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January/February 2022 Quasilinear double phase problems in the whole space via perturbation methods
Bin Ge, Patrizia Pucci
Adv. Differential Equations 27(1/2): 1-30 (January/February 2022).

Abstract

We are concerned with the following double phase problems in the whole space $$ \begin{aligned} -{\rm div}(|\nabla u|^{p-2}\nabla u + & \mu(x)|\nabla u|^{q-2}\nabla u) \\ & +|u|^{p-2}u+\mu(x)|u|^{q-2}u= f(x,u)\;{\rm in}\; \mathbb{R}^N. \end{aligned}$$ The nonlinearity is super-linear but does not satisfy the Ambrosetti-Rabinowitz type condition. The main difficulty is that weak limits of $(PS)$ sequences are not always weak solutions of the problem. To overcome this difficulty, we add {a} potential term and, using the mountain pass theorem, we get weak solutions $u_\lambda$ of the perturbed equations. First, we prove that $u_\lambda\rightharpoonup u$ as $\lambda\rightarrow 0$. Then, via a vanishing lemma, we get that $u$ is a nontrivial solution of the original problem.

Citation

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Bin Ge. Patrizia Pucci. "Quasilinear double phase problems in the whole space via perturbation methods." Adv. Differential Equations 27 (1/2) 1 - 30, January/February 2022.

Information

Published: January/February 2022
First available in Project Euclid: 6 January 2022

Subjects:
Primary: 35J60 , 35J70 , 58E05

Rights: Copyright © 2022 Khayyam Publishing, Inc.

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Vol.27 • No. 1/2 • January/February 2022
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