deng et al. (Adv. Nonlinear Stud., 2023, 23: No. 20220049) with the mountain pass lemma and Brézis‐Lieb's lemma, it is proved that either the norm of the sequence of approximation solution goes to infinity or the problem admits a nontrivial weak solution, under appropriate assumptions on the parameters.

Quasilinear elliptic problem in anisotropic Orlicz–Sobolev space on unbounded domain

AbstractWe study a quasilinear elliptic problem $$-\text {div} (\nabla \Phi (\nabla u))+V(x)N'(u)=f(u)$$ - div ( ∇ Φ ( ∇ u ) ) + V ( x ) N ′ ( u ) = f ( u ) with anisotropic convex function $$\Phi $$ Φ on the whole $$\mathbb {R}^n$$ R n . To prove existence of a nontrivial weak solution we use the mountain pass theorem for a functional defined on anisotropic Orlicz–Sobolev space $${{{\,\mathrm{\textbf{W}}\,}}^1}{{\,\mathrm{\textbf{L}}\,}}^{{\Phi }} (\mathbb {R}^n)$$ W 1 L Φ ( R n ) . As the domain is unbounded we need to use Lions type lemma formulated for Young functions. Our assumptions broaden the class of considered functions $$\Phi $$ Φ so our result generalizes earlier analogous results proved in isotropic setting.

A multiplicity result for critical elliptic problems involving differences of local and nonlocal operators

We study some critical elliptic problems involving the difference of two nonlocal operators, or the difference of a local operator and a nonlocal operator. The main result is the existence of two nontrivial weak solutions, one with negative energy and the other with positive energy, for all sufficiently small values of a parameter. The proof is based on an abstract result recently obtained in \cite{MR4293883}.

Robin problem involving the $p(x)$-Laplacian operator without Ambrosetti-Rabinowizt condition

The paper deals with the following Robin problem$$\left\lbrace\begin{aligned}- \mathcal{M} \left( \int _{\Omega} \frac{1}{p(x)} \vert \nabla u \vert ^{p(x)} dx + \int _{\partial \Omega } \frac{a(x)}{p(x)} \vert \nabla u \vert ^{p(x)} d \sigma \right) \mathop{\rm div} (\vert \nabla u \vert ^{p(x)-2} \nabla u) = \lambda h(x,u) \ \ \text{ in } \Omega,\\\vert \nabla u \vert ^{p(x)-2} \frac{\partial u}{\partial \nu} + a(x) \vert u \vert ^{p(x)-2} u &=0 \quad \quad \quad \ \text{ on } \partial \Omega .\end{aligned}\right.$$The goal is to determine the precise positive interval of $\lambda $ for which the problem admits at least two nontrivial solutions via variational approach for the above problem without assuming the Ambrosetti-Rabinowitz condition. Next, we give a result on the existence of an unbounded sequence of nontrivial weak solutions by employing the fountain theoreom with Cerami condition.

On a p(x)-biharmonic singular problem with changing sign weight and with no-flux boundary condition

In the present paper, we study $p(x)-$biharmonic problem involving $q(x)-$Hardy type potential with no-flux boundary condition. By using the mountain pass type theorem and Ekeland variatoinal principle, we obtain at least two nontrivial weak solutions.

Nonlinear Eigenvalue Problems for (p, 2)-Equations and Laplace Equations with Perturbations

We consider two types of nonlinear eigenvalue problems involving Laplace and p-Laplace operators (p>2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(p>2)$$\\end{document}. The main result establishes the existence of at least two nontrivial weak solutions in the case of the perturbed equation and the existence of a continuous spectrum in the case of the (p, 2)-equation. In both cases, variational methods play a central role in our arguments.

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.

On the Pohozaev identity for the fractional p$p$‐Laplacian operator in RN$\mathbb {R}^N$

AbstractIn this paper, we show the existence of a nontrivial weak solution for a nonlinear problem involving the fractional ‐Laplacian operator and a Berestycki–Lions type nonlinearity. This solution satisfies a Pohozaev identity. Moreover, we prove that any sufficiently smooth solution fulfills the Pohozaev identity.

On a fractional (p,q)-Laplacian equation with critical nonlinearities

In this paper, we consider the following fractional ( p , q ) -Laplacian equations with critical Hardy-Sobolev exponents { ( − Δ ) p s 1 u + ( − Δ ) q s 2 u = λ | u | r − 2 u + μ | u | p s 1 ∗ ( α ) − 2 u | x | α in Ω , u = 0 in R N ∖Ω , where 0 < s 2 < s 1 < 1 < q < r ≤ p < N s 1 , 0 $ ]]> λ , μ > 0 are two parameters, 0 ≤ α < p s 1 and p s 1 ∗ ( α ) = p ( N − α ) N − p s 1 is the fractional Hardy-Sobolev critical exponent, Ω ⊂ R N is an open bounded domain with smooth boundary. By using variational methods, we show that the problem has a nontrivial nonnegative weak solution.

Nonlocal τ(m)‐Laplacian‐like problem with logarithmic nonlinearity and without Ambrosetti–Rabinowitz condition on compact Riemannian manifolds

By means of variational methods, we study the existence and multiplicity of nontrivial weak solutions for a nonlocal ‐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.

Nonexistence results of solutions for some fractional p -Laplacian equations in R N

In the present paper, we study the nonexistence of nontrivial weak solutions to a class of fractional p -Laplacian equation in two cases which are s p &gt; N and s p &lt; N . In each of these cases, by using fractional Laplacian theory and inequality techniques, we obtain concrete range of parameter for which nontrivial weak solution of the problem does not exist. Our work complements the known nonexistence results in this direction.

Anisotropic $ (\vec{p}, \vec{q}) $-Laplacian problems with superlinear nonlinearities

&lt;abstract&gt;&lt;p&gt;In this paper we consider a class of anisotropic $ (\vec{p}, \vec{q}) $-Laplacian problems with nonlinear right-hand sides that are superlinear at $ \pm\infty $. We prove the existence of two nontrivial weak solutions to this kind of problem by applying an abstract critical point theorem under very general assumptions on the data without supposing the Ambrosetti-Rabinowitz condition.&lt;/p&gt;&lt;/abstract&gt;

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.

The stability of a class of 2D non-newtonian fluid equations with unbounded delays

ABSTRACT We address the stability of stationary solutions to a class of 2D non-newtonian fluid equations, when the external force contains hereditary characteristics involving unbounded delays. Firstly, when the unbounded variable delay is driven by a continuously differential function, we establish the stability of nontrivial weak stationary solutions and the asymptotic stability of trivial stationary solution. Then when the general unbounded delay is continuous with respect to time, the stability of nontrivial strong stationary solutions is also obtained. Eventually, when the proportional delay is considered, the polynomial stability of trivial stationary solution is verified.

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.

Nonexistence of Nontrivial Weak Solutions of Some Nonlinear Inequalities with Gradient Nonlinearity

Nonexistence of Nontrivial Weak Solutions of Some Nonlinear Inequalities with Gradient Nonlinearity

Existence of the solution for a class of the semilinear degenerate elliptic equation involving the Grushin operator in R2$\mathbb {R}^2$: The interaction between Grushin's critical exponent and exponential growth

AbstractIn this work, we study the existence of the solution for a class of semilinear degenerate elliptic equations in the whole space involving the Grushin operator and a nonlinearity that involves the Grushin's critical growth and the exponential growth. The existence of at least one nontrivial weak solution is done by combining the mountain‐pass theorem, Trudinger–Moser inequality, and a version of a result due to Lions for the exponential growth in in the corresponding weighted Sobolev space.