Nontrivial Weak Solutions Research Articles

Non‐homogeneous p‐Laplacian equations on the Sierpinski gasket

Let be the Sierpiński gasket in and denote the boundary of . In this paper, we study the following non‐homogeneous ‐Laplacian equation where are real numbers such that and the function is suitably chosen. The existence of at least two nontrivial weak solutions to the above non‐homogeneous equation on the Sierpiński gasket will be established.

Multiplicity Results for Weak Solutions of a Semilinear Dirichlet Elliptic Problem with a Parametric Nonlinearity

This paper deals with the existence of weak solutions to a Dirichlet problem for a semilinear elliptic equation involving the difference of two main nonlinearities functions that depends on a real parameter λ . According to the values of λ , we give both nonexistence and multiplicity results by using variational methods. In particular, we first exhibit a critical positive value such that the problem admits at least a nontrivial non-negative weak solution if and only if λ is greater than or equal to this critical value. Furthermore, for λ greater than a second critical positive value, we show the existence of two independent nontrivial non-negative weak solutions to the problem.

The nontrivial weak solution of Boussinesq system in high dimensions

The nontrivial weak solution of Boussinesq system in high dimensions

H-principle for the 2-dimensional incompressible porous media equation with viscosity jump

In this work we extend the results in [6,32] on the 2D IPM system with constant viscosity (Atwood number $A_{\mu}=0$) to the case of viscosity jump ($|A_{\mu}|<1$). We prove a h-principle whereby (infinitely many) weak solutions in $C_tL_{w^*}^{\infty}$ are recovered via convex integration whenever a subsolution is provided. As a first example, non-trivial weak solutions with compact support in time are obtained. Secondly, we construct mixing solutions to the unstable Muskat problem with initial flat interface. As a byproduct, we check that the connection, established by Sz\'ekelyhidi for $A_{\mu}=0$, between the subsolution and the Lagrangian relaxed solution of Otto, holds for $|A_{\mu}|<1$ too. For different viscosities, we show how a pinch singularity in the relaxation prevents the two fluids from mixing wherever there is neither Rayleigh-Taylor nor vorticity at the interface.

On a Class of Schrödinger System Problem in Orlicz–Sobolev Spaces

Using the mountain pass theorem, we obtain the existence of a nontrivial and nonnegative weak solution of a quasi-linear Schrödinger system driven by the ω ⋅ -Laplacian operator in Orlicz–Sobolev spaces.

Multiple solutions for p(x)-Kirchhoff type problems with Robin boundary conditions

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].

Fractional p(·)-Kirchhoff Type Problems Involving Variable Exponent Logarithmic Nonlinearity

In this paper, we investigate a fractional p(·)-Kirchhoff type problem involving variable exponent logarithmic nonlinearity. With the help of the Nehari manifold approach, the existence and multiplicity of nontrivial weak solutions for the above problem are obtained. The main aspect and challenges of this paper are the presence of double non-local terms and logarithmic nonlinearity.

Ground state solutions of a fractional advection–dispersion equation

In this paper, a class of fractional Sturm–Liouville advection–dispersion equations with instantaneous and noninstantaneous impulsive boundary conditions is considered. At the beginning, the existence of at least one nontrivial ground state solution is proved by the method of Nehari manifold without the Ambrosetti–Rabinowitz condition. Then, the existence of infinitely many nontrivial weak solutions is obtained by using the genus properties. Finally, an example is given to illustrate the main results obtained in this paper.

Existence of solutions for critical (p,q)-Laplacian equations in ℝN

In this paper, we are mainly interested in existence properties for a class of nonlinear PDEs driven by the ([Formula: see text])-Laplace operator where the reaction combines a power-type nonlinearity at critical level with a subcritical term. In addition, nonnegative nontrivial weights and a positive parameter [Formula: see text] are included in the nonlinearity. An important role in the analysis developed is played by the two potentials. Precisely, under suitable conditions on the exponents of the nonlinearity, first a detailed proof of the tight convergence of a sequence of measures is given, then the existence of a nontrivial weak solution is obtained provided that the parameter [Formula: see text] is far from [Formula: see text]. Our proofs use concentration compactness principles by Lions and Mountain Pass Theorem by Ambrosetti and Rabinowitz.

Multiplicity Results of Solutions to Non-Local Magnetic Schrödinger–Kirchhoff Type Equations in RN

In this paper, we establish the existence of a nontrivial weak solution to Schrödinger-kirchhoff type equations with the fractional magnetic field without Ambrosetti and Rabinowitz condition using mountain pass theorem under a suitable assumption of the external force. Furthermore, we prove the existence of infinitely many large- or small-energy solutions to this problem with Ambrosetti and Rabinowitz condition. The strategy of the proof for these results is to approach the problem by applying the variational methods, that is, the fountain and the dual fountain theorem with Cerami condition.

Multiple solutions for a class of bi-nonlocal problems with nonlinear Neumann boundary conditions

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.

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.

Solutions for nonhomogeneous singular fractional p-Laplacian equations via fixed point theorem

In this paper, we study the following fractional p-Laplacian equations with exponential growth and Hardy term: where the nonlinear term satisfies exponential growth, , , λ is a positive parameter, is the fractional p-Laplacian, is a scalar potential, the weight function either falls into some Lebesgue space or is a convolution function, and g is a perturbation term. With the help of a fixed point result and singular Trudinger–Moser inequality in , we obtain the existence of a nontrivial weak solution for the above equations.

Infinitely many solutions for a p(x)-Kirchhoff type equation with Steklov boundary value

In the present article deal with the existence and multiplicity of solutions to a class of p(x)-Kirchhoff type problem with Steklov boundary-value. By variational approach and theory of the variable exponent Sobolev spaces, under appropriate assumptions on f , we obtain existence of infinitely solutions and at least one nontrivial weak solution.

Instantaneous blow-up for nonlinear Sobolev type equations with potentials on Riemannian manifolds

&lt;p style='text-indent:20px;'&gt;We investigate Cauchy problems for two classes of nonlinear Sobolev type equations with potentials defined on complete noncompact Riemannian manifolds. The first one involves a polynomial nonlinearity and the second one involves a gradient nonlinearity. Namely, we derive sufficient conditions depending on the geometry of the manifold, the power nonlinearity, the behavior of the potential at infinity, and the initial data, for which the considered problems admit no nontrivial local weak solutions, i.e., an instantaneous blow-up occurs.&lt;/p&gt;

Variational approach for a Steklov problem involving nonstandard growth conditions

&lt;abstract&gt;&lt;p&gt;The aim of this paper is to study the multiplicity of solutions for a nonlocal $ p(x) $-Kirchhoff type problem with Steklov boundary value in variable exponent Sobolev spaces. We prove the existence of at least three solutions and a nontrivial weak solution of the problem, using the Ricceri's three critical points theorem together with Mountain Pass theorem.&lt;/p&gt;&lt;/abstract&gt;

The existence of one non-trivial weak solution of generalized Yamabe equations

The existence of one non-trivial weak solution of generalized Yamabe equations

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

In this article, we modify the results obtained by Mitidieri and Pohozaev on sufficient conditions for the absence of nontrivial weak solutions to nonlinear inequalities and systems with integer powers of|the Laplace operator and with a nonlinear term of the form a(x)|(mu)|q+ b(x)|u|s. We obtainoptimal a priori estimates by applying the nonlinear capacity method with an appropriate choice of testfunctions. As a result, we prove the absence of nontrivial weak solutions to nonlinear inequalities and systems by contradiction.

Non-uniqueness of Steady-State Weak Solutions to the Surface Quasi-Geostrophic Equations

We show the existence of nontrivial stationary weak solutions to the surface quasi-geostrophic equations on the two dimensional periodic torus.

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.