Compactness results and applications to some “zero mass” elliptic problems. (English) Zbl 1159.35022

In this paper the existence of solutions for the semilinear elliptic equation \(-\Delta u=f'(u)\) in an unbounded subdomain of \(\mathbb{R}^3\) is considered. Specifically, the “zero mass” case is treated, that is, \(f'(0)\) and \(f''(0)\) vanish.
To use the variational structure of this class of problems, the natural space to work in is \(\mathcal{D}^{1,2}(\Omega)\), where \(\Omega\subseteq\mathbb{R}^3\) is unbounded. One also considers the space sum \(L^p+L^q(\Omega)\) with the norm
\[ \| v\| _{L^p+L^q(\Omega)}:=\inf\{\| v_1\| _{L^p(\Omega)}+\| v_2\| _{L^q(\Omega)}\mid (v_1,v_2)\in L^p(\Omega)\times L^q(\Omega),\;v=v_1+v_2\}. \]
If \(1<p<6<q\) then it is known that there is a continuous embedding \(\mathcal{D}^{1,2}(\Omega) \hookrightarrow L^p+L^q(\Omega)\).
Let \(\Omega\) have suitable symmetries and denote by \(\mathcal{D}_{\text{s}}^{1,2}(\Omega)\) a subspace of suitably symmetric functions of \(\mathcal{D}^{1,2}(\Omega)\). One of the aims of the authors is to prove the compactness of the embedding \(\mathcal{D}_{\text{s}}^{1,2}(\Omega) \hookrightarrow L^p+L^q(\Omega)\) in various situations. This information can be used to prove the relative compactness of Palais-Smale sequences for the application of variational methods.
In the first application, consider \(f\in C^1(\mathbb{C},\mathbb{R})\), satisfying \(f(0)=0\), \(f(M)>0\) for some \(M>0\), \(| f'(\xi)| \leq C\min\{| \xi| ^{p-1},| \xi| ^{q-1}\}\) for some constant \(C>0\) and all \(\xi\in\mathbb{C}\), and \(f(\xi) = f(| \xi| )\) for all \(\xi\in\mathbb{C}\). In particular, \(f\) has supercritical growth at \(0\) and subcritical growth at \(\infty\). It is proved that then for every \(n\in\mathbb{Z}\) the equation \(-\Delta v=f'(v)\) has a complex valued solution \(v^{(n)}\in\mathcal{D}^{1,2}(\mathbb{R}^3)\) of the form \(v^{(n)}(x,y,z)=u^{(n)}(r,z)\text{e}^{\text{i}n\theta}\), where \((r,\theta,z)\) represent cylindrical coordinates and \(u^{(n)}(r,z)\in\mathbb{R}\).
The second application is the equation \(-\Delta v=f'(v)\) posed on \(\mathbb{R}^2\times I\), where \(I\) is an open bounded interval in \(\mathbb{R}\). Here \(f\in C^1(\mathbb{R},\mathbb{R})\) satisfies \(f(0)=0\), \(f(\xi)\geq C_1\min\{| \xi| ^p,| \xi| ^q\}\) and \(f'(\xi)\leq C_2\min\{| \xi| ^{p-1},| \xi| ^{q-1}\}\) for some positive constants \(C_1,C_2\) and all \(\xi\in\mathbb{R}\), and the weak Ambrosetti-Rabinowitz condition \(\alpha f(\xi)\leq f'(\xi)\xi\) with some constant \(\alpha\geq2\) and for all \(\xi\in\mathbb{R}\). Then existence of infinitely many cylindrically symmetric solutions is proved.


35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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