Existence Of Nontrivial Weak Solutions Research Articles

Weak solutions for the singular potential wave system

We investigate the existence of weak solutions for a class of the system of wave equations with singular potential nonlinearity. We obtain a theorem which shows the existence of nontrivial weak solution for a class of the wave system with singular potential nonlinearity and the Dirichlet boundary condition. We obtain this result by using the variational method and critical point theory for indefinite functional.

Morse theory and local linking for a nonlinear degenerate problem arising in the theory of electrorheological fluids

Many electrorheological fluids are suspensions of solid particles that are exposed to a strong electric field. This causes a dramatic increase of their effective viscosity. In this paper we are concerned with a mathematical problem that is related with this non-Newtonian behavior. More precisely, we study the nonlinear stationary equation −div(|∇u|p(x)−2∇u)+|u|p(x)−2u=f(x,u) in Ω, under Dirichlet boundary conditions, where Ω is a smooth bounded domain in Rn, p>1 is a continuous function, and f(x,u) has a sublinear growth near the origin. Under various natural assumptions, by using the Morse theory in combination with local linking arguments, we obtain the existence of nontrivial weak solutions.

Existence of Weak Solutions for a Nonlocal Problem Involving the p(x)-Laplace Operator

This paper deals with the existence of weak solutions for some nonlocal problem involving the p (x)- Laplace operator. Using a direct variational method and the theory of the variable exponent Sobolev spaces, we set some conditions that ensures the existence of nontrivial weak solutions. p(x) p(x) dx)∆p(x)u = λa (x) |u| q(x)−2 u in Ω, u = 0 on ∂Ω, (P)

EXISTENCE OF SOLUTIONS FOR AN ELLIPTIC EQUATION WITH NONSTANDARD GROWTH

This paper deals with the existence of solutions for some elliptic equations with nonstandard growth under zero Dirichlet boundary condition. Using a direct variational method and the theory of the variable exponent Sobolev spaces, we set some conditions that ensures the existence of nontrivial weak solutions.

Existence of Nontrivial Weak Solutions to Quasi-linear Elliptic Equations with Exponential Growth

In this paper, we study the existence of nontrivial weak solutions to the following quasi-linear elliptic equations −△nu+V(x)|u| u= f (x,u) |x|β , x∈Rn (n>2), where−△nu=−div(|∇u|∇u), 06β<n, V :Rn→R is a continuous function, f (x,u) is continuous in Rn×R and behaves like eαu n n−1 as u →+∞. AMS Subject Classifications: 35J20, 35J60 Chinese Library Classifications: O175

Existence of nontrivial weak solutions for a quasilinear elliptic systems with concave-convex nonlinearities

Existence of nontrivial weak solutions for a quasilinear elliptic systems with concave-convex nonlinearities

Dirichlet problem for semilinear edge-degenerate elliptic equations with singular potential term

In this paper, we introduce weighted p-Sobolev spaces on manifolds with edge singularities. We give the proof for the corresponding edge type Sobolev inequality, Poincaré inequality and Hardy inequality. As an application of these inequalities, we prove the existence of nontrivial weak solutions for the Dirichlet problem of semilinear elliptic equations with singular potentials on manifolds with edge singularities.

A variational approach to a class of quasilinear elliptic equations not in divergence form

The aim of this paper is to use a variational approach in order to obtain the existence of non-trivial weak solutions of a quasilinear elliptic equation not in divergence form, in dimension $N=3$. Moreover, we prove that our solution is $C^{1, \alpha}(\overline\Omega)$ and also locally $C^{2, \alpha}(\overline\Omega)$ for a suitable $\alpha\in (0,1)$.

Adams type inequalities and related elliptic partial differential equations in dimension four

Motivated by Ruf–Saniʼs recent work, we prove an Adams type inequality and a singular Adams type inequality in the whole four-dimensional Euclidean space. As applications of those inequalities, a class of elliptic partial differential equations are considered. Existence of nontrivial weak solutions and multiplicity results are obtained via the mountain-pass theorem and the Ekelandʼs variational principle. This is a continuation of our previous work about singular Trudinger–Moser type inequality.

Existence of Weak Solutions for A Class of Nonuniformly Nonlinear Elliptic Equations of $p$-Laplacian Type

The goal of this paper is to study the existence of non-trivial weak solutions for the nonuniformly nonlinear elliptic equation in an unbounded domain. The solution will be obtained in a subspace of the Sobolev space and the proofs rely essentially on a variation of the mountain pass theorem.

Existence of nontrivial solutions for quasilinear elliptic equations at critical growth

Abstract In this paper, we study the existence of nontrivial weak solutions for the following quasilinear elliptic equation with natural growth - D j ( a ij ( x , u ) D i u ) + 1 2 ∂ s a ij ( x , u ) D i uD j u = f ( x , u ) , in a bounded smooth domain Ω ⊂ R N , where aij(x, u) are functions possibly unbounded with respect to u.

Cone Sobolev inequality and Dirichlet problem for nonlinear elliptic equations on a manifold with conical singularities

In present work, we first establish the corresponding Sobolev inequality and Poincare inequality on the cone Sobolev spaces, and then, as an application of such inequalities, we prove the existence of non-trivial weak solution for Dirichlet boundary value problem for a class of non-linear elliptic equation on manifolds with conical singularities.

EXISTENCE RESULTS FOR A CLASS OF NON-UNIFORMLY ELLIPTIC EQUATIONS OF p-LAPLACIAN TYPE

In this paper, we establish the existence of non-trivial weak solutions in [Formula: see text], 1 &lt; p &lt; ∞, to a class of non-uniformly elliptic equations of the form [Formula: see text] in a bounded domain Ω of ℝN. Here a satisfies [Formula: see text] for all ξ ∈ ℝN, a.e. x ∈ Ω, [Formula: see text], [Formula: see text], h0(x) ≧ 0, h1(x) ≧ 1 for a.e. x in Ω.

On the Solvability of Superlinear and Nonhomogeneous Quasilinear Equations

Using Mountain Pass Lemma, we obtain the existence of nontrivial weak solutions for a class of superlinear and nonhomogeneous quasilinear equations. The key factor in this paper is to use the new idea of near p-homogeneity in conjunction with variational techniques to obtain a new multiplicity result for a vast set of nonlinear equations, such as the mean curvature equation and so on.

Existence of weak solutions for a class of nonuniformly nonlinear elliptic equations in unbounded domains

The goal of this paper is to study the existence of non-trivial weak solutions for the nonuniformly nonlinear elliptic equation − div ( h ( x ) ∇ u ) + q ( x ) u = f ( x , u ) in an unbounded domain Ω ⊂ R N ( N ≥ 3 ) , where h ( x ) ∈ L loc 1 ( Ω ) . The solutions will be obtained in a subspace of the Sobolev space H 0 1 ( Ω ) and the proofs rely essentially on a variation of the Mountain pass theorem in [D.M. Duc, Nonlinear singular elliptic equations, J. London. Math. Soc. 40 (2) (1989) 420–440].

Existence results for a superlinear p-Laplacian equation with indefinite weights

In this paper, using Mountain Pass Lemma and Linking Argument, we prove the existence of nontrivial weak solutions for the Dirichlet problem of the superlinear p-Laplacian equation with indefinite weights in the case where the eigenvalue parameter λ∈(0, λ 2), λ 2 is the second positive eigenvalue of the p-Laplacian with indefinite weights.

Über das homogene Dirichlet‐Problem bei nichtlinearen partiellen Differentialgleichungen vom Typ der Boussinesq‐Gleichung

AbstractSemilinear equations of Boussinesq type, e.g. utt + uxx − uxxxx + (u2)xx = 0, utt + uxx − uxxxx + uxuxx = 0, or certain equations containing the squared wave operator, e.g. uxxtt − uk = 0, k ϵ N k ≥ 2, are studied. A generalized boundary value problem on bounded domains can be treated using Hilbert space methods. The linear parts of these equations are not elliptic, the latter not even hypoelliptic. A mountain pass lemma is used to prove the existence of nontrivial weak solutions. These solutions are obtained in anisotropic Sobolev spaces.