## 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.

### MSC:

 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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