Critical Sobolev Nonlinearities Research Articles

Existence of Solutions for Quasilinear Coupled Systemswith Multiple Critical Nonlinearities

This paper is dedicated to investigating a quasilinear elliptic systemwith p- Laplacian in R N , which involves critical Hardy-Littlewood-Sobolev nonlinearities and critical Sobolev nonlinearities. Based upon the Hardy-Littlewood-Sobolev inequality and variational methods, we obtain the attainability of the corresponding best constants and the existence of nontrivial solutions.

Fractional Kirchhoff Hardy problems with singular and critical Sobolev nonlinearities

Fractional Kirchhoff Hardy problems with singular and critical Sobolev nonlinearities

Existence and nonexistence results for critical biharmonic systems involving multiple singularities

In this paper, a system of biharmonic equations is investigated, which involves critical Sobolev nonlinearities and multiple singular points. By variational methods and analytical techniques, the best Soblev constant corresponding to the system is investigated, and the existence (nonexistence) of ground state solutions to the system is established.

Ground state solutions to biharmonic equations involving critical nonlinearities and multiple singular potentials

In this paper, a biharmonic problem is investigated, which involves critical Sobolev nonlinearity and multiple Rellich-type terms. By complicated asymptotic analysis and variational arguments, the existence and nonexistence of ground state solution to the problem are established.

Compact embedding results of Sobolev spaces and positive solutions to an elliptic equation

Using a regular Borel measure μ ⩾ 0 we derive a proper subspace of the commonly used Sobolev space D1(ℝN) when N ⩾ 3. The space resembles the standard Sobolev space H1(Ω) when Ω is a bounded region with a compact Lipschitz boundary ∂Ω. An equivalence characterization and an example are provided that guarantee that is compactly embedded into L1(RN). In addition, as an application we prove an existence result of positive solutions to an elliptic equation in ℝN that involves the Laplace operator with the critical Sobolev nonlinearity, or with a general nonlinear term that has a subcritical and superlinear growth. We also briefly discuss the compact embedding of to Lp(ℝN) when N ⩾ 2 and 2 ⩽ p ⩽ N.

Addendum to: Positive solutions of elliptic problems involving both critical Sobolev nonlinearities on exterior regions

Addendum to: Positive solutions of elliptic problems involving both critical Sobolev nonlinearities on exterior regions

Positive solutions of elliptic problems involving both critical Sobolev nonlinearities on exterior regions

This paper is mainly devoted to the study of the existence of positive solutions to an elliptic problem that involves the Laplace operator and both critical Sobolev nonlinearities on an exterior region $$U$$ , one appearing in the equation and the other in the boundary terms. Some extensions to the $$p$$ -Laplacian problems are briefly discussed. To carry out our proofs, a space $$M^{q,p}(U)$$ that is a natural generalization of the standard Sobolev space $$W^{1,p}(U)$$ is described, and its associated (compact) (trace) embedding results are investigated.

A perturbation method for spinorial Yamabe type equations on $$S^m$$ and its application

For \(m\ge 2\), we prove the existence of non-trivial solutions for a certain kind of nonlinear Dirac equations with critical Sobolev nonlinearities on \(S^m\) via a perturbative variational method. For the special case \(m=2\), this establishes the existence of a conformal immersion \(S^2\rightarrow \mathbb R ^3\) with prescribed mean curvature \(H\) which is close to a positive constant under an index counting condition on \(H\).

On a critical Neumann problem with a perturbation of lower order

We investigate the solvability of the Neumann problem (1.1) involving the critical Sobolev nonlinearity and a term of lower order. We allow a coefficient of u in equation (1.1) to be unbounded. We prove the existence of a solution in a weighted Sobolev space.

The critical Neumann problem for semilinear elliptic equations with the Hardy potential

We investigate the solvability of the Neumann problem (1.1) involving the critical Sobolev nonlinearity with an indefinite weight function and the Hardy potential. We prove that there exists $\lambda^*>0$ such that for $\lambda \in (0,\lambda^*)$, problem (1.1) admits at least two distinct solutions.

Critical Neumann Problem with Competing Hardy Potentials

In this paper we investigate the solvability of the nonlinear Neumann problem involving a critical Sobolev nonlinearity and two competing Hardy potentials in a bounded domain. We examine the common effect of the shape of the graph of the weight function, the mean curvature of the boundary and Hardy potentials on the existence of solutions of this problem. We are mainly interested in the existence of positive solutions. We also obtain the existence of sign-changing solutions.

Positive solutions of a Neumann problem with competing critical nonlinearities

We present existence results for a Neumann problem involving critical Sobolev nonlinearities both on the right hand side of the equation and at the boundary condition.. Positive solutions are obtained through constrained minimization on the Nehari manifold. Our approach is based on the concentration 'compactness principle of P. L. Lions and M. Struwe.

On the exterior Neumann problem with critical growth

In this paper we investigate the solvability of the nonlinear Neumann problem (1.1) involving a critical Sobolev nonlinearity in an exterior domain. We examine the common effect of the shape of the graph of the weight function, the mean curvature of the boundary and a nonlinear perturbation of lower order on the existence of solutions of problem (1.1).

On positive solutions of nonlinear elliptic equations involving concave and critical nonlinearities

We study the existence of nonnegative solutions of elliptic equations involving concave and critical Sobolev nonlinearities. Applying various variational principles we obtain the existence of at least two nonnegative solutions.

On the nonlinear Neumann problem at resonance with critical Sobolev nonlinearity

We consider the Neumann problem for the equation $-{\mit \Delta } u-\lambda u = Q(x)|u|^{2^{*}-2}u$, $u\in H^1({\mit \Omega })$, where $Q$ is a positive and continuous coefficient on $\hskip 1.8pt\overline {\hskip -1.8pt{\mit \Omega }\hskip -.2pt}\hskip .

On the Neumann problem for an elliptic system of equations involving the critical Sobolev exponent

We consider the Neumann problem for an elliptic system of two equations involving the critical Sobolev nonlinearity. Our main objective is to study the effect of the coefficient of the critical Sobolev nonlinearity on the existence and nonexistence of lea