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Cited by 17 publications
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We consider the existence of solutions of the following weighted problem: { L : = - d i v ( ρ ( x ) | ∇ u | N - 2 ∇ u ) + ξ ( x ) | u | N - 2 u = f ( x , u ) i n B u > 0 i n B u = 0 o n ∂ B , \left\{ {\matrix{{L: = - div\left( {\rho \left( x \right){{\left| {\nabla u} \right|}^{N - 2}}\nabla u} \right) + \xi \left( x \right){{\left| u \right|}^{N - 2}}} \hfill & {u = f\left( {x,u} \right)} \hfill & {in} \hfill & B \hfill \cr {} \hfill & {u > 0} \hfill & {in} \hfill & B \hfill \cr {} \hfill & {u = 0} \hfill & {on} \hfill & {\partial B,} \hfill \cr } } \right. where B is the unit ball of ℝ N , N #62; 2, ρ ( x ) = ( log e | x | ) N - 1 \rho \left( x \right) = {\left( {\log {e \over {\left| x \right|}}} \right)^{N - 1}} the singular logarithm weight with the limiting exponent N − 1 in the Trudinger-Moser embedding, and ξ(x) is a positif continuous potential. The nonlinearities are critical or subcritical growth in view of Trudinger-Moser inequalities of double exponential type. We prove the existence of positive solution by using Mountain Pass theorem. In the critical case, the function of Euler Lagrange does not fulfil the requirements of Palais-Smale conditions at all levels. We dodge this problem by using adapted test functions to identify this level of compactness.
We consider the existence of solutions of the following weighted problem: { L : = - d i v ( ρ ( x ) | ∇ u | N - 2 ∇ u ) + ξ ( x ) | u | N - 2 u = f ( x , u ) i n B u > 0 i n B u = 0 o n ∂ B , \left\{ {\matrix{{L: = - div\left( {\rho \left( x \right){{\left| {\nabla u} \right|}^{N - 2}}\nabla u} \right) + \xi \left( x \right){{\left| u \right|}^{N - 2}}} \hfill & {u = f\left( {x,u} \right)} \hfill & {in} \hfill & B \hfill \cr {} \hfill & {u > 0} \hfill & {in} \hfill & B \hfill \cr {} \hfill & {u = 0} \hfill & {on} \hfill & {\partial B,} \hfill \cr } } \right. where B is the unit ball of ℝ N , N #62; 2, ρ ( x ) = ( log e | x | ) N - 1 \rho \left( x \right) = {\left( {\log {e \over {\left| x \right|}}} \right)^{N - 1}} the singular logarithm weight with the limiting exponent N − 1 in the Trudinger-Moser embedding, and ξ(x) is a positif continuous potential. The nonlinearities are critical or subcritical growth in view of Trudinger-Moser inequalities of double exponential type. We prove the existence of positive solution by using Mountain Pass theorem. In the critical case, the function of Euler Lagrange does not fulfil the requirements of Palais-Smale conditions at all levels. We dodge this problem by using adapted test functions to identify this level of compactness.
We deal with the nonlinear weighted elliptic problemwhere B is the unit ball of R 4 and w(x) = log e |x| β , β ∈ (0, 1) a singular logarithm weight. The nonlinearity is critical in view of Adam's inequalities in the weighted Sobolev space W 2,2 0 (B, w). We prove the existence of non trivial solutions via the critical point theory. The main difficulty is the loss of compactness due to the critical exponential growth of the nonlinear term f . We give a new growth condition and we point out its importance for checking the Palais-Smale compactness condition.
We discuss some Trudinger-Moser inequalities with weighted Sobolev norms. Suitable logarithmic weights in these norms allow an improvement in the maximal growth for integrability, when one restricts to radial functions. The main results concern the application of these inequalities to the existence of solutions for certain mean-field equations of Liouville-type. Sharp critical thresholds are found such that for parameters below these thresholds the corresponding functionals are coercive and hence solutions are obtained as global minima of these functionals. In the critical cases the functionals are no longer coercive and solutions may not exist. We also discuss a limiting case, in which the allowed growth is of double exponential type. Surprisingly, we are able to show that in this case a local minimum persists to exist for critical and also for slightly supercritical parameters. This allows to obtain the existence of a second (mountain-pass) solution, for almost all slightly supercritical parameters, using the Struwe monotonicity trick. This result is in contrast to the non-weighted case, where positive solutions do not exist (in star-shaped domains) in the critical and supercritical case.
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