We prove the existence of positive solutions for a nonlocal problem (1.2) with Neumann boundary conditions. We distinguish two cases: 2 < p < 2* (subcritical) and p = 2* (critical). The existence of solutions is established by variational methods.

We establish the existence of solutions of nonlinear elliptic boundary value problems involving a positive parameter on the boundary. We also examine a profile of solutions of problem (1.2) when a parameter λ tends to 0.

We investigate the solvability of the Neumann problem involving two critical exponents: Sobolev and Hardy-Sobolev. We establish the existence of a solution in three cases: ( i) 2 < p + 1 < 2* ( s), ( ii) p + 1 = 2* ( s) and ( iii) 2* ( s) < p + 1 <= 2*, where 2*(s) = 2 (N-s)/N-2, 0 < s < 2, and 2*= 2N/N-2 denote the critical Hardy-Sobolev exponent and the critical Sobolev exponent, respectively.

We investigate the solvability of the inhomogeneous Neumann problem involving the critical Sobolev exponent. In particular, we discuss the impact of the shape of the graph of the coefficient of the critical exponent on the existence of a solution. We prove the existence of at least two solutions for belonging to a bounded interval . We establish the existence of at least two solutions for a modified problem, that is, with the operator replaced by .

We establish the existence of ground states on Euclidean space for the Laplace operator involving the Hardy type potential. This gives rise to the existence of the principal eigenfunctions for the Laplace operator involving weighted Hardy potentials. We also obtain a higher integrability property for the principal eigenfunction. This is used to examine the behaviour of the principal eigenfunction around 0.

We establish the existence of principal eigenfunctions for the Laplace operator involving weighted Hardy potentials. We consider the Dirichlet and Neumann boundary conditions.

In this note we consider a class of semilinear elliptic variational inequalities on H 1 (Ω) space. With the aid of the mountain-pass principle and the Ekeland variational principle we prove the existence of solutions.

We establish compact embeddings of the radial Sobolev space H(rad)(1,p) (R(N)) into weighted Lebesgue spaces L(w)(q) (R(N)) under various assumptions on the weight function w. We use these, along with a variety of variational techniques, to prove the existence of nontrivial nonnegative solutions for a class of nonlinear Schrodinger type equations.

We establish the existence of solutions for the Neumann problem for a system of two equations involving a homogeneous nonlinearity of a critical degree. The existence of a solution is obtained by a constrained minimization with the aid of P.-L. Lions' concentration-compactness principle.

In this paper we construct a minimizing sequence for the problem (1). In particular, we show that for any subsolution of the Hamilton-Jacobi equation (∗) there exists a minimizing sequence weakly convergent to this subsolution. The variational problem (1) arises from the theory of computer vision equations.

We investigate the solvability of a singular equation of Caffarelli-Kohn-Nirenberg type having a critical-like nonlinearity with a sign-changing weight function. We shall examine how the properties of the Nehari manifold and the fibering maps affect the question of existence of positive solutions.

We establish the existence of solutions for the Neumann problem involving two Hardy - Sobolev potentials with singularities at two distinct points.

We study the existence and non-existence of ground states for the Schrodinger equations -Delta u - lambda Sigma(i<j) u/vertical bar x(i) - x(j)vertical bar(2) = vertical bar u vertical bar(2)* -(2)u, x = (x(1), ... , x(m)) is an element of R-mN, and -Delta u -lambda u/vertical bar y vertical bar(2) = vertical bar u vertical bar(2*) (-2)u, x = (y, z) is an element of R-N. In both cases we assume lambda not equal 0 and lambda < (lambda) over bar, where (lambda) over bar is the Hardy constant corresponding to the problem.

We investigate the solvability of the Neumann problem \((1.1)\) involving the nonlinearity depending on the gradient. We prove the existence of a solution when the right hand side \(f\) of the equation belongs to \(L^m(\Omega )\) with \(1 \leq m \lt 2\).

We establish the existence of two distinct solutions for problem (1.1) for small values of a parameter λ > 0 in a subcritical case. This is obtained as a combination of approximation and variational methods. In a critical case we show the existence of at least one solution.

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.

We investigate the solvability of the Neumann problem（1.1）involving the critical Sobolev nonlin- earity 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.