Sum of weighted Lebesgue spaces and nonlinear elliptic equations (Q640381)

The authors consider the existence and multiplicity of solutions of nonlinear \(p\)-Laplacian equations of the form \[ -\Delta _p u + V(| x | ) | u | ^{p-2}u =f(| x | ,u ) \quad \mathrm{in}\,\,\, {\mathbb R}^N \tag{1} \] where \(1<p<N\) and \(\Delta _p= \mathrm{div}\, (| \nabla u | ^{p-2} \nabla u ) \). Assume that \(V: (0,+\infty )\to [0,+\infty )\) and \(f: (0,+\infty ) \times {\mathbb R} \to [0,+\infty )\) are, respectively, a measurable and a Carathéodory function, both nonnegative. Define \[ W^{1,p}({\mathbb R}^N ,V):= \left\{ u \in D^{1,p}({\mathbb R}^N); \int _{{\mathbb R}^N} V(| x | ) | u | ^p dx< + \infty \right\}. \] Denoting \(F(x,t):= \int _0^t f(x,s)ds\) and \(p^*:=pN/(N-p)\), they impose the following hypotheses. {(\textbf{V}) \(V\in L^1((a,b))\) for some open bounded interval with \(b>a>0\). (\textbf{f}) There exists \(\gamma >p\) such that for almost every \(r>0\) and all \(t\geq 0\), one has \[ 0\leq \gamma F(r,t)\leq f(x,t) t \leq M \max \{r^{\theta _1},r^{\theta _2}\}\min \{t^{q_1},t^{q_2}\} \] for some constant \(M>0\) and \(\theta _1,\theta _2,q_1,q_2 \in {\mathbb R} \) such that \[ p<q_1<p^* + \frac{\theta _1p}{N-p}\leq p^* + \frac{\theta _2p}{N-p}<q_2. \] Note that the condition (\textbf{f}) means the double-power growth condition. } The authors prove the following two theorems. \textbf{Theorem 1.1} Assume that (\textbf{V}) and (\textbf{f}) hold. If there exists \(t_*>0\) such that \(F(r,t)>0\) for almost every \(r>0\) and all \(t\geq t_*\), then (1) has a nontrivial nonnegative radial weak solution. \textbf{Theorem 1.2} Assume that (\textbf{V}) and (\textbf{f}) hold. If for almost every \(r>0\) and all \(t\geq 0\) one has \(f(r,t) = -f(r,-t)\) and \[ F(r,t)\geq m \max \{r ^{\theta _1},r^{\theta _2}\} \min \{t^{q_1},t^{q_2}\} \] for some constant \(m>0\), then (1) has infinitely many radial weak solutions. Here \(u \in W^{1,p}({\mathbb R}^N, V)\) is a weak solution of (1) if and only if \[ \int _{{\mathbb R}^N}(| \nabla u | ^{p-2} \nabla u \cdot \nabla h + V(| x | )| u | ^{p-2}uh )dx = \int _{{\mathbb R}} f(| x | ,u )h dx \] for all \(h \in W ^{1,p}({\mathbb R}^N,V)\).