Nontrivial Weak Solutions Research Articles

Nonexistence of Global Positive Solutions for p-Laplacian Equations with Non-Linear Memory

The Cauchy problem in Rd,d≥1, for a non-local in time p-Laplacian equations is considered. The nonexistence of nontrivial global weak solutions by using the test function method is obtained.

Combined effects of singular and superlinear nonlinearities in singular double phase problems in [formula omitted

This paper is concerned with multiplicity results for parametric singular double phase problems in RN via the Nehari manifold approach. It is shown that the problem under consideration has at least two nontrivial weak solutions provided the parameter is sufficiently small. The idea is to split the Nehari manifold into three disjoint parts minimizing the energy functional on two of them. The third set turns out to be the empty set for small values of the parameter.

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 weak solution for a (p(x),q(x))-Laplacian elliptic problem with a singular term

Here, we consider the following elliptic problem with variable components: −a(x)Δp(x)u−b(x)Δq(x)u+u|u|s−2|x|s=λf(x,u),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ -a(x)\\Delta _{p(x)}u - b(x) \\Delta _{q(x)}u+ \\frac{u \\vert u \\vert ^{s-2}}{|x|^{s}}= \\lambda f(x,u), $$\\end{document} with Dirichlet boundary condition in a bounded domain in mathbb{R}^{N} with a smooth boundary. By applying the variational method, we prove the existence of at least one nontrivial weak solution to the problem.

The fractional p(.,.)-Neumann boundary conditions for the nonlocal p(.,.)-Laplacian operator

In this paper, we deal with the fractional -Laplacian problem with nonlocal Neumann boundary conditions here are parameters and are two nonnegative weighted functions. The domain is smooth and bounded, are continuous bounded functions. Applying the Ekeland principle and the variational method, under appropriate assumptions, we show that the above problem has nontrivial weak solutions.

Embedding and extension results in fractional Musielak–Sobolev spaces

In this paper, we are concerned with some qualitative properties of the new fractional Musielak–Sobolev spaces such that the generalized Poincaré type inequality and some continuous and compact embedding results. Moreover, we prove that any function in may be extended to a function in , with is a bounded domain of class . In addition, we establish a result that relates to the complemented subspace in . 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.

Positive solution to singular semilinear elliptic problems

In this paper, we study the existence and regularity of a non-trivial weak solution to the following semilinear elliptic problem with singular weights and singular nonlinearity: under assumptions and .

The Multiplicity of Nontrivial Solutions for a New p x -Kirchhoff-Type Elliptic Problem

In the paper, we study the existence of weak solutions for a class of new nonlocal problems involving a p x -Laplacian operator. By using Ekeland’s variational principle and mountain pass theorem, we prove that the new p x -Kirchhoff problem has at least two nontrivial weak solutions.

On a critical biharmonic system involving [formula omitted]-Laplacian and Hardy potential

In this paper, I establish a useful embedding inequality in D2,2(Rn,R2). Benefiting from the previous inequality, I also obtain a nontrivial weak solution to a critical biharmonic system involving p-Laplacian and Hardy potential via variational methods.

Small perturbations of Robin problems driven by the $p$-Laplacian plus a positive potential

We consider a quasilinear Robin problem driven by the $p$-Laplacian plus a positive potential and with a small perturbation. We assume that the main term in the equation has an Ekeland structure but we do not suppose any growth condition for the perturbation term. Applying variational methods, we prove the existence of at least one nontrivial weak solution.

Singular elliptic equations with quadratic gradient term

We establish the existence of a non-trivial weak solution to the following singular quasilinear equation with Hardy potential and singular quadratic gradient term:{−Δu−μu|x|2=|∇u|2u+f(x,u)inΩ,u>0inΩ,u=0on∂Ω, where Ω⊂RN is a smooth bounded domain, μ>0,0∈Ω,N≥3. We show that there exists a solution u∈H01(Ω) to the above problem. The notable characteristic of this problem is that it includes quadratic gradient nonlinearity and strong singularity.

An existence result for singular Finsler double phase problems

In this paper, we study a class of singular double phase problems defined on Minkowski spaces in terms of Finsler manifolds and with right-hand sides that allow a certain type of critical growth for such problems. Under very general assumptions and based on variational tools we prove the existence of at least one nontrivial weak solution for such a problem. This is the first work dealing with a Finsler double phase operator even in the nonsingular case.

Ground state solutions for quasilinear scalar field equations arising in nonlinear optics

In this paper, we study a class of quasilinear elliptic equations which appears in nonlinear optics. By using the mountain pass theorem together with a technique of adding one dimension of space (Hirata et al. in Topol Methods Nonlinear Anal 35:253–276, 2010; Jeanjean in Nonlinear Anal Theory Methods Appl 28:1633–1659, 1997), we prove the existence of a non-trivial weak solution for general nonlinear terms of Berestycki–Lions’ type. The existence of a radial ground state solution and a ground state solution is also established under stronger assumptions on the quasilinear term.

A System of p-Laplacian Equations on the Sierpiński Gasket

In this paper, we study a system of boundary value problems involving weak p-Laplacian on the Sierpiński gasket in $$\mathbb {R}^2$$ . Parameters $$\lambda , \gamma , \alpha , \beta $$ are real and $$1<q<p<\alpha +\beta .$$ Functions $$a,b,h : \mathcal {S} \rightarrow \mathbb {R}$$ are suitably chosen. For $$p>1,$$ we show the existence of at least two nontrivial weak solutions to the system of equations for some $$(\lambda ,\gamma ) \in \mathbb {R}^2.$$

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.

On a p(x)-biharmonic Kirchhoff type problem with indefinite weight and no flux boundary condition

In this paper we study the existence and the multiplicity of nontrivial weak solutions for a fourth order variable exponent Kirchhoff type problem involving p(x)-biharmonic operator with changing sign weight and with no flux boundary condition. By using variational approach and the theory of variable exponent Sobolev spaces, we determine an interval of parameters for which this problem admits at least two nontrivial weak solutions.

Neumann and Robin type boundary conditions in Fractional Orlicz-Sobolev spaces

In the first part of this article we deal with the existence of at least three non-trivial weak solutions of a nonlocal problem with nonstandard growth involving a nonlocal Robin type boundary condition. The second part of the article is devoted to study eigenvalues and minimizers of several nonlocal problems for the fractionalg-Laplacian (-Δg)swith different boundary conditions, namely, Dirichlet, Neumann and Robin.

Existence and concentration of nontrivial solutions for an elastic beam equation with local nonlinearity

&lt;abstract&gt;&lt;p&gt;In this paper, by using the mountain pass lemma and the skill of truncation function, we investigate the existence and concentration phenomenon of nontrivial weak solutions for a class of elastic beam differential equation with two parameters $ \lambda $ and $ \mu $ when the nonlinear term satisfies some growth conditions only near the origin. In particular, we obtain a concrete lower bound of the parameter $ \lambda $, and analyze the relationship between $ \lambda $ and $ \mu $. In the end, we investigate the concentration phenomenon of solutions when $ \mu\to 0 $, and obtain a specific lower bound of the parameter $ \lambda $ which is independent of $ \mu $.&lt;/p&gt;&lt;/abstract&gt;

Existence of multiple solutions to semilinear Dirichlet problem for subelliptic operator

We study the existence of multiple solutions to the Dirichlet problem for a class of semilinear subelliptic equations associated with the Hormander type operator. By using mountain pass theorem and Ekeland’s variational principle respectively, we obtain two different nontrivial weak solutions for the problem.

Nonexistence of Global Weak Solutions for Evolution Equations with Fractional Laplacian

A nonlocal nonlinear parabolic equation with fractional Laplacian is considered. By means of the method of test functions, the nonexistence of nontrivial global weak solutions is demonstrated. Simultaneously, the nonexistence of nontrivial weak solutions for the corresponding elliptic case is established.