Webinário Equações de evolução e aplicações
On generalized subcritical elliptic problems with non-power nonlinearity
Profa. Rosa Pardo (Univ. Complutense de Madrid)

We consider a slightly subcritical Dirichlet problem with a non-power nonlinearity in a bounded smooth domain. Specifically, the nonlinearity is the critical power divided by a logarithm raised to a certain exponent.
For this problem, standard compact embeddings cannot be used to guarantee the existence of solutions as in the case of power-type nonlinearities. Instead, we provide a priori bounds and through the Leray-Schauder degree, we show existence of positive solutions. Our arguments rely on the moving planes method, a Pohozaev identity, W1,q regularity results for q > N, and Morrey’s Theorem. This technique is available in a certain range of the exponent, far away from zero, see [1, 3, 4] for the semilinear case, the quasi linear case and for systems respectively.
On the other hand, when the exponent goes to zero, we use a Ljapunov-Schmidt reduction method to show that there is a positive solution which concentrates at a non-degenerate critical point of the Robin function, see [2].

[1] A. Castro, R. Pardo, A priori bounds for positive solutions of subcritical elliptic equations. Rev. Mat. Complut. 28: 715–731, 2015.
[2] M. Clapp, R. Pardo, A. Pistoia, and A. Saldaa, A solution to a slightly subcritical elliptic problem with non-power nonlinearity Preprint .
[3] L. Damascelli and R. Pardo, A priori estimates for some elliptic equations involving the p-Laplacian. Nonlinear Anal., 41: 475 – 496, 2018.
[4] N. Mavinga and R. Pardo, A priori bounds and existence of positive solutions for subcritical
semilinear elliptic systems. J. Math. Anal. Appl., 449 (2): 1172–1188, 2017.





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