Abstract
In this paper, we study the the following problem
The term f could be exponential growth at \(+\infty \). The convection term involved with \(\nabla u\) makes the problem (0.1) nonvariational and the variational methods are not applicable. Under suitable conditions imposed on f, the approximation scheme is employed to obtain the existence of positive solutions for all \(\lambda \in (0,\lambda ^*]\) with \(\lambda ^*>0\).
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Partially supported by National Natural Science Foundation of China (Grant No. 12071266, 12101376, 12026218, 11801338) and Research Project Supported by Shanxi Scholarship Council of China, 2020-005.
Appendix
Appendix
Lemma A.1
Assume that sequence \(\{u_n\}\subset W_0^{1,N}(\Omega )\) satisfies \(u_n\rightarrow u\) in \(W_0^{1,N}(\Omega )\) as \(n\rightarrow \infty \). Then there exists a subsequence \(\{u_{n_j}\}\) and \(\omega \in W_0^{1,N}(\Omega )\) such that
and
Proof
Since \(u_n\rightarrow u\) in \(W_0^{1,N}(\Omega )\), going if necessary to a subsequence, we have
There exists a subsequence \(\{u_{n_j}\}\) of \(\{u_n\}\) such that
Set
It is clear that \(w\in W_0^{1,N}(\Omega )\) and \(|u_{n_{j}}(x)|,|u(x)|\leqslant w(x)\) a.e. \(x\in \Omega \). \(\square \)
Lemma A.2
Suppose condition \(\mathrm{(F)}\) holds. Let \(u_n\rightarrow u\) in \(W_0^{1,N}(\Omega )\) as \(n\rightarrow \infty \). Then
Proof
According to \(u_n\rightarrow u\) in \(W_0^{1,N}(\Omega )\) and Lemma A.1, going to a subsequence \(\{u_{n_j}\}\) we have
On the one hand, by (A.2) and \(f\in C(\Omega \times {\mathbb {R}})\), we get
On the other hand, by the condition \(\mathrm{(F)}\), we have
By the Hölder inequality and Lemma 2.6, we have
Therefore, by the Lebesgue dominated convergence theorem, we obtain
\(\square \)
Now, for \(m\in {\mathbb {N}}\), we consider the mapping \(A_m:W_m\rightarrow W_m^*\) defined by
for \(u,\phi \in W_m\).
Lemma A.3
Under the condition (F), \(A_m\) is continuous.
Proof
Suppose \(u_n\rightarrow u\) in \(W_m\). For any \(\phi \in W_m\) we get
By the Hölder inequality and the Sobolev embedding inequality, we deduce that
Thus,
Since \(u_n\rightarrow u\) in \(W_m\), going if necessary to subsequences, we have
By the Lebesgue dominated convergence theorem and Lemma A.2, we have
Therefore,
\(\square \)
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Liu, W. Existence of Positive Solutions for (N, p)-Laplace Equation with Exponential Nonlinearity Term and Convection Term in Dimension \({\mathbb {N}}\). Qual. Theory Dyn. Syst. 21, 71 (2022). https://doi.org/10.1007/s12346-022-00604-y
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DOI: https://doi.org/10.1007/s12346-022-00604-y