We investigate the solvability of the quasilinear Neumann problem (1.1) with sub- and supercritical exponents in an unbounded domain Ω. Under some integrability conditions on the coefficients we establish embedding theorems of weighted Sobolev spaces into weighted Lebesgue spaces. This is used to obtain solutions through a global minimization of a variational functional.
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We investigate the solvability of the linear Neumann problem (1.1) with L¹ data. The results are applied to obtain existence theorems for a semilinear Neumann problem.
We investigate the solvability of the Neumann problem (1.1) involving a critical Sobolev exponent. In the first part of this work it is assumed that the coefficients Q and h are at least continuous. Moreover Q is positive on Ω̅ and λ > 0 is a parameter. We examine the common effect of the mean curvature and the shape of the graphs of the coefficients Q and h on the existence of low energy solutions. In the second part of this work we consider the same problem with Q replaced by -Q. In this case the problem can be supercritical and the existence results depend on integrability conditions on Q and h.
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We discuss the existence of solutions for a system of elliptic equations involving a coupling nonlinearity containing a critical and subcritical Sobolev exponent. We establish the existence of ground state solutions. The concentration of solutions is also established as a parameter λ becomes large.
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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 least energy solutions. As a by-product we obtain a new weighted Sobolev inequality.
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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.
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We consider the Neumann problem for the equation $-Δu - λu = Q(x)|u|^{2*-2}u$, u ∈ H¹(Ω), where Q is a positive and continuous coefficient on Ω̅ and λ is a parameter between two consecutive eigenvalues $λ_{k-1}$ and $λ_{k}$. Applying a min-max principle based on topological linking we prove the existence of a solution.
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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 investigate the effect of the topology of the boundary ∂Ω and of the graph topology of the coefficient Q on the number of solutions of the nonlinear Neumann problem $(1_d)$.
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