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Search: MSC category 35J60 ( Nonlinear elliptic equations )

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1. CMB Online first

Mihăilescu, Mihai; Moroşanu, Gheorghe
Eigenvalues of $-\Delta_p -\Delta_q$ under Neumann boundary condition
The eigenvalue problem $-\Delta_p u-\Delta_q u=\lambda|u|^{q-2}u$ with $p\in(1,\infty)$, $q\in(2,\infty)$, $p\neq q$ subject to the corresponding homogeneous Neumann boundary condition is investigated on a bounded open set with smooth boundary from $\mathbb{R}^N$ with $N\geq 2$. A careful analysis of this problem leads us to a complete description of the set of eigenvalues as being a precise interval $(\lambda_1, +\infty )$ plus an isolated point $\lambda =0$. This comprehensive result is strongly related to our framework which is complementary to the well-known case $p=q\neq 2$ for which a full description of the set of eigenvalues is still unavailable.

Keywords:eigenvalue problem, Sobolev space, Nehari manifold, variational methods
Categories:35J60, 35J92, 46E30, 49R05

2. CMB 2011 (vol 55 pp. 537)

Kang, Dongsheng
Asymptotic Properties of Solutions to Semilinear Equations Involving Multiple Critical Exponents
In this paper, we investigate a semilinear elliptic equation that involves multiple Hardy-type terms and critical Hardy-Sobolev exponents. By the Moser iteration method and analytic techniques, the asymptotic properties of its nontrivial solutions at the singular points are investigated.

Keywords:elliptic problem, solution, Hardy-Sobolev inequality, singularity, Moser iteration
Categories:35B33, 35B40, 35J60

3. CMB 2007 (vol 50 pp. 356)

Filippakis, Michael E.; Papageorgiou, Nikolaos S.
Existence of Positive Solutions for Nonlinear Noncoercive Hemivariational Inequalities
In this paper we investigate the existence of positive solutions for nonlinear elliptic problems driven by the $p$-Laplacian with a nonsmooth potential (hemivariational inequality). Under asymptotic conditions that make the Euler functional indefinite and incorporate in our framework the asymptotically linear problems, using a variational approach based on nonsmooth critical point theory, we obtain positive smooth solutions. Our analysis also leads naturally to multiplicity results.

Keywords:$p$-Laplacian, locally Lipschitz potential, nonsmooth critical point theory, principal eigenvalue, positive solutions, nonsmooth Mountain Pass Theorem
Categories:35J20, 35J60, 35J85

4. CMB 2001 (vol 44 pp. 346)

Wang, Wei
Positive Solution of a Subelliptic Nonlinear Equation on the Heisenberg Group
In this paper, we establish the existence of positive solution of a nonlinear subelliptic equation involving the critical Sobolev exponent on the Heisenberg group, which generalizes a result of Brezis and Nirenberg in the Euclidean case.

Keywords:Heisenberg group, subLapacian, critical Sobolev exponent, extremals
Categories:35J20, 35J60

5. CMB 2001 (vol 44 pp. 210)

Leung, Man Chun
Growth Estimates on Positive Solutions of the Equation $\Delta u+K u^{\frac{n+2}{n-2}}=0$ in $\R^n$
We construct unbounded positive $C^2$-solutions of the equation $\Delta u + K u^{(n + 2)/(n - 2)} = 0$ in $\R^n$ (equipped with Euclidean metric $g_o$) such that $K$ is bounded between two positive numbers in $\R^n$, the conformal metric $g=u^{4/(n-2)}g_o$ is complete, and the volume growth of $g$ can be arbitrarily fast or reasonably slow according to the constructions. By imposing natural conditions on $u$, we obtain growth estimate on the $L^{2n/(n-2)}$-norm of the solution and show that it has slow decay.

Keywords:positive solution, conformal scalar curvature equation, growth estimate
Categories:35J60, 58G03

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